Log-normal distribution

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class="vector-toc-list"> </ul> </li> <li id="toc-Characteristic_function_and_moment_generating_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characteristic_function_and_moment_generating_function"> <div class="vector-toc-text"> 1.5 Characteristic function and moment generating function </div> </a> <ul id="toc-Characteristic_function_and_moment_generating_function-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 2 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Probability_in_different_domains" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_in_different_domains"> <div class="vector-toc-text"> 2.1 Probability in different domains </div> </a> <ul id="toc-Probability_in_different_domains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probabilities_of_functions_of_a_log-normal_variable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probabilities_of_functions_of_a_log-normal_variable"> <div class="vector-toc-text"> 2.2 Probabilities of functions of a log-normal variable </div> </a> <ul id="toc-Probabilities_of_functions_of_a_log-normal_variable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_or_multiplicative_moments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_or_multiplicative_moments"> <div class="vector-toc-text"> 2.3 Geometric or multiplicative moments </div> </a> <ul id="toc-Geometric_or_multiplicative_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic_moments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic_moments"> <div class="vector-toc-text"> 2.4 Arithmetic moments </div> </a> <ul id="toc-Arithmetic_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mode,_median,_quantiles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mode,_median,_quantiles"> <div class="vector-toc-text"> 2.5 Mode, median, quantiles </div> </a> <ul id="toc-Mode,_median,_quantiles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Partial_expectation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Partial_expectation"> <div class="vector-toc-text"> 2.6 Partial expectation </div> </a> <ul id="toc-Partial_expectation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditional_expectation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conditional_expectation"> <div class="vector-toc-text"> 2.7 Conditional expectation </div> </a> <ul id="toc-Conditional_expectation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_parameterizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_parameterizations"> <div class="vector-toc-text"> 2.8 Alternative parameterizations </div> </a> <ul id="toc-Alternative_parameterizations-sublist" class="vector-toc-list"> <li id="toc-Examples_for_re-parameterization" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_for_re-parameterization"> <div class="vector-toc-text"> 2.8.1 Examples for re-parameterization </div> </a> <ul id="toc-Examples_for_re-parameterization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Multiple,_reciprocal,_power" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple,_reciprocal,_power"> <div class="vector-toc-text"> 2.9 Multiple, reciprocal, power </div> </a> <ul id="toc-Multiple,_reciprocal,_power-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplication_and_division_of_independent,_log-normal_random_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplication_and_division_of_independent,_log-normal_random_variables"> <div class="vector-toc-text"> 2.10 Multiplication and division of independent, log-normal random variables </div> </a> <ul id="toc-Multiplication_and_division_of_independent,_log-normal_random_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplicative_central_limit_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplicative_central_limit_theorem"> <div class="vector-toc-text"> 2.11 Multiplicative central limit theorem </div> </a> <ul id="toc-Multiplicative_central_limit_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other"> <div class="vector-toc-text"> 2.12 Other </div> </a> <ul id="toc-Other-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_distributions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_distributions"> <div class="vector-toc-text"> 3 Related distributions </div> </a> <ul id="toc-Related_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistical_inference" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Statistical_inference"> <div class="vector-toc-text"> 4 Statistical inference </div> </a> <button aria-controls="toc-Statistical_inference-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Statistical inference subsection </button> <ul id="toc-Statistical_inference-sublist" class="vector-toc-list"> <li id="toc-Estimation_of_parameters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimation_of_parameters"> <div class="vector-toc-text"> 4.1 Estimation of parameters </div> </a> <ul id="toc-Estimation_of_parameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interval_estimates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interval_estimates"> <div class="vector-toc-text"> 4.2 Interval estimates </div> </a> <ul id="toc-Interval_estimates-sublist" class="vector-toc-list"> <li id="toc-Prediction_intervals" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Prediction_intervals"> <div class="vector-toc-text"> 4.2.1 Prediction intervals </div> </a> <ul id="toc-Prediction_intervals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Confidence_interval_for_eμ" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Confidence_interval_for_eμ"> <div class="vector-toc-text"> 4.2.2 Confidence interval for eμ </div> </a> <ul id="toc-Confidence_interval_for_eμ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Confidence_interval_for_E(X)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Confidence_interval_for_E(X)"> <div class="vector-toc-text"> 4.2.3 Confidence interval for E(X) </div> </a> <ul id="toc-Confidence_interval_for_E(X)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Confidence_interval_for_comparing_two_log_normals" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Confidence_interval_for_comparing_two_log_normals"> <div class="vector-toc-text"> 4.2.4 Confidence interval for comparing two log normals </div> </a> <ul id="toc-Confidence_interval_for_comparing_two_log_normals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extremal_principle_of_entropy_to_fix_the_free_parameter_σ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extremal_principle_of_entropy_to_fix_the_free_parameter_σ"> <div class="vector-toc-text"> 4.3 Extremal principle of entropy to fix the free parameter σ </div> </a> <ul id="toc-Extremal_principle_of_entropy_to_fix_the_free_parameter_σ-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Heavy-tailness_of_the_Log-Normal" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Heavy-tailness_of_the_Log-Normal"> <div class="vector-toc-text"> 5 Heavy-tailness of the Log-Normal </div> </a> <ul id="toc-Heavy-tailness_of_the_Log-Normal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Occurrence_and_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Occurrence_and_applications"> <div class="vector-toc-text"> 6 Occurrence and applications </div> </a> <button aria-controls="toc-Occurrence_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Occurrence and applications subsection </button> <ul id="toc-Occurrence_and_applications-sublist" class="vector-toc-list"> <li id="toc-Human_behavior" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Human_behavior"> <div class="vector-toc-text"> 6.1 Human behavior </div> </a> <ul id="toc-Human_behavior-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology_and_medicine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biology_and_medicine"> <div class="vector-toc-text"> 6.2 Biology and medicine </div> </a> <ul id="toc-Biology_and_medicine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chemistry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chemistry"> <div class="vector-toc-text"> 6.3 Chemistry </div> </a> <ul id="toc-Chemistry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hydrology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hydrology"> <div class="vector-toc-text"> 6.4 Hydrology </div> </a> <ul id="toc-Hydrology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Social_sciences_and_demographics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Social_sciences_and_demographics"> <div class="vector-toc-text"> 6.5 Social sciences and demographics </div> </a> <ul id="toc-Social_sciences_and_demographics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Technology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Technology"> <div class="vector-toc-text"> 6.6 Technology </div> </a> <ul id="toc-Technology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 8 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 10 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 11 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Log-normal distribution</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 26 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-26" aria-hidden="true" > 26 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%BD%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%BE_%D1%80%D0%B0%D0%B7%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Логнормално разпределение – Bulgarian" lang="bg" hreflang="bg" data-title="Логнормално разпределение" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_log-normal" title="Distribució log-normal – Catalan" lang="ca" hreflang="ca" data-title="Distribució log-normal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Logaritmicko-norm%C3%A1ln%C3%AD_rozd%C4%9Blen%C3%AD" title="Logaritmicko-normální rozdělení – Czech" lang="cs" hreflang="cs" data-title="Logaritmicko-normální rozdělení" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Logarithmische_Normalverteilung" title="Logarithmische Normalverteilung – German" lang="de" hreflang="de" data-title="Logarithmische Normalverteilung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distribuci%C3%B3n_log-normal" title="Distribución log-normal – Spanish" lang="es" hreflang="es" data-title="Distribución log-normal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%DB%8C%D8%B9_%D9%84%D8%A7%DA%AF-%D9%86%D8%B1%D9%85%D8%A7%D9%84" title="توزیع لاگ-نرمال – Persian" lang="fa" hreflang="fa" data-title="توزیع لاگ-نرمال" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Loi_log-normale" title="Loi log-normale – French" lang="fr" hreflang="fr" data-title="Loi log-normale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distribuci%C3%B3n_lognormal" title="Distribución lognormal – Galician" lang="gl" hreflang="gl" data-title="Distribución lognormal" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%9C%EA%B7%B8_%EC%A0%95%EA%B7%9C_%EB%B6%84%ED%8F%AC" title="로그 정규 분포 – Korean" lang="ko" hreflang="ko" data-title="로그 정규 분포" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distribuzione_lognormale" title="Distribuzione lognormale – Italian" lang="it" hreflang="it" data-title="Distribuzione lognormale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%A4%D7%9C%D7%92%D7%95%D7%AA_%D7%9C%D7%95%D7%92-%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" title="התפלגות לוג-נורמלית – Hebrew" lang="he" hreflang="he" data-title="התפלגות לוג-נורמלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Log-norm%C3%A1lis_eloszl%C3%A1s" title="Log-normális eloszlás – Hungarian" lang="hu" hreflang="hu" data-title="Log-normális eloszlás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lognormale_verdeling" title="Lognormale verdeling – Dutch" lang="nl" hreflang="nl" data-title="Lognormale verdeling" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AF%BE%E6%95%B0%E6%AD%A3%E8%A6%8F%E5%88%86%E5%B8%83" title="対数正規分布 – Japanese" lang="ja" hreflang="ja" data-title="対数正規分布" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_logarytmicznie_normalny" title="Rozkład logarytmicznie normalny – Polish" lang="pl" hreflang="pl" data-title="Rozkład logarytmicznie normalny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Distribui%C3%A7%C3%A3o_log-normal" title="Distribuição log-normal – Portuguese" lang="pt" hreflang="pt" data-title="Distribuição log-normal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%BD%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Логнормальное распределение – Russian" lang="ru" hreflang="ru" data-title="Логнормальное распределение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Shp%C3%ABrndarja_log-normale" title="Shpërndarja log-normale – Albanian" lang="sq" hreflang="sq" data-title="Shpërndarja log-normale" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Logaritemsko_normalna_porazdelitev" title="Logaritemsko normalna porazdelitev – Slovenian" lang="sl" hreflang="sl" data-title="Logaritemsko normalna porazdelitev" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Sebaran_Log-normal" title="Sebaran Log-normal – Sundanese" lang="su" hreflang="su" data-title="Sebaran Log-normal" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Log-normaalijakauma" title="Log-normaalijakauma – Finnish" lang="fi" hreflang="fi" data-title="Log-normaalijakauma" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Lognormalf%C3%B6rdelning" title="Lognormalfördelning – Swedish" lang="sv" hreflang="sv" data-title="Lognormalfördelning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Log-normal_da%C4%9F%C4%B1l%C4%B1m" title="Log-normal dağılım – Turkish" lang="tr" hreflang="tr" data-title="Log-normal dağılım" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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searchaux" style="display:none">Probability distribution</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><style data-mw-deduplicate="TemplateStyles:r1259743904">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid var(--border-color-base,#a2a9b1);padding:0.3em 0.4em}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:var(--background-color-neutral,#eaecf0);font-weight:bold;text-align:center}</style><table class="infobox infobox-table ib-prob-dist"><caption class="infobox-title">Log-normal distribution</caption><tbody><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Probability density function</div><a href="/wiki/File:Log-normal-pdfs.png" class="mw-file-description" title="Plot of the Lognormal PDF"><img alt="Plot of the Lognormal PDF" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Log-normal-pdfs.png/300px-Log-normal-pdfs.png" decoding="async" width="300" height="286" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Log-normal-pdfs.png/450px-Log-normal-pdfs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Log-normal-pdfs.png/600px-Log-normal-pdfs.png 2x" data-file-width="1552" data-file-height="1481" /></a> Identical parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mu \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>μ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mu \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8260e0b7db3b377710a5275c1657f20ee1735f63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.563ex; height:2.176ex;" alt="{\displaystyle \ \mu \ }"> but differing parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></td></tr><tr><td colspan="4" class="infobox-image"> <div class="ib-prob-dist-image">Cumulative distribution function</div><a href="/wiki/File:Log-normal-cdfs.png" class="mw-file-description" title="Plot of the Lognormal CDF"><img alt="Plot of the Lognormal CDF" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Log-normal-cdfs.png/300px-Log-normal-cdfs.png" decoding="async" width="300" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Log-normal-cdfs.png/450px-Log-normal-cdfs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Log-normal-cdfs.png/600px-Log-normal-cdfs.png 2x" data-file-width="1592" data-file-height="1504" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mu =0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mu =0\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f328e7286a778242e340049db82e656e15a686fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.824ex; height:2.676ex;" alt="{\displaystyle \ \mu =0\ }"></td></tr><tr><th scope="row" class="infobox-label">Notation</th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \operatorname {Lognormal} \left(\ \mu ,\,\sigma ^{2}\ \right)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Lognormal</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mi>μ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \operatorname {Lognormal} \left(\ \mu ,\,\sigma ^{2}\ \right)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d399777fa57273886b432f528135143b2cc597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.936ex; height:3.343ex;" alt="{\displaystyle \ \operatorname {Lognormal} \left(\ \mu ,\,\sigma ^{2}\ \right)\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mu \in (\ -\infty ,+\infty \ )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>μ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mtext> </mtext> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mtext> </mtext> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mu \in (\ -\infty ,+\infty \ )\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04fd6b178060954f7b6fe88e3d86e402de336f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.704ex; height:2.843ex;" alt="{\displaystyle \ \mu \in (\ -\infty ,+\infty \ )\ }"> (logarithm of <a href="/wiki/Location_parameter" title="Location parameter">location</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sigma >0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>σ</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sigma >0\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c359548e94e2288bf11593aa2398e3047db6831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.752ex; height:2.176ex;" alt="{\displaystyle \ \sigma >0\ }"> (logarithm of <a href="/wiki/Scale_parameter" title="Scale parameter">scale</a>)</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x\in (\ 0,+\infty \ )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mtext> </mtext> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mtext> </mtext> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x\in (\ 0,+\infty \ )\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/172ce2da3ac05d91219015cbb6642a2d110f8dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.631ex; height:2.843ex;" alt="{\displaystyle \ x\in (\ 0,+\infty \ )\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_density_function" title="Probability density function">PDF</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {1}{\ x\sigma {\sqrt {2\pi \ }}\ }}\ \exp \left(-{\frac {\left(\ln x-\mu \ \right)^{2}}{2\sigma ^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>x</mi> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\frac {1}{\ x\sigma {\sqrt {2\pi \ }}\ }}\ \exp \left(-{\frac {\left(\ln x-\mu \ \right)^{2}}{2\sigma ^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3496c13256744a487141ec0a5fc3643eb1f77eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.437ex; height:7.509ex;" alt="{\displaystyle \ {\frac {1}{\ x\sigma {\sqrt {2\pi \ }}\ }}\ \exp \left(-{\frac {\left(\ln x-\mu \ \right)^{2}}{2\sigma ^{2}}}\right)}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {\ 1\ }{2}}\left[1+\operatorname {erf} \left({\frac {\ \ln x-\mu \ }{\sigma {\sqrt {2\ }}}}\right)\right]=\Phi \left({\frac {\ln(x)-\mu }{\sigma }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>erf</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mtext> </mtext> </mrow> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mtext> </mtext> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\frac {\ 1\ }{2}}\left[1+\operatorname {erf} \left({\frac {\ \ln x-\mu \ }{\sigma {\sqrt {2\ }}}}\right)\right]=\Phi \left({\frac {\ln(x)-\mu }{\sigma }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c51991c6608ba25efced3e0a4107e57c9b0c42d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:45.687ex; height:6.676ex;" alt="{\displaystyle \ {\frac {\ 1\ }{2}}\left[1+\operatorname {erf} \left({\frac {\ \ln x-\mu \ }{\sigma {\sqrt {2\ }}}}\right)\right]=\Phi \left({\frac {\ln(x)-\mu }{\sigma }}\right)}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Quantile_function" title="Quantile function">Quantile</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \exp \left(\mu +{\sqrt {2\sigma ^{2}}}\operatorname {erf} ^{-1}(2p-1)\right)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <msup> <mi>erf</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \exp \left(\mu +{\sqrt {2\sigma ^{2}}}\operatorname {erf} ^{-1}(2p-1)\right)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e96a516ca1ec52b97e6fc5ddb3fce9eb904f14ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.509ex; height:4.843ex;" alt="{\displaystyle \ \exp \left(\mu +{\sqrt {2\sigma ^{2}}}\operatorname {erf} ^{-1}(2p-1)\right)\ }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\exp(\mu +\sigma \Phi ^{-1}(p))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\exp(\mu +\sigma \Phi ^{-1}(p))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f730d10b42f7c10a3625457178db052bea3d800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.376ex; height:3.176ex;" alt="{\displaystyle =\exp(\mu +\sigma \Phi ^{-1}(p))}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \exp \left(\ \mu +{\frac {\sigma ^{2}}{2}}\ \right)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \exp \left(\ \mu +{\frac {\sigma ^{2}}{2}}\ \right)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0a1c206987c9941fda520b0fda98af85b5268f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.146ex; height:6.343ex;" alt="{\displaystyle \ \exp \left(\ \mu +{\frac {\sigma ^{2}}{2}}\ \right)\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Median" title="Median">Median</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \exp(\ \mu \ )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mtext> </mtext> <mi>μ</mi> <mtext> </mtext> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \exp(\ \mu \ )\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26cbcffbfa136b2fb3eacc5bc879d9204b1c8f42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.473ex; height:2.843ex;" alt="{\displaystyle \ \exp(\ \mu \ )\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \exp \left(\ \mu -\sigma ^{2}\ \right)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mi>μ</mi> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \exp \left(\ \mu -\sigma ^{2}\ \right)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08452cbe245900f6b7872b80a7ae468699c1030a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.019ex; height:3.343ex;" alt="{\displaystyle \ \exp \left(\ \mu -\sigma ^{2}\ \right)\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left[\ \exp(\sigma ^{2})-1\ \right]\ \exp \left(2\ \mu +\sigma ^{2}\right)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mtext> </mtext> <mi>μ</mi> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left[\ \exp(\sigma ^{2})-1\ \right]\ \exp \left(2\ \mu +\sigma ^{2}\right)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a3c017965f320e31686d9dca9506e857b83166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.192ex; height:3.343ex;" alt="{\displaystyle \ \left[\ \exp(\sigma ^{2})-1\ \right]\ \exp \left(2\ \mu +\sigma ^{2}\right)\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \left[\ \exp \left(\sigma ^{2}\right)+2\ \right]{\sqrt {\exp(\sigma ^{2})-1\;}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mspace width="thickmathspace" /> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left[\ \exp \left(\sigma ^{2}\right)+2\ \right]{\sqrt {\exp(\sigma ^{2})-1\;}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1657430133ee87501a9b184a9f8cd9f01d00fa12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:31.631ex; height:4.843ex;" alt="{\displaystyle \ \left[\ \exp \left(\sigma ^{2}\right)+2\ \right]{\sqrt {\exp(\sigma ^{2})-1\;}}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Excess_kurtosis" class="mw-redirect" title="Excess kurtosis">Excess kurtosis</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 1\ \exp \left(4\ \sigma ^{2}\right)+2\ \exp \left(3\ \sigma ^{2}\right)+3\ \exp \left(2\sigma ^{2}\right)-6\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>1</mn> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>3</mn> <mtext> </mtext> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>6</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 1\ \exp \left(4\ \sigma ^{2}\right)+2\ \exp \left(3\ \sigma ^{2}\right)+3\ \exp \left(2\sigma ^{2}\right)-6\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79f0d65f3e6732589dbd7d7c6a9451f5610cd338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.085ex; height:3.343ex;" alt="{\displaystyle \ 1\ \exp \left(4\ \sigma ^{2}\right)+2\ \exp \left(3\ \sigma ^{2}\right)+3\ \exp \left(2\sigma ^{2}\right)-6\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Entropy</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \log _{2}\left(\ {\sqrt {2\pi \ }}\ \sigma \ e^{\mu +{\tfrac {1}{2}}}\ \right)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mi>σ</mi> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> </msup> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \log _{2}\left(\ {\sqrt {2\pi \ }}\ \sigma \ e^{\mu +{\tfrac {1}{2}}}\ \right)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e91fcf2352e4403ce3795f9c7058372f5adee50e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.902ex; height:6.176ex;" alt="{\displaystyle \ \log _{2}\left(\ {\sqrt {2\pi \ }}\ \sigma \ e^{\mu +{\tfrac {1}{2}}}\ \right)\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Moment-generating_function" title="Moment-generating function">MGF</a></th><td colspan="3" class="infobox-data">  defined only for numbers with a  non-positive real part, see text</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td colspan="3" class="infobox-data">  representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sum _{n=0}^{\infty }{\frac {\ (i\ t)^{n}\ }{n!}}e^{\ n\mu +n^{2}\sigma ^{2}/2}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>i</mi> <mtext> </mtext> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi>n</mi> <mi>μ</mi> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sum _{n=0}^{\infty }{\frac {\ (i\ t)^{n}\ }{n!}}e^{\ n\mu +n^{2}\sigma ^{2}/2}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4f75c743409fbb0a1390bd5634197d24c3728b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.925ex; height:6.843ex;" alt="{\displaystyle \ \sum _{n=0}^{\infty }{\frac {\ (i\ t)^{n}\ }{n!}}e^{\ n\mu +n^{2}\sigma ^{2}/2}\ }">  is asymptotically divergent, but adequate  for most numerical purposes</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\begin{pmatrix}{\frac {1}{\ \sigma ^{2}\ }}&0\\0&{\frac {2}{\ \sigma ^{2}\ }}\end{pmatrix}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\begin{pmatrix}{\frac {1}{\ \sigma ^{2}\ }}&0\\0&{\frac {2}{\ \sigma ^{2}\ }}\end{pmatrix}}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1168f75d050f4a5819e6fe1f924d7e5bde2c301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.842ex; height:8.176ex;" alt="{\displaystyle \ {\begin{pmatrix}{\frac {1}{\ \sigma ^{2}\ }}&0\\0&{\frac {2}{\ \sigma ^{2}\ }}\end{pmatrix}}\ }"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mu =\log \left({\frac {\operatorname {\mathbb {E} } [X]\ }{\ {\sqrt {{\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ }}\ }}\right)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mtext> </mtext> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mu =\log \left({\frac {\operatorname {\mathbb {E} } [X]\ }{\ {\sqrt {{\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ }}\ }}\right)\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55074cffa7b5b5f102b010be57837a25279bf580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:30.231ex; height:13.176ex;" alt="{\displaystyle \ \mu =\log \left({\frac {\operatorname {\mathbb {E} } [X]\ }{\ {\sqrt {{\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ }}\ }}\right)\ ,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sigma ={\sqrt {\log \left({\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ \right)\ }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mtext> </mtext> <mtext> </mtext> </mrow> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sigma ={\sqrt {\log \left({\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ \right)\ }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46349b67d9ab6239b10e582d5c92ad410254a97e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.946ex; height:7.509ex;" alt="{\displaystyle \ \sigma ={\sqrt {\log \left({\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ \right)\ }}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_shortfall" title="Expected shortfall">Expected shortfall</a></th><td colspan="3" class="infobox-data"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\frac {1}{2}}e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {\ 1+\operatorname {erf} \left({\frac {\sigma }{\ {\sqrt {2\ }}\ }}+\operatorname {erf} ^{-1}(2p-1)\right)\ }{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mn>1</mn> <mo>+</mo> <mi>erf</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi>erf</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mrow> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\frac {1}{2}}e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {\ 1+\operatorname {erf} \left({\frac {\sigma }{\ {\sqrt {2\ }}\ }}+\operatorname {erf} ^{-1}(2p-1)\right)\ }{p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9d37c56f5c487d2c26cc03264019eb265cbf54d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.739ex; height:8.009ex;" alt="{\displaystyle \ {\frac {1}{2}}e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {\ 1+\operatorname {erf} \left({\frac {\sigma }{\ {\sqrt {2\ }}\ }}+\operatorname {erf} ^{-1}(2p-1)\right)\ }{p}}}"><a href="#cite_note-norton-1">[1]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ =e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {1}{1-p}}(1-\Phi (\Phi ^{-1}(p)-\sigma ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ =e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {1}{1-p}}(1-\Phi (\Phi ^{-1}(p)-\sigma ))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e498ad131a88eb8fde258ce308276a5cf2aff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.133ex; height:6.009ex;" alt="{\displaystyle \ =e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {1}{1-p}}(1-\Phi (\Phi ^{-1}(p)-\sigma ))}"></td></tr></tbody></table> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, a log-normal (or lognormal) distribution is a continuous <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of a <a href="/wiki/Random_variable" title="Random variable">random variable</a> whose <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> is <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a>. Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution.<a href="#cite_note-:1-2">[2]</a><a href="#cite_note-:2-3">[3]</a> Equivalently, if Y has a normal distribution, then the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> of Y, X = exp(Y) , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and <a href="/wiki/Engineering" title="Engineering">engineering</a> sciences, as well as <a href="/wiki/Medicine" title="Medicine">medicine</a>, <a href="/wiki/Economics" title="Economics">economics</a> and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics). The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a>.<a href="#cite_note-JKB-4">[4]</a> The log-normal distribution has also been associated with other names, such as <a href="/wiki/Donald_MacAlister#log-normal" title="Donald MacAlister">McAlister</a>, <a href="/wiki/Gibrat%27s_law" title="Gibrat's law">Gibrat</a> and <a href="/wiki/Cobb%E2%80%93Douglas" class="mw-redirect" title="Cobb–Douglas">Cobb–Douglas</a>.<a href="#cite_note-JKB-4">[4]</a> A log-normal process is the statistical realization of the multiplicative <a href="/wiki/Mathematical_product" class="mw-redirect" title="Mathematical product">product</a> of many <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> <a href="/wiki/Random_variable" title="Random variable">random variables</a>, each of which is positive. This is justified by considering the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> in the log domain (sometimes called <a href="/wiki/Gibrat%27s_law" title="Gibrat's law">Gibrat's law</a>). The log-normal distribution is the <a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum entropy probability distribution</a> for a random variate X—for which the mean and variance of ln(X) are specified.<a href="#cite_note-5">[5]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=1" title="Edit section: Definitions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Generation_and_parameters">Generation and parameters</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=2" title="Edit section: Generation and parameters">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ Z\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Z</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Z\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175e98486f427d21166a91a0e1bf6607150d182a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.842ex; height:2.176ex;" alt="{\displaystyle \ Z\ }"> be a <a href="/wiki/Normal_distribution#Standard_normal_distribution" title="Normal distribution">standard normal variable</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> be two real numbers, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma >0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/762ecd0f0905dd0d4d7a07f80fa8bfb324b9b021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma >0}">. Then, the distribution of the random variable <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=e^{\mu +\sigma Z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <mi>Z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=e^{\mu +\sigma Z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f28880971cb6d8b20b541b4943e518f190528bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.792ex; height:2.676ex;" alt="{\displaystyle X=e^{\mu +\sigma Z}}"></dd></dl> is called the log-normal distribution with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">. These are the <a href="/wiki/Expected_value" title="Expected value">expected value</a> (or <a href="/wiki/Mean" title="Mean">mean</a>) and <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> of the variable's natural <a href="/wiki/Logarithm" title="Logarithm">logarithm</a>, not the expectation and standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3f6c335ba3fd217820a6dd6cd310c49a39d08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.141ex; height:2.176ex;" alt="{\displaystyle \ X\ }"> itself. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Lognormal_Distribution.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Lognormal_Distribution.svg/330px-Lognormal_Distribution.svg.png" decoding="async" width="330" height="244" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Lognormal_Distribution.svg/495px-Lognormal_Distribution.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Lognormal_Distribution.svg/660px-Lognormal_Distribution.svg.png 2x" data-file-width="629" data-file-height="465" /></a><figcaption>Relation between normal and log-normal distribution. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ Y=\mu +\sigma Z\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Y</mi> <mo>=</mo> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <mi>Z</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Y=\mu +\sigma Z\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16acf7ffded2254c78cfb2a0fbf66bc80391e6e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.285ex; height:2.676ex;" alt="{\displaystyle \ Y=\mu +\sigma Z\ }"> is normally distributed, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X\sim e^{Y}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mo>∼</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X\sim e^{Y}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c035bb8302179f1f4485218f00336e5931faa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.809ex; height:2.676ex;" alt="{\displaystyle \ X\sim e^{Y}\ }"> is log-normally distributed.</figcaption></figure> This relationship is true regardless of the base of the logarithmic or exponential function: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \log _{a}(X)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \log _{a}(X)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d037d5a8117ed312a3ef12281365666edfe4c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.411ex; height:2.843ex;" alt="{\displaystyle \ \log _{a}(X)\ }"> is normally distributed, then so is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \log _{b}(X)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \log _{b}(X)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce699f758d747f309db2f1e6d9846453922dd7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.247ex; height:2.843ex;" alt="{\displaystyle \ \log _{b}(X)\ }"> for any two positive numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ a,b\neq 1~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>≠</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a,b\neq 1~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74f2778186f5ce515b60541fa2d8ad041bdc1604" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.33ex; height:2.676ex;" alt="{\displaystyle \ a,b\neq 1~.}"> Likewise, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ e^{Y}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ e^{Y}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1e443d4fef746788b995d717cbfbfcf5d6cc50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.731ex; height:2.676ex;" alt="{\displaystyle \ e^{Y}\ }"> is log-normally distributed, then so is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ a^{Y}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a^{Y}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1acf1335a220edf0c225cb17c70826ad30b186b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.524ex; height:3.009ex;" alt="{\displaystyle \ a^{Y}\ ,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<a\neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a\neq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be67e475dacb968ed2a92f7d9374e67c4991227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.752ex; height:2.676ex;" alt="{\displaystyle 0<a\neq 1}">. In order to produce a distribution with desired mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{X}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfe6d3f115b8d6cb595119ea9bc7962a11db65a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.034ex; height:2.176ex;" alt="{\displaystyle \mu _{X}}"> and variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sigma _{X}^{2}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sigma _{X}^{2}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b77bf24f3807b5c91552842683286cd5716280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.768ex; height:3.176ex;" alt="{\displaystyle \ \sigma _{X}^{2}\ ,}"> one uses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mu =\ln \left({\frac {\mu _{X}^{2}}{\ {\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}\ }}\ }}\right)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mu =\ln \left({\frac {\mu _{X}^{2}}{\ {\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}\ }}\ }}\right)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4be50a035ec3dd4a80585c0eac5e6de47e601e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:25.404ex; height:10.509ex;" alt="{\displaystyle \ \mu =\ln \left({\frac {\mu _{X}^{2}}{\ {\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}\ }}\ }}\right)\ }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sigma ^{2}=\ln \left(1+{\frac {\ \sigma _{X}^{2}\ }{\mu _{X}^{2}}}\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sigma ^{2}=\ln \left(1+{\frac {\ \sigma _{X}^{2}\ }{\mu _{X}^{2}}}\right)~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/688ca1db3b2d617ca73ea383482e5dfb05caecc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.872ex; height:7.509ex;" alt="{\displaystyle \ \sigma ^{2}=\ln \left(1+{\frac {\ \sigma _{X}^{2}\ }{\mu _{X}^{2}}}\right)~.}"> Alternatively, the "multiplicative" or "geometric" parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mu ^{*}=e^{\mu }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mu ^{*}=e^{\mu }\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f9e2d9074a9fa1772e4d98ac4bba531e1dc041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.023ex; height:2.843ex;" alt="{\displaystyle \ \mu ^{*}=e^{\mu }\ }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sigma ^{*}=e^{\sigma }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sigma ^{*}=e^{\sigma }\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93496bb0f907e05b896f5d12c3e433442ac4f56f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.9ex; height:2.343ex;" alt="{\displaystyle \ \sigma ^{*}=e^{\sigma }\ }"> can be used. They have a more direct interpretation: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \mu ^{*}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mu ^{*}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8ac7c9f5bf3fe8ed5e60697f2ba0535e9edcd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.617ex; height:2.843ex;" alt="{\displaystyle \ \mu ^{*}\ }"> is the <a href="/wiki/Median" title="Median">median</a> of the distribution, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sigma ^{*}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sigma ^{*}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a77df7f918a3a0d73c4852bbabedebbf5f376ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.546ex; height:2.343ex;" alt="{\displaystyle \ \sigma ^{*}\ }"> is useful for determining "scatter" intervals, see below. <div class="mw-heading mw-heading3"><h3 id="Probability_density_function">Probability density function</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=3" title="Edit section: Probability density function">edit</a>]</div> A positive random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3f6c335ba3fd217820a6dd6cd310c49a39d08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.141ex; height:2.176ex;" alt="{\displaystyle \ X\ }"> is log-normally distributed (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X\sim \operatorname {Lognormal} \left(\ \mu ,\sigma ^{2}\ \right)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mtext> </mtext> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X\sim \operatorname {Lognormal} \left(\ \mu ,\sigma ^{2}\ \right)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6660778741a0008c77513902651dd67789d3871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.241ex; height:3.343ex;" alt="{\displaystyle \ X\sim \operatorname {Lognormal} \left(\ \mu ,\sigma ^{2}\ \right)\ }">), if the natural logarithm of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc3f6c335ba3fd217820a6dd6cd310c49a39d08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.141ex; height:2.176ex;" alt="{\displaystyle \ X\ }"> is normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> and variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \sigma ^{2}\ :}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \sigma ^{2}\ :}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b2124cde3fe52f72b984bb1384d15452b852ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.838ex; height:2.676ex;" alt="{\displaystyle \ \sigma ^{2}\ :}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b92c953f2ecbfccfb70cfa5e80fa3b71ac06b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.732ex; height:3.176ex;" alt="{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})}"></dd></dl> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Phi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Φ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Phi \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7cceb6296aeb0b11e9529b1efa8aab6e651147a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.839ex; height:2.176ex;" alt="{\displaystyle \ \Phi \ }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \varphi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>φ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \varphi \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/783cc6d0a6afcecc3e2c31fbb82b40d0616c05b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.681ex; height:2.176ex;" alt="{\displaystyle \ \varphi \ }"> be respectively the cumulative probability distribution function and the probability density function of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathcal {N}}(\ 0,1\ )\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mtext> </mtext> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mtext> </mtext> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {N}}(\ 0,1\ )\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83414e55cf4ea8ea7b4e1d3f489ee8fee859a172" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.766ex; height:3.009ex;" alt="{\displaystyle \ {\mathcal {N}}(\ 0,1\ )\ }"> standard normal distribution, then we have that<a href="#cite_note-:1-2">[2]</a><a href="#cite_note-JKB-4">[4]</a> the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> of the log-normal distribution is given by: <dl><dd><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f_{X}(x)&={\frac {\rm {d}}{{\rm {d}}x}}\ \operatorname {\mathbb {P} _{\mathit {X}}} \,\!{\bigl [}\ X\leq x\ {\bigr ]}\\[6pt]&={\frac {\rm {d}}{{\rm {d}}x}}\ \operatorname {\mathbb {P} _{\mathit {X}}} \,\!{\bigl [}\ \ln X\leq \ln x\ {\bigr ]}\\[6pt]&={\frac {\rm {d}}{{\rm {d}}x}}\operatorname {\Phi } \!\!\left({\frac {\ \ln x-\mu \ }{\sigma }}\right)\\[6pt]&=\operatorname {\varphi } \!\left({\frac {\ln x-\mu }{\sigma }}\right){\frac {\rm {d}}{{\rm {d}}x}}\left({\frac {\ \ln x-\mu \ }{\sigma }}\right)\\[6pt]&=\operatorname {\varphi } \!\left({\frac {\ \ln x-\mu \ }{\sigma }}\right){\frac {1}{\ \sigma \ x\ }}\\[6pt]&={\frac {1}{\ x\ \sigma {\sqrt {2\ \pi \ }}\ }}\exp \left(-{\frac {\ (\ln x-\mu )^{2}\ }{2\ \sigma ^{2}}}\right)~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.9em 0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">X</mi> </mrow> </mrow> </msub> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mtext> </mtext> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">X</mi> </mrow> </mrow> </msub> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mtext> </mtext> <mi>ln</mi> <mo>⁡</mo> <mi>X</mi> <mo>≤</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Φ</mi> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mtext> </mtext> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>φ</mi> </mrow> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mtext> </mtext> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>φ</mi> </mrow> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <mtext> </mtext> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>σ</mi> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>x</mi> <mtext> </mtext> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mtext> </mtext> <mi>π</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn>2</mn> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{X}(x)&={\frac {\rm {d}}{{\rm {d}}x}}\ \operatorname {\mathbb {P} _{\mathit {X}}} \,\!{\bigl [}\ X\leq x\ {\bigr ]}\\[6pt]&={\frac {\rm {d}}{{\rm {d}}x}}\ \operatorname {\mathbb {P} _{\mathit {X}}} \,\!{\bigl [}\ \ln X\leq \ln x\ {\bigr ]}\\[6pt]&={\frac {\rm {d}}{{\rm {d}}x}}\operatorname {\Phi } \!\!\left({\frac {\ \ln x-\mu \ }{\sigma }}\right)\\[6pt]&=\operatorname {\varphi } \!\left({\frac {\ln x-\mu }{\sigma }}\right){\frac {\rm {d}}{{\rm {d}}x}}\left({\frac {\ \ln x-\mu \ }{\sigma }}\right)\\[6pt]&=\operatorname {\varphi } \!\left({\frac {\ \ln x-\mu \ }{\sigma }}\right){\frac {1}{\ \sigma \ x\ }}\\[6pt]&={\frac {1}{\ x\ \sigma {\sqrt {2\ \pi \ }}\ }}\exp \left(-{\frac {\ (\ln x-\mu )^{2}\ }{2\ \sigma ^{2}}}\right)~.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10dc2a3d559ddc86174a24365d121314f6131cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -21.671ex; width:44.005ex; height:44.509ex;" alt="{\displaystyle {\begin{aligned}f_{X}(x)&={\frac {\rm {d}}{{\rm {d}}x}}\ \operatorname {\mathbb {P} _{\mathit {X}}} \,\!{\bigl [}\ X\leq x\ {\bigr ]}\\[6pt]&={\frac {\rm {d}}{{\rm {d}}x}}\ \operatorname {\mathbb {P} _{\mathit {X}}} \,\!{\bigl [}\ \ln X\leq \ln x\ {\bigr ]}\\[6pt]&={\frac {\rm {d}}{{\rm {d}}x}}\operatorname {\Phi } \!\!\left({\frac {\ \ln x-\mu \ }{\sigma }}\right)\\[6pt]&=\operatorname {\varphi } \!\left({\frac {\ln x-\mu }{\sigma }}\right){\frac {\rm {d}}{{\rm {d}}x}}\left({\frac {\ \ln x-\mu \ }{\sigma }}\right)\\[6pt]&=\operatorname {\varphi } \!\left({\frac {\ \ln x-\mu \ }{\sigma }}\right){\frac {1}{\ \sigma \ x\ }}\\[6pt]&={\frac {1}{\ x\ \sigma {\sqrt {2\ \pi \ }}\ }}\exp \left(-{\frac {\ (\ln x-\mu )^{2}\ }{2\ \sigma ^{2}}}\right)~.\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Cumulative_distribution_function">Cumulative distribution function</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=4" title="Edit section: Cumulative distribution function">edit</a>]</div> The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x)=\Phi \left({\frac {(\ln x)-\mu }{\sigma }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)=\Phi \left({\frac {(\ln x)-\mu }{\sigma }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29095d9cbd6539833d549c59149b9fc5bd06339b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.394ex; height:6.343ex;" alt="{\displaystyle F_{X}(x)=\Phi \left({\frac {(\ln x)-\mu }{\sigma }}\right)}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \Phi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Φ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Phi \ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7cceb6296aeb0b11e9529b1efa8aab6e651147a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.839ex; height:2.176ex;" alt="{\displaystyle \ \Phi \ }"> is the cumulative distribution function of the standard normal distribution (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \operatorname {\mathcal {N}} (\ 0,\ 1)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mtext> </mtext> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \operatorname {\mathcal {N}} (\ 0,\ 1)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ca15d9274da562ab7724e9193dc113b6fac7ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.153ex; height:3.009ex;" alt="{\displaystyle \ \operatorname {\mathcal {N}} (\ 0,\ 1)\ }">). This may also be expressed as follows:<a href="#cite_note-:1-2">[2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)\right]={\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>erf</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>erfc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> </mrow> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)\right]={\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7373f66d2a24f5817a8bc2f2f44836941b79118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:46.791ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)\right]={\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)}"></dd></dl> where erfc is the <a href="/wiki/Complementary_error_function" class="mw-redirect" title="Complementary error function">complementary error function</a>. <div class="mw-heading mw-heading3"><h3 id="Multivariate_log-normal">Multivariate log-normal</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=5" title="Edit section: Multivariate log-normal">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {X}}\sim {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">X</mi> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {X}}\sim {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68a7228540f7ae0eaf24737f822bed65c276575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.395ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {X}}\sim {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}"> is a <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}=\exp(X_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}=\exp(X_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ba8de606b15c819f3d279c76ade97f7ecf9e58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.334ex; height:2.843ex;" alt="{\displaystyle Y_{i}=\exp(X_{i})}"> has a multivariate log-normal distribution.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a> The exponential is applied elementwise to the random vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {X}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/899e933d518eefcbbd0c48512cc7887ee117d040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.215ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {X}}}">. The mean of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {Y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">Y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {Y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026ecbeeef99cb5ee8eaef9ec2f59bf7a8fce8fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.036ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {Y}}}"> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [{\boldsymbol {Y}}]_{i}=e^{\mu _{i}+{\frac {1}{2}}\Sigma _{ii}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">Y</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [{\boldsymbol {Y}}]_{i}=e^{\mu _{i}+{\frac {1}{2}}\Sigma _{ii}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488f8b7b6e5331b3d4b257c87b40752a01ee6293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.443ex; height:4.009ex;" alt="{\displaystyle \operatorname {E} [{\boldsymbol {Y}}]_{i}=e^{\mu _{i}+{\frac {1}{2}}\Sigma _{ii}},}"></dd></dl> and its <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} [{\boldsymbol {Y}}]_{ij}=e^{\mu _{i}+\mu _{j}+{\frac {1}{2}}(\Sigma _{ii}+\Sigma _{jj})}(e^{\Sigma _{ij}}-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">Y</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} [{\boldsymbol {Y}}]_{ij}=e^{\mu _{i}+\mu _{j}+{\frac {1}{2}}(\Sigma _{ii}+\Sigma _{jj})}(e^{\Sigma _{ij}}-1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11b3d9175a3f442f40eb4687f58014c3efdfa7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.837ex; height:4.176ex;" alt="{\displaystyle \operatorname {Var} [{\boldsymbol {Y}}]_{ij}=e^{\mu _{i}+\mu _{j}+{\frac {1}{2}}(\Sigma _{ii}+\Sigma _{jj})}(e^{\Sigma _{ij}}-1).}"></dd></dl> Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the <a href="/wiki/Univariate_distribution" title="Univariate distribution">univariate distribution</a>. <div class="mw-heading mw-heading3"><h3 id="Characteristic_function_and_moment_generating_function">Characteristic function and moment generating function</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=6" title="Edit section: Characteristic function and moment generating function">edit</a>]</div> All moments of the log-normal distribution exist and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +n^{2}\sigma ^{2}/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>μ</mi> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +n^{2}\sigma ^{2}/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec49bbb5852b6e735f0a6a49468771db326b7bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.996ex; height:3.509ex;" alt="{\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +n^{2}\sigma ^{2}/2}}"></dd></dl> This can be derived by letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z={\tfrac {\ln(x)-(\mu +n\sigma ^{2})}{\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>+</mo> <mi>n</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mi>σ</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\tfrac {\ln(x)-(\mu +n\sigma ^{2})}{\sigma }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cadf7a6ab2c73ee906f76ece5d7a2eff61e8097b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.2ex; height:4.176ex;" alt="{\displaystyle z={\tfrac {\ln(x)-(\mu +n\sigma ^{2})}{\sigma }}}"> within the integral. However, the log-normal distribution is not determined by its moments.<a href="#cite_note-Heyde-8">[8]</a> This implies that it cannot have a defined moment generating function in a neighborhood of zero.<a href="#cite_note-9">[9]</a> Indeed, the expected value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [e^{tX}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [e^{tX}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0379eb85a8f71d1d2e06107ba42758bc26c355b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.186ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [e^{tX}]}"> is not defined for any positive value of the argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">, since the defining integral diverges. The <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [e^{itX}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [e^{itX}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33bdf53bdb972f0154a057c687c9545db5e7ff7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.753ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} [e^{itX}]}"> is defined for real values of t, but is not defined for any complex value of t that has a negative imaginary part, and hence the characteristic function is not <a href="/wiki/Analytic_function" title="Analytic function">analytic</a> at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.<a href="#cite_note-Holgate-10">[10]</a> In particular, its Taylor <a href="/wiki/Formal_series" class="mw-redirect" title="Formal series">formal series</a> diverges: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>μ</mi> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6733a3f8e5325d8f77495404c70168cdaa7cfab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.054ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}}"></dd></dl> However, a number of alternative <a href="/wiki/Divergent_series" title="Divergent series">divergent series</a> representations have been obtained.<a href="#cite_note-Holgate-10">[10]</a><a href="#cite_note-Barakat-11">[11]</a><a href="#cite_note-Barouch-12">[12]</a><a href="#cite_note-Leipnik-13">[13]</a> A closed-form formula for the characteristic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by<a href="#cite_note-Asmussen-14">[14]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)\approx {\frac {\exp \left(-{\frac {W^{2}(-it\sigma ^{2}e^{\mu })+2W(-it\sigma ^{2}e^{\mu })}{2\sigma ^{2}}}\right)}{\sqrt {1+W(-it\sigma ^{2}e^{\mu })}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>t</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>W</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>t</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi>t</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (t)\approx {\frac {\exp \left(-{\frac {W^{2}(-it\sigma ^{2}e^{\mu })+2W(-it\sigma ^{2}e^{\mu })}{2\sigma ^{2}}}\right)}{\sqrt {1+W(-it\sigma ^{2}e^{\mu })}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/943de3767423ad4dc55d86c1a8a090a70506e036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.602ex; height:10.176ex;" alt="{\displaystyle \varphi (t)\approx {\frac {\exp \left(-{\frac {W^{2}(-it\sigma ^{2}e^{\mu })+2W(-it\sigma ^{2}e^{\mu })}{2\sigma ^{2}}}\right)}{\sqrt {1+W(-it\sigma ^{2}e^{\mu })}}}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"> is the <a href="/wiki/Lambert_W_function" title="Lambert W function">Lambert W function</a>. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }">. <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=7" title="Edit section: Properties">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Probabilities_of_log_normal.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Probabilities_of_log_normal.png/220px-Probabilities_of_log_normal.png" decoding="async" width="220" height="709" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Probabilities_of_log_normal.png/330px-Probabilities_of_log_normal.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Probabilities_of_log_normal.png/440px-Probabilities_of_log_normal.png 2x" data-file-width="816" data-file-height="2628" /></a><figcaption>a. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> is a log-normal variable with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =1,\sigma =0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>σ</mi> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =1,\sigma =0.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13231b6f8be3cd163edbca85ecdd5b2341d5171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.096ex; height:2.676ex;" alt="{\displaystyle \mu =1,\sigma =0.5}">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\sin y>0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\sin y>0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef1b69b54b91029cae332e187dd5ed0ee69f412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.727ex; height:2.843ex;" alt="{\displaystyle p(\sin y>0)}"> is computed by transforming to the normal variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\ln y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\ln y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e659eb94c8baee9b640f8cbf8f379c0d1503350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.91ex; height:2.509ex;" alt="{\displaystyle x=\ln y}">, then integrating its density over the domain defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin e^{x}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin e^{x}>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0af4dc8ec13b4cc4111f7f70e78049c7f93f978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.76ex; height:2.343ex;" alt="{\displaystyle \sin e^{x}>0}"> (blue regions), using the numerical method of ray-tracing.<a href="#cite_note-Das-15">[15]</a> b & c. The pdf and cdf of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bef72df24eb14edf9eef7d1c2b1c5a702b34f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.398ex; height:2.509ex;" alt="{\displaystyle \sin y}"> of the log-normal variable can also be computed in this way.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Probability_in_different_domains">Probability in different domains</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=8" title="Edit section: Probability in different domains">edit</a>]</div> The probability content of a log-normal distribution in any arbitrary domain can be computed to desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method.<a href="#cite_note-Das-15">[15]</a> (<a rel="nofollow" class="external text" href="https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions">Matlab code</a>) <div class="mw-heading mw-heading3"><h3 id="Probabilities_of_functions_of_a_log-normal_variable">Probabilities of functions of a log-normal variable</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=9" title="Edit section: Probabilities of functions of a log-normal variable">edit</a>]</div> Since the probability of a log-normal can be computed in any domain, this means that the cdf (and consequently pdf and inverse cdf) of any function of a log-normal variable can also be computed.<a href="#cite_note-Das-15">[15]</a> (<a rel="nofollow" class="external text" href="https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions">Matlab code</a>) <div class="mw-heading mw-heading3"><h3 id="Geometric_or_multiplicative_moments">Geometric or multiplicative moments</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=10" title="Edit section: Geometric or multiplicative moments">edit</a>]</div> The <a href="/wiki/Geometric_mean" title="Geometric mean">geometric or multiplicative mean</a> of the log-normal distribution is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GM} [X]=e^{\mu }=\mu ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GM</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GM} [X]=e^{\mu }=\mu ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9445c2ca179a4932cdadf4b511c0348c3449a4ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.189ex; height:2.843ex;" alt="{\displaystyle \operatorname {GM} [X]=e^{\mu }=\mu ^{*}}">. It equals the median. The <a href="/wiki/Geometric_standard_deviation" title="Geometric standard deviation">geometric or multiplicative standard deviation</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GSD} [X]=e^{\sigma }=\sigma ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GSD</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GSD} [X]=e^{\sigma }=\sigma ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d39b3e467b86b861d3c19285f10f6d7dc8ec923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.004ex; height:2.843ex;" alt="{\displaystyle \operatorname {GSD} [X]=e^{\sigma }=\sigma ^{*}}">.<a href="#cite_note-ReferenceA-16">[16]</a><a href="#cite_note-17">[17]</a> By analogy with the arithmetic statistics, one can define a geometric variance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GVar} [X]=e^{\sigma ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GVar</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GVar} [X]=e^{\sigma ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8446c1bc836c47f03e578df8e5971015871417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.102ex; height:3.509ex;" alt="{\displaystyle \operatorname {GVar} [X]=e^{\sigma ^{2}}}">, and a <a href="/wiki/Coefficient_of_variation#Log-normal_data" title="Coefficient of variation">geometric coefficient of variation</a>,<a href="#cite_note-ReferenceA-16">[16]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GCV} [X]=e^{\sigma }-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GCV</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GCV} [X]=e^{\sigma }-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/521d8430003df46e507169d1e2fd3ee976b4105e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.877ex; height:2.843ex;" alt="{\displaystyle \operatorname {GCV} [X]=e^{\sigma }-1}">, has been proposed. This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CV} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CV</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {CV} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5455da15e231c252c84a8390aade81336fc790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.421ex; height:2.176ex;" alt="{\displaystyle \operatorname {CV} }"> itself (see also <a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a>). Note that the geometric mean is smaller than the arithmetic mean. This is due to the <a href="/wiki/AM%E2%80%93GM_inequality" title="AM–GM inequality">AM–GM inequality</a> and is a consequence of the logarithm being a <a href="/wiki/Concave_function" title="Concave function">concave function</a>. In fact, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]=e^{\mu +{\frac {1}{2}}\sigma ^{2}}=e^{\mu }\cdot {\sqrt {e^{\sigma ^{2}}}}=\operatorname {GM} [X]\cdot {\sqrt {\operatorname {GVar} [X]}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mi>GM</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>GVar</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]=e^{\mu +{\frac {1}{2}}\sigma ^{2}}=e^{\mu }\cdot {\sqrt {e^{\sigma ^{2}}}}=\operatorname {GM} [X]\cdot {\sqrt {\operatorname {GVar} [X]}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e16c0e50545da50c815d65794d7589b9bf513be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.205ex; height:5.343ex;" alt="{\displaystyle \operatorname {E} [X]=e^{\mu +{\frac {1}{2}}\sigma ^{2}}=e^{\mu }\cdot {\sqrt {e^{\sigma ^{2}}}}=\operatorname {GM} [X]\cdot {\sqrt {\operatorname {GVar} [X]}}.}"><a href="#cite_note-Acoustic_Stimuli_Revisited_2016-18">[18]</a></dd></dl> In finance, the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {1}{2}}\sigma ^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-{\frac {1}{2}}\sigma ^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7eb38e2f48456c820bf84ab317a286f42e9c5c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.87ex; height:3.509ex;" alt="{\displaystyle e^{-{\frac {1}{2}}\sigma ^{2}}}"> is sometimes interpreted as a <a href="/wiki/Convexity_correction" class="mw-redirect" title="Convexity correction">convexity correction</a>. From the point of view of <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">stochastic calculus</a>, this is the same correction term as in <a href="/wiki/It%C5%8D%27s_lemma#Geometric_Brownian_motion" class="mw-redirect" title="Itō's lemma">Itō's lemma for geometric Brownian motion</a>. <div class="mw-heading mw-heading3"><h3 id="Arithmetic_moments">Arithmetic moments</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=11" title="Edit section: Arithmetic moments">edit</a>]</div> For any real or complex number n, the n-th <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moment</a> of a log-normally distributed variable X is given by<a href="#cite_note-JKB-4">[4]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +{\frac {1}{2}}n^{2}\sigma ^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +{\frac {1}{2}}n^{2}\sigma ^{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b6efe7347f26a8054654edcdfb03eb8b28bbf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.502ex; height:4.009ex;" alt="{\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +{\frac {1}{2}}n^{2}\sigma ^{2}}.}"></dd></dl> Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are respectively given by:<a href="#cite_note-:1-2">[2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} [X]&=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}},\\[4pt]\operatorname {E} [X^{2}]&=e^{2\mu +2\sigma ^{2}},\\[4pt]\operatorname {Var} [X]&=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=(\operatorname {E} [X])^{2}(e^{\sigma ^{2}}-1)=e^{2\mu +\sigma ^{2}}(e^{\sigma ^{2}}-1),\\[4pt]\operatorname {SD} [X]&={\sqrt {\operatorname {Var} [X]}}=\operatorname {E} [X]{\sqrt {e^{\sigma ^{2}}-1}}=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}{\sqrt {e^{\sigma ^{2}}-1}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>μ</mi> <mo>+</mo> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>μ</mi> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>SD</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </msqrt> </mrow> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [X]&=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}},\\[4pt]\operatorname {E} [X^{2}]&=e^{2\mu +2\sigma ^{2}},\\[4pt]\operatorname {Var} [X]&=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=(\operatorname {E} [X])^{2}(e^{\sigma ^{2}}-1)=e^{2\mu +\sigma ^{2}}(e^{\sigma ^{2}}-1),\\[4pt]\operatorname {SD} [X]&={\sqrt {\operatorname {Var} [X]}}=\operatorname {E} [X]{\sqrt {e^{\sigma ^{2}}-1}}=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}{\sqrt {e^{\sigma ^{2}}-1}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b59e2bead4a03f70fcf34a610106ae8704959a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:64.164ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [X]&=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}},\\[4pt]\operatorname {E} [X^{2}]&=e^{2\mu +2\sigma ^{2}},\\[4pt]\operatorname {Var} [X]&=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}=(\operatorname {E} [X])^{2}(e^{\sigma ^{2}}-1)=e^{2\mu +\sigma ^{2}}(e^{\sigma ^{2}}-1),\\[4pt]\operatorname {SD} [X]&={\sqrt {\operatorname {Var} [X]}}=\operatorname {E} [X]{\sqrt {e^{\sigma ^{2}}-1}}=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}{\sqrt {e^{\sigma ^{2}}-1}},\end{aligned}}}"></dd></dl> The arithmetic <a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">coefficient of variation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CV} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CV</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {CV} [X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89fe40c7a2788b7bb2797aeda4b90c1f53be8ce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.695ex; height:2.843ex;" alt="{\displaystyle \operatorname {CV} [X]}"> is the ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\operatorname {SD} [X]}{\operatorname {E} [X]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>SD</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\operatorname {SD} [X]}{\operatorname {E} [X]}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1c3719bd3e6716f973e3ab735e695e19df4a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.32ex; height:4.843ex;" alt="{\displaystyle {\tfrac {\operatorname {SD} [X]}{\operatorname {E} [X]}}}">. For a log-normal distribution it is equal to<a href="#cite_note-:2-3">[3]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CV} [X]={\sqrt {e^{\sigma ^{2}}-1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CV</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {CV} [X]={\sqrt {e^{\sigma ^{2}}-1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cad386f192fe53b9e0525951f5423f46e03e36d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.855ex; height:4.843ex;" alt="{\displaystyle \operatorname {CV} [X]={\sqrt {e^{\sigma ^{2}}-1}}.}"></dd></dl> This estimate is sometimes referred to as the "geometric CV" (GCV),<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean. The parameters μ and σ can be obtained, if the arithmetic mean and the arithmetic variance are known: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mu &=\ln \left({\frac {\operatorname {E} [X]^{2}}{\sqrt {\operatorname {E} [X^{2}]}}}\right)=\ln \left({\frac {\operatorname {E} [X]^{2}}{\sqrt {\operatorname {Var} [X]+\operatorname {E} [X]^{2}}}}\right),\\[4pt]\sigma ^{2}&=\ln \left({\frac {\operatorname {E} [X^{2}]}{\operatorname {E} [X]^{2}}}\right)=\ln \left(1+{\frac {\operatorname {Var} [X]}{\operatorname {E} [X]^{2}}}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>μ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msqrt> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mu &=\ln \left({\frac {\operatorname {E} [X]^{2}}{\sqrt {\operatorname {E} [X^{2}]}}}\right)=\ln \left({\frac {\operatorname {E} [X]^{2}}{\sqrt {\operatorname {Var} [X]+\operatorname {E} [X]^{2}}}}\right),\\[4pt]\sigma ^{2}&=\ln \left({\frac {\operatorname {E} [X^{2}]}{\operatorname {E} [X]^{2}}}\right)=\ln \left(1+{\frac {\operatorname {Var} [X]}{\operatorname {E} [X]^{2}}}\right).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ede6a785b6ed56d35a478e9927963cea65ba96e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:49.311ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\mu &=\ln \left({\frac {\operatorname {E} [X]^{2}}{\sqrt {\operatorname {E} [X^{2}]}}}\right)=\ln \left({\frac {\operatorname {E} [X]^{2}}{\sqrt {\operatorname {Var} [X]+\operatorname {E} [X]^{2}}}}\right),\\[4pt]\sigma ^{2}&=\ln \left({\frac {\operatorname {E} [X^{2}]}{\operatorname {E} [X]^{2}}}\right)=\ln \left(1+{\frac {\operatorname {Var} [X]}{\operatorname {E} [X]^{2}}}\right).\end{aligned}}}"></dd></dl> A probability distribution is not uniquely determined by the moments E[Xn] = enμ + <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/2⁠n2σ2 for n ≥ 1. That is, there exist other distributions with the same set of moments.<a href="#cite_note-JKB-4">[4]</a> In fact, there is a whole family of distributions with the same moments as the log-normal distribution.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Mode,_median,_quantiles">Mode, median, quantiles</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=12" title="Edit section: Mode, median, quantiles">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Comparison_mean_median_mode.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Comparison_mean_median_mode.svg/280px-Comparison_mean_median_mode.svg.png" decoding="async" width="280" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Comparison_mean_median_mode.svg/420px-Comparison_mean_median_mode.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Comparison_mean_median_mode.svg/560px-Comparison_mean_median_mode.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>Comparison of <a href="/wiki/Mean" title="Mean">mean</a>, <a href="/wiki/Median" title="Median">median</a> and <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a> of two log-normal distributions with different <a href="/wiki/Skewness" title="Skewness">skewness</a>.</figcaption></figure> The <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a> is the point of global maximum of the probability density function. In particular, by solving the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\ln f)'=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\ln f)'=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/381bbe52c7caeb8f9570316e05d4686ea192ff95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.36ex; height:3.009ex;" alt="{\displaystyle (\ln f)'=0}">, we get that: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mode} [X]=e^{\mu -\sigma ^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mode</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mode} [X]=e^{\mu -\sigma ^{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696ae3ee691abe8666911db6b83228e86d685f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.995ex; height:3.509ex;" alt="{\displaystyle \operatorname {Mode} [X]=e^{\mu -\sigma ^{2}}.}"></dd></dl> Since the <a href="/wiki/Logarithm_transformation" class="mw-redirect" title="Logarithm transformation">log-transformed</a> variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\ln X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\ln X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55025bac0098995f281804a9ec37e22027af1d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.178ex; height:2.176ex;" alt="{\displaystyle Y=\ln X}"> has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{X}(\alpha )=e^{\mu +\sigma q_{\Phi }(\alpha )}=\mu ^{*}(\sigma ^{*})^{q_{\Phi }(\alpha )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Φ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Φ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{X}(\alpha )=e^{\mu +\sigma q_{\Phi }(\alpha )}=\mu ^{*}(\sigma ^{*})^{q_{\Phi }(\alpha )},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88deb2beb3d16d702584061ed5ee4cf6a7b6fb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.602ex; height:3.343ex;" alt="{\displaystyle q_{X}(\alpha )=e^{\mu +\sigma q_{\Phi }(\alpha )}=\mu ^{*}(\sigma ^{*})^{q_{\Phi }(\alpha )},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\Phi }(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Φ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\Phi }(\alpha )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f191baa04077412d76630710ddeef397b37d0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.753ex; height:2.843ex;" alt="{\displaystyle q_{\Phi }(\alpha )}"> is the quantile of the standard normal distribution. Specifically, the median of a log-normal distribution is equal to its multiplicative mean,<a href="#cite_note-21">[21]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Med} [X]=e^{\mu }=\mu ^{*}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Med</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Med} [X]=e^{\mu }=\mu ^{*}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca3cf66c48a40e98cfbbad81df359a85e2898f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.917ex; height:2.843ex;" alt="{\displaystyle \operatorname {Med} [X]=e^{\mu }=\mu ^{*}~.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Partial_expectation">Partial expectation</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=13" title="Edit section: Partial expectation">edit</a>]</div> The partial expectation of a random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> with respect to a threshold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k)=\int _{k}^{\infty }xf_{X}(x\mid X>k)\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>x</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∣</mo> <mi>X</mi> <mo>></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(k)=\int _{k}^{\infty }xf_{X}(x\mid X>k)\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e351e627ea05b75f73d7ada7de7f405576715cdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.008ex; height:5.843ex;" alt="{\displaystyle g(k)=\int _{k}^{\infty }xf_{X}(x\mid X>k)\,dx.}"></dd></dl> Alternatively, by using the definition of <a href="/wiki/Conditional_expectation" title="Conditional expectation">conditional expectation</a>, it can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k)=\operatorname {E} [X\mid X>k]P(X>k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>∣</mo> <mi>X</mi> <mo>></mo> <mi>k</mi> <mo stretchy="false">]</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>></mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(k)=\operatorname {E} [X\mid X>k]P(X>k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc597040aafacac51656980faecab241210cd32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.163ex; height:2.843ex;" alt="{\displaystyle g(k)=\operatorname {E} [X\mid X>k]P(X>k)}">. For a log-normal random variable, the partial expectation is given by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(k)=\int _{k}^{\infty }xf_{X}(x\mid X>k)\,dx=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}\,\Phi \!\left({\frac {\mu +\sigma ^{2}-\ln k}{\sigma }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>x</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∣</mo> <mi>X</mi> <mo>></mo> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>k</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(k)=\int _{k}^{\infty }xf_{X}(x\mid X>k)\,dx=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}\,\Phi \!\left({\frac {\mu +\sigma ^{2}-\ln k}{\sigma }}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533a86dbaa387973efeaa76154517b9f1f2d703f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.803ex; height:6.343ex;" alt="{\displaystyle g(k)=\int _{k}^{\infty }xf_{X}(x\mid X>k)\,dx=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}\,\Phi \!\left({\frac {\mu +\sigma ^{2}-\ln k}{\sigma }}\right)}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> is the <a href="/wiki/Normal_cumulative_distribution_function" class="mw-redirect" title="Normal cumulative distribution function">normal cumulative distribution function</a>. The derivation of the formula is provided in the <a href="/wiki/Talk:Log-normal_distribution" title="Talk:Log-normal distribution">Talk page</a>. The partial expectation formula has applications in <a href="/wiki/Insurance" title="Insurance">insurance</a> and <a href="/wiki/Economics" title="Economics">economics</a>, it is used in solving the partial differential equation leading to the <a href="/wiki/Black%E2%80%93Scholes_formula" class="mw-redirect" title="Black–Scholes formula">Black–Scholes formula</a>. <div class="mw-heading mw-heading3"><h3 id="Conditional_expectation">Conditional expectation</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=14" title="Edit section: Conditional expectation">edit</a>]</div> The conditional expectation of a log-normal random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">—with respect to a threshold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">—is its partial expectation divided by the cumulative probability of being in that range: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E[X\mid X<k]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\ln(k)-\mu -\sigma ^{2}}{\sigma }}\right]}{\Phi \left[{\frac {\ln(k)-\mu }{\sigma }}\right]}}\\[8pt]E[X\mid X\geqslant k]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\mu +\sigma ^{2}-\ln(k)}{\sigma }}\right]}{1-\Phi \left[{\frac {\ln(k)-\mu }{\sigma }}\right]}}\\[8pt]E[X\mid X\in [k_{1},k_{2}]]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\ln(k_{2})-\mu -\sigma ^{2}}{\sigma }}\right]-\Phi \left[{\frac {\ln(k_{1})-\mu -\sigma ^{2}}{\sigma }}\right]}{\Phi \left[{\frac {\ln(k_{2})-\mu }{\sigma }}\right]-\Phi \left[{\frac {\ln(k_{1})-\mu }{\sigma }}\right]}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo>∣</mo> <mi>X</mi> <mo><</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Φ</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">Φ</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo>∣</mo> <mi>X</mi> <mo>⩾</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Φ</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo>+</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>E</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo>∣</mo> <mi>X</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Φ</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">Φ</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E[X\mid X<k]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\ln(k)-\mu -\sigma ^{2}}{\sigma }}\right]}{\Phi \left[{\frac {\ln(k)-\mu }{\sigma }}\right]}}\\[8pt]E[X\mid X\geqslant k]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\mu +\sigma ^{2}-\ln(k)}{\sigma }}\right]}{1-\Phi \left[{\frac {\ln(k)-\mu }{\sigma }}\right]}}\\[8pt]E[X\mid X\in [k_{1},k_{2}]]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\ln(k_{2})-\mu -\sigma ^{2}}{\sigma }}\right]-\Phi \left[{\frac {\ln(k_{1})-\mu -\sigma ^{2}}{\sigma }}\right]}{\Phi \left[{\frac {\ln(k_{2})-\mu }{\sigma }}\right]-\Phi \left[{\frac {\ln(k_{1})-\mu }{\sigma }}\right]}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183bad844b619e2b3e49c9a51d7021120b124865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.671ex; width:63.889ex; height:38.509ex;" alt="{\displaystyle {\begin{aligned}E[X\mid X<k]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\ln(k)-\mu -\sigma ^{2}}{\sigma }}\right]}{\Phi \left[{\frac {\ln(k)-\mu }{\sigma }}\right]}}\\[8pt]E[X\mid X\geqslant k]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\mu +\sigma ^{2}-\ln(k)}{\sigma }}\right]}{1-\Phi \left[{\frac {\ln(k)-\mu }{\sigma }}\right]}}\\[8pt]E[X\mid X\in [k_{1},k_{2}]]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi \left[{\frac {\ln(k_{2})-\mu -\sigma ^{2}}{\sigma }}\right]-\Phi \left[{\frac {\ln(k_{1})-\mu -\sigma ^{2}}{\sigma }}\right]}{\Phi \left[{\frac {\ln(k_{2})-\mu }{\sigma }}\right]-\Phi \left[{\frac {\ln(k_{1})-\mu }{\sigma }}\right]}}\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Alternative_parameterizations">Alternative parameterizations</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=15" title="Edit section: Alternative parameterizations">edit</a>]</div> In addition to the characterization by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ,\sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89a7381d3e0a26af97c80f7f003f03595e74645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.176ex;" alt="{\displaystyle \mu ,\sigma }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{*},\sigma ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{*},\sigma ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584417f6136aa66cdcdb2035648904b4cf3190a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.875ex; height:2.843ex;" alt="{\displaystyle \mu ^{*},\sigma ^{*}}">, here are multiple ways how the log-normal distribution can be parameterized. <a href="/wiki/ProbOnto" title="ProbOnto">ProbOnto</a>, the knowledge base and ontology of <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a><a href="#cite_note-22">[22]</a><a href="#cite_note-23">[23]</a> lists seven such forms: <figure typeof="mw:File/Thumb"><a href="/wiki/File:LogNormal17.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/LogNormal17.jpg/400px-LogNormal17.jpg" decoding="async" width="400" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/LogNormal17.jpg/600px-LogNormal17.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/LogNormal17.jpg/800px-LogNormal17.jpg 2x" data-file-width="1130" data-file-height="550" /></a><figcaption>Overview of parameterizations of the log-normal distributions.</figcaption></figure> <ul><li>LogNormal1(μ,σ) with <a href="/wiki/Mean" title="Mean">mean</a>, μ, and <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>, σ, both on the log-scale <a href="#cite_note-Forbes-24">[24]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d254929914b40b8fa2329e6a02fa53353ed7fa07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.273ex; height:7.509ex;" alt="{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}\right]}"></dd></dl></li> <li>LogNormal2(μ,υ) with mean, μ, and variance, υ, both on the log-scale <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {v}})={\frac {1}{x{\sqrt {v}}{\sqrt {2\pi }}}}\exp \left[-{\frac {(\ln x-\mu )^{2}}{2v}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>v</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>v</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {v}})={\frac {1}{x{\sqrt {v}}{\sqrt {2\pi }}}}\exp \left[-{\frac {(\ln x-\mu )^{2}}{2v}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47756e2ba5dcec56108d985f54ced5802726cb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.73ex; height:7.509ex;" alt="{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {v}})={\frac {1}{x{\sqrt {v}}{\sqrt {2\pi }}}}\exp \left[-{\frac {(\ln x-\mu )^{2}}{2v}}\right]}"></dd></dl></li> <li>LogNormal3(m,σ) with <a href="/wiki/Median" title="Median">median</a>, m, on the natural scale and standard deviation, σ, on the log-scale<a href="#cite_note-Forbes-24">[24]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\sigma ^{2}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">m</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\sigma ^{2}}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81ab16e30f347597540cda86ccb2702b0be5f85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.599ex; height:7.509ex;" alt="{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\sigma ^{2}}}\right]}"></dd></dl></li> <li>LogNormal4(m,cv) with median, m, and <a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">coefficient of variation</a>, cv, both on the natural scale <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {cv}})={\frac {1}{x{\sqrt {\ln(cv^{2}+1)}}{\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\ln(cv^{2}+1)}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">m</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">c</mi> <mi mathvariant="bold-italic">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {cv}})={\frac {1}{x{\sqrt {\ln(cv^{2}+1)}}{\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\ln(cv^{2}+1)}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34339307b7039935ae1071b1fb0ca1a04b51e0a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.604ex; height:7.509ex;" alt="{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {cv}})={\frac {1}{x{\sqrt {\ln(cv^{2}+1)}}{\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\ln(cv^{2}+1)}}\right]}"></dd></dl></li> <li>LogNormal5(μ,τ) with mean, μ, and precision, τ, both on the log-scale<a href="#cite_note-25">[25]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\tau }})={\sqrt {\frac {\tau }{2\pi }}}{\frac {1}{x}}\exp \left[-{\frac {\tau }{2}}(\ln x-\mu )^{2}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>τ</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>τ</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\tau }})={\sqrt {\frac {\tau }{2\pi }}}{\frac {1}{x}}\exp \left[-{\frac {\tau }{2}}(\ln x-\mu )^{2}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ded50fe9521e9c21996167b6261b45fa2270849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.677ex; height:6.343ex;" alt="{\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\tau }})={\sqrt {\frac {\tau }{2\pi }}}{\frac {1}{x}}\exp \left[-{\frac {\tau }{2}}(\ln x-\mu )^{2}\right]}"></dd></dl></li> <li>LogNormal6(m,σg) with median, m, and <a href="/wiki/Geometric_standard_deviation" title="Geometric standard deviation">geometric standard deviation</a>, σg, both on the natural scale<a href="#cite_note-26">[26]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma _{g}}})={\frac {1}{x\ln(\sigma _{g}){\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\ln ^{2}(\sigma _{g})}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">m</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold-italic">σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">g</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msup> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma _{g}}})={\frac {1}{x\ln(\sigma _{g}){\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\ln ^{2}(\sigma _{g})}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d63692975723d89e040953aefd71ec18822b86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.882ex; height:7.509ex;" alt="{\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma _{g}}})={\frac {1}{x\ln(\sigma _{g}){\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\ln ^{2}(\sigma _{g})}}\right]}"></dd></dl></li> <li>LogNormal7(μN,σN) with mean, μN, and standard deviation, σN, both on the natural scale<a href="#cite_note-27">[27]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x;{\boldsymbol {\mu _{N}}},{\boldsymbol {\sigma _{N}}})={\frac {1}{x{\sqrt {2\pi \ln \left(1+\sigma _{N}^{2}/\mu _{N}^{2}\right)}}}}\exp \left(-{\frac {{\Big [}\ln x-\ln {\frac {\mu _{N}}{\sqrt {1+\sigma _{N}^{2}/\mu _{N}^{2}}}}{\Big ]}^{2}}{2\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold-italic">μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">N</mi> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold-italic">σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">N</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msqrt> <mn>1</mn> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x;{\boldsymbol {\mu _{N}}},{\boldsymbol {\sigma _{N}}})={\frac {1}{x{\sqrt {2\pi \ln \left(1+\sigma _{N}^{2}/\mu _{N}^{2}\right)}}}}\exp \left(-{\frac {{\Big [}\ln x-\ln {\frac {\mu _{N}}{\sqrt {1+\sigma _{N}^{2}/\mu _{N}^{2}}}}{\Big ]}^{2}}{2\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c50f543829dadcdbc41b007ca74be9029c563e56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:72.987ex; height:13.843ex;" alt="{\displaystyle P(x;{\boldsymbol {\mu _{N}}},{\boldsymbol {\sigma _{N}}})={\frac {1}{x{\sqrt {2\pi \ln \left(1+\sigma _{N}^{2}/\mu _{N}^{2}\right)}}}}\exp \left(-{\frac {{\Big [}\ln x-\ln {\frac {\mu _{N}}{\sqrt {1+\sigma _{N}^{2}/\mu _{N}^{2}}}}{\Big ]}^{2}}{2\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})}}\right)}"></dd></dl></li></ul> <div class="mw-heading mw-heading4"><h4 id="Examples_for_re-parameterization">Examples for re-parameterization</h4>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=16" title="Edit section: Examples for re-parameterization">edit</a>]</div> Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM<a href="#cite_note-28">[28]</a> and PopED.<a href="#cite_note-29">[29]</a> The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results. For the transition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {LN2} (\mu ,v)\to \operatorname {LN7} (\mu _{N},\sigma _{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>LN2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>LN7</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {LN2} (\mu ,v)\to \operatorname {LN7} (\mu _{N},\sigma _{N})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b57a6de2e1c6e133b1d88f9090d0e93b2aa8e5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.658ex; height:2.843ex;" alt="{\displaystyle \operatorname {LN2} (\mu ,v)\to \operatorname {LN7} (\mu _{N},\sigma _{N})}"> following formulas hold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mu _{N}=\exp(\mu +v/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mu _{N}=\exp(\mu +v/2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbeae4089532c4e0562fd01740114549afb2f76e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.248ex; height:2.843ex;" alt="{\textstyle \mu _{N}=\exp(\mu +v/2)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma _{N}=\exp(\mu +v/2){\sqrt {\exp(v)-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma _{N}=\exp(\mu +v/2){\sqrt {\exp(v)-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da6aabb604b607ec8d317ceea541b7f82f910bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.99ex; height:3.343ex;" alt="{\textstyle \sigma _{N}=\exp(\mu +v/2){\sqrt {\exp(v)-1}}}">. For the transition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {LN7} (\mu _{N},\sigma _{N})\to \operatorname {LN2} (\mu ,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>LN7</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>LN2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {LN7} (\mu _{N},\sigma _{N})\to \operatorname {LN2} (\mu ,v)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80be213d3c8a410035598bdd72dc84cdb2399eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.658ex; height:2.843ex;" alt="{\displaystyle \operatorname {LN7} (\mu _{N},\sigma _{N})\to \operatorname {LN2} (\mu ,v)}"> following formulas hold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mu =\ln \left(\mu _{N}/{\sqrt {1+\sigma _{N}^{2}/\mu _{N}^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mu =\ln \left(\mu _{N}/{\sqrt {1+\sigma _{N}^{2}/\mu _{N}^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0a87e2a4b06d5766b8cfef10b3eb7307ec3f74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.072ex; height:4.843ex;" alt="{\textstyle \mu =\ln \left(\mu _{N}/{\sqrt {1+\sigma _{N}^{2}/\mu _{N}^{2}}}\right)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle v=\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle v=\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1eeb2fcdeb9fd07bc91295dc8ac2590144af7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.252ex; height:3.176ex;" alt="{\textstyle v=\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})}">. All remaining re-parameterisation formulas can be found in the specification document on the project website.<a href="#cite_note-probontoWebsite-30">[30]</a> <div class="mw-heading mw-heading3"><h3 id="Multiple,_reciprocal,_power">Multiple, reciprocal, power</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=17" title="Edit section: Multiple, reciprocal, power">edit</a>]</div> <ul><li>Multiplication by a constant: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c08a8fc58091f7804e20c5b9052f31893d430dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.598ex; height:3.176ex;" alt="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aX\sim \operatorname {Lognormal} (\mu +\ln a,\ \sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>X</mi> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aX\sim \operatorname {Lognormal} (\mu +\ln a,\ \sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e4a8af99ad2973635cd66fd4660d9d27e20a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.805ex; height:3.176ex;" alt="{\displaystyle aX\sim \operatorname {Lognormal} (\mu +\ln a,\ \sigma ^{2})}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47586c689085690d15621968c72a61073b32357e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.138ex; height:2.176ex;" alt="{\displaystyle a>0.}"></li> <li>Reciprocal: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c08a8fc58091f7804e20c5b9052f31893d430dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.598ex; height:3.176ex;" alt="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Lognormal} (-\mu ,\ \sigma ^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>X</mi> </mfrac> </mstyle> </mrow> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>μ</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Lognormal} (-\mu ,\ \sigma ^{2}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c53c2ed61fdb1d2a6fc8989ce5666138782a547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.889ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{X}}\sim \operatorname {Lognormal} (-\mu ,\ \sigma ^{2}).}"></li> <li>Power: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c08a8fc58091f7804e20c5b9052f31893d430dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.598ex; height:3.176ex;" alt="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{a}\sim \operatorname {Lognormal} (a\mu ,\ a^{2}\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>μ</mi> <mo>,</mo> <mtext> </mtext> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{a}\sim \operatorname {Lognormal} (a\mu ,\ a^{2}\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f615ae4c73722b82cb5c75632359e29368be2d22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.811ex; height:3.176ex;" alt="{\displaystyle X^{a}\sim \operatorname {Lognormal} (a\mu ,\ a^{2}\sigma ^{2})}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4be4ca48aafe304408dc86889e3c578a7be30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.138ex; height:2.676ex;" alt="{\displaystyle a\neq 0.}"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Multiplication_and_division_of_independent,_log-normal_random_variables">Multiplication and division of independent, log-normal random variables</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=18" title="Edit section: Multiplication and division of independent, log-normal random variables">edit</a>]</div> If two <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>, log-normal variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}"> are multiplied [divided], the product [ratio] is again log-normal, with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu _{1}+\mu _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{1}+\mu _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94f01934f43180e55fd6c60c80c2a9f4cefb538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.252ex; height:2.509ex;" alt="{\displaystyle \mu =\mu _{1}+\mu _{2}}"> [<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu _{1}-\mu _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{1}-\mu _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0cf5b32496b3473135448b548ca1aaeb546cf8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.252ex; height:2.509ex;" alt="{\displaystyle \mu =\mu _{1}-\mu _{2}}">] and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=\sigma _{1}^{2}+\sigma _{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=\sigma _{1}^{2}+\sigma _{2}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90fda1167d13dce4bbff4038f2e01077489ef1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.093ex; height:3.343ex;" alt="{\displaystyle \sigma ^{2}=\sigma _{1}^{2}+\sigma _{2}^{2}}">. This is easily generalized to the product of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> such variables. More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716b3b7beaa8e122d0a848092319968194a90344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.361ex; height:3.509ex;" alt="{\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})}"> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> independent, log-normally distributed variables, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\textstyle \prod _{j=1}^{n}X_{j}\sim \operatorname {Lognormal} {\Big (}\textstyle \sum _{j=1}^{n}\mu _{j},\ \sum _{j=1}^{n}\sigma _{j}^{2}{\Big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\textstyle \prod _{j=1}^{n}X_{j}\sim \operatorname {Lognormal} {\Big (}\textstyle \sum _{j=1}^{n}\mu _{j},\ \sum _{j=1}^{n}\sigma _{j}^{2}{\Big )}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9968221e2045e4f3481ec1e8c1bbc038797b99d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.881ex; height:4.843ex;" alt="{\displaystyle Y=\textstyle \prod _{j=1}^{n}X_{j}\sim \operatorname {Lognormal} {\Big (}\textstyle \sum _{j=1}^{n}\mu _{j},\ \sum _{j=1}^{n}\sigma _{j}^{2}{\Big )}.}"> <div class="mw-heading mw-heading3"><h3 id="Multiplicative_central_limit_theorem">Multiplicative central limit theorem</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=19" title="Edit section: Multiplicative central limit theorem">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Gibrat%27s_law" title="Gibrat's law">Gibrat's law</a></div> The geometric or multiplicative mean of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> independent, identically distributed, positive random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"> shows, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }">, approximately a log-normal distribution with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =E[\ln(X_{i})]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>E</mi> <mo stretchy="false">[</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =E[\ln(X_{i})]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a724b374b7dbb96f1b3a40018c88d0011d859e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.042ex; height:2.843ex;" alt="{\displaystyle \mu =E[\ln(X_{i})]}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}={\mbox{var}}[\ln(X_{i})]/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>var</mtext> </mstyle> </mrow> <mo stretchy="false">[</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}={\mbox{var}}[\ln(X_{i})]/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb61e99822586c483b382dd80770ed7df53680d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.108ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}={\mbox{var}}[\ln(X_{i})]/n}">, assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"> is finite. In fact, the random variables do not have to be identically distributed. It is enough for the distributions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(X_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(X_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f43cf1bf401537a5b6b1cb7dedea2d4fa2ae20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.473ex; height:2.843ex;" alt="{\displaystyle \ln(X_{i})}"> to all have finite variance and satisfy the other conditions of any of the many variants of the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>. This is commonly known as <a href="/wiki/Gibrat%27s_law" title="Gibrat's law">Gibrat's law</a>. <div class="mw-heading mw-heading3"><h3 id="Other">Other</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=20" title="Edit section: Other">edit</a>]</div> A set of data that arises from the log-normal distribution has a symmetric <a href="/wiki/Lorenz_curve" title="Lorenz curve">Lorenz curve</a> (see also <a href="/wiki/Lorenz_asymmetry_coefficient" title="Lorenz asymmetry coefficient">Lorenz asymmetry coefficient</a>).<a href="#cite_note-EcolgyArticle-31">[31]</a> The harmonic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}">, geometric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"> and arithmetic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> means of this distribution are related;<a href="#cite_note-Rossman1990-32">[32]</a> such relation is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\frac {G^{2}}{A}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>A</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\frac {G^{2}}{A}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc5402a99bd6fe40d0dc047f7ed3c9d6d5eef8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.526ex; height:5.843ex;" alt="{\displaystyle H={\frac {G^{2}}{A}}.}"></dd></dl> Log-normal distributions are <a href="/wiki/Infinite_divisibility_(probability)" title="Infinite divisibility (probability)">infinitely divisible</a>,<a href="#cite_note-OlofThorin1978LNInfDivi-33">[33]</a> but they are not <a href="/wiki/Stable_distribution" title="Stable distribution">stable distributions</a>, which can be easily drawn from.<a href="#cite_note-Gao-34">[34]</a> <div class="mw-heading mw-heading2"><h2 id="Related_distributions">Related distributions</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=21" title="Edit section: Related distributions">edit</a>]</div> <ul><li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e49a012c102388008a926ef3e2e28d099d539751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.983ex; height:3.176ex;" alt="{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}"> is a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(X)\sim \operatorname {Lognormal} (\mu ,\sigma ^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(X)\sim \operatorname {Lognormal} (\mu ,\sigma ^{2}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97328d8a9edc37d6d62c2d056df41979eca84840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.606ex; height:3.176ex;" alt="{\displaystyle \exp(X)\sim \operatorname {Lognormal} (\mu ,\sigma ^{2}).}"></li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c08a8fc58091f7804e20c5b9052f31893d430dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.598ex; height:3.176ex;" alt="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> is distributed log-normally, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b92c953f2ecbfccfb70cfa5e80fa3b71ac06b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.732ex; height:3.176ex;" alt="{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})}"> is a normal random variable.</li> <li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716b3b7beaa8e122d0a848092319968194a90344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.361ex; height:3.509ex;" alt="{\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})}"> be independent log-normally distributed variables with possibly varying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> parameters, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Y=\sum _{j=1}^{n}X_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Y=\sum _{j=1}^{n}X_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56637ca4806beda63934012a76114bfb55fb93d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.557ex; height:3.509ex;" alt="{\textstyle Y=\sum _{j=1}^{n}X_{j}}">. The distribution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> has no closed-form expression, but can be reasonably approximated by another log-normal distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"> at the right tail.<a href="#cite_note-Asmussen2-35">[35]</a> Its probability density function at the neighborhood of 0 has been characterized<a href="#cite_note-Gao-34">[34]</a> and it does not resemble any log-normal distribution. A commonly used approximation due to L.F. Fenton (but previously stated by R.I. Wilkinson and mathematically justified by Marlow<a href="#cite_note-Marlow-36">[36]</a>) is obtained by matching the mean and variance of another log-normal distribution: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left[{\frac {\sum e^{2\mu _{j}+\sigma _{j}^{2}}(e^{\sigma _{j}^{2}}-1)}{(\sum e^{\mu _{j}+\sigma _{j}^{2}/2})^{2}}}+1\right],\\\mu _{Z}&=\ln \!\left[\sum e^{\mu _{j}+\sigma _{j}^{2}/2}\right]-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mo>∑</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <mo>∑</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left[{\frac {\sum e^{2\mu _{j}+\sigma _{j}^{2}}(e^{\sigma _{j}^{2}}-1)}{(\sum e^{\mu _{j}+\sigma _{j}^{2}/2})^{2}}}+1\right],\\\mu _{Z}&=\ln \!\left[\sum e^{\mu _{j}+\sigma _{j}^{2}/2}\right]-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc943ff6dcd032b6a82e022dd853316e4e77307" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:35.83ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left[{\frac {\sum e^{2\mu _{j}+\sigma _{j}^{2}}(e^{\sigma _{j}^{2}}-1)}{(\sum e^{\mu _{j}+\sigma _{j}^{2}/2})^{2}}}+1\right],\\\mu _{Z}&=\ln \!\left[\sum e^{\mu _{j}+\sigma _{j}^{2}/2}\right]-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}}"> In the case that all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}"> have the same variance parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{j}=\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{j}=\sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c74a1ee09c7ce4bc56587a4a293d2d04e4444a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.665ex; height:2.343ex;" alt="{\displaystyle \sigma _{j}=\sigma }">, these formulas simplify to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left[(e^{\sigma ^{2}}-1){\frac {\sum e^{2\mu _{j}}}{(\sum e^{\mu _{j}})^{2}}}+1\right],\\\mu _{Z}&=\ln \!\left[\sum e^{\mu _{j}}\right]+{\frac {\sigma ^{2}}{2}}-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mo>∑</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <mo>∑</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left[(e^{\sigma ^{2}}-1){\frac {\sum e^{2\mu _{j}}}{(\sum e^{\mu _{j}})^{2}}}+1\right],\\\mu _{Z}&=\ln \!\left[\sum e^{\mu _{j}}\right]+{\frac {\sigma ^{2}}{2}}-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3403a57bdac83cd433bc61aacd2206067d27bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.974ex; margin-bottom: -0.197ex; width:34.821ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left[(e^{\sigma ^{2}}-1){\frac {\sum e^{2\mu _{j}}}{(\sum e^{\mu _{j}})^{2}}}+1\right],\\\mu _{Z}&=\ln \!\left[\sum e^{\mu _{j}}\right]+{\frac {\sigma ^{2}}{2}}-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}}"></li></ul> For a more accurate approximation, one can use the <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a> to estimate the cumulative distribution function, the pdf and the right tail.<a href="#cite_note-BotLec2017-37">[37]</a><a href="#cite_note-AGL2016-38">[38]</a> The sum of correlated log-normally distributed random variables can also be approximated by a log-normal distribution[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S_{+}&=\operatorname {E} \left[\sum _{i}X_{i}\right]=\sum _{i}\operatorname {E} [X_{i}]=\sum _{i}e^{\mu _{i}+\sigma _{i}^{2}/2}\\\sigma _{Z}^{2}&=1/S_{+}^{2}\,\sum _{i,j}\operatorname {cor} _{ij}\sigma _{i}\sigma _{j}\operatorname {E} [X_{i}]\operatorname {E} [X_{j}]=1/S_{+}^{2}\,\sum _{i,j}\operatorname {cor} _{ij}\sigma _{i}\sigma _{j}e^{\mu _{i}+\sigma _{i}^{2}/2}e^{\mu _{j}+\sigma _{j}^{2}/2}\\\mu _{Z}&=\ln \left(S_{+}\right)-\sigma _{Z}^{2}/2\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>cor</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>⁡</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>cor</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>⁡</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>)</mo> </mrow> <mo>−</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{+}&=\operatorname {E} \left[\sum _{i}X_{i}\right]=\sum _{i}\operatorname {E} [X_{i}]=\sum _{i}e^{\mu _{i}+\sigma _{i}^{2}/2}\\\sigma _{Z}^{2}&=1/S_{+}^{2}\,\sum _{i,j}\operatorname {cor} _{ij}\sigma _{i}\sigma _{j}\operatorname {E} [X_{i}]\operatorname {E} [X_{j}]=1/S_{+}^{2}\,\sum _{i,j}\operatorname {cor} _{ij}\sigma _{i}\sigma _{j}e^{\mu _{i}+\sigma _{i}^{2}/2}e^{\mu _{j}+\sigma _{j}^{2}/2}\\\mu _{Z}&=\ln \left(S_{+}\right)-\sigma _{Z}^{2}/2\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ce906d9079b01314fe41adcaab5b630462d164" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:75.535ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}S_{+}&=\operatorname {E} \left[\sum _{i}X_{i}\right]=\sum _{i}\operatorname {E} [X_{i}]=\sum _{i}e^{\mu _{i}+\sigma _{i}^{2}/2}\\\sigma _{Z}^{2}&=1/S_{+}^{2}\,\sum _{i,j}\operatorname {cor} _{ij}\sigma _{i}\sigma _{j}\operatorname {E} [X_{i}]\operatorname {E} [X_{j}]=1/S_{+}^{2}\,\sum _{i,j}\operatorname {cor} _{ij}\sigma _{i}\sigma _{j}e^{\mu _{i}+\sigma _{i}^{2}/2}e^{\mu _{j}+\sigma _{j}^{2}/2}\\\mu _{Z}&=\ln \left(S_{+}\right)-\sigma _{Z}^{2}/2\end{aligned}}}"> <ul><li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c08a8fc58091f7804e20c5b9052f31893d430dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.598ex; height:3.176ex;" alt="{\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X+c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da22f3c54e0cc4d40f0a5ef44b27ee36cb1569e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.827ex; height:2.343ex;" alt="{\displaystyle X+c}"> is said to have a Three-parameter log-normal distribution with support <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in (c,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (c,+\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325b07f5a012969475c05bce74d25141dc82e865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.152ex; height:2.843ex;" alt="{\displaystyle x\in (c,+\infty )}">.<a href="#cite_note-Sangal1970-39">[39]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X+c]=\operatorname {E} [X]+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X+c]=\operatorname {E} [X]+c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bccff99c9c6a0829010eafc025c7a24c33fe6e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.506ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X+c]=\operatorname {E} [X]+c}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} [X+c]=\operatorname {Var} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} [X+c]=\operatorname {Var} [X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3cc065bfe4de4faaf4facb23f8fa2891ea72c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.127ex; height:2.843ex;" alt="{\displaystyle \operatorname {Var} [X+c]=\operatorname {Var} [X]}">.</li> <li>The log-normal distribution is a special case of the semi-bounded <a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's SU-distribution</a>.<a href="#cite_note-Johnson1949-40">[40]</a></li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\mid Y\sim \operatorname {Rayleigh} (Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∣</mo> <mi>Y</mi> <mo>∼</mo> <mi>Rayleigh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\mid Y\sim \operatorname {Rayleigh} (Y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64810465165066c96919bcd87a3bbf65ecc8186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.253ex; height:2.843ex;" alt="{\displaystyle X\mid Y\sim \operatorname {Rayleigh} (Y)}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>∼</mo> <mi>Lognormal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57862f49db834dec0c769a6e89a0dab61559882a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.391ex; height:3.176ex;" alt="{\displaystyle Y\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim \operatorname {Suzuki} (\mu ,\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∼</mo> <mi>Suzuki</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\sim \operatorname {Suzuki} (\mu ,\sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f24cc579afa881a66a738370a77f0a0ef77b7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.437ex; height:2.843ex;" alt="{\displaystyle X\sim \operatorname {Suzuki} (\mu ,\sigma )}"> (<a href="/w/index.php?title=Suzuki_distribution&action=edit&redlink=1" class="new" title="Suzuki distribution (page does not exist)">Suzuki distribution</a>).</li> <li>A substitute for the log-normal whose integral can be expressed in terms of more elementary functions<a href="#cite_note-41">[41]</a> can be obtained based on the <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic distribution</a> to get an approximation for the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x;\mu ,\sigma )=\left[\left({\frac {e^{\mu }}{x}}\right)^{\pi /(\sigma {\sqrt {3}})}+1\right]^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x;\mu ,\sigma )=\left[\left({\frac {e^{\mu }}{x}}\right)^{\pi /(\sigma {\sqrt {3}})}+1\right]^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e28d7a1ba703b5e772530f62f55f314b9ba007bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.441ex; height:8.009ex;" alt="{\displaystyle F(x;\mu ,\sigma )=\left[\left({\frac {e^{\mu }}{x}}\right)^{\pi /(\sigma {\sqrt {3}})}+1\right]^{-1}.}"> This is a <a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">log-logistic distribution</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Statistical_inference">Statistical inference</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=22" title="Edit section: Statistical inference">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Estimation_of_parameters">Estimation of parameters</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=23" title="Edit section: Estimation of parameters">edit</a>]</div> For determining the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimators of the log-normal distribution parameters μ and σ, we can use the <a href="/wiki/Normal_distribution#Estimation_of_parameters" title="Normal distribution">same procedure</a> as for the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>. Note that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mu ,\sigma )=\prod _{i=1}^{n}{\frac {1}{x_{i}}}\varphi _{\mu ,\sigma }(\ln x_{i}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(\mu ,\sigma )=\prod _{i=1}^{n}{\frac {1}{x_{i}}}\varphi _{\mu ,\sigma }(\ln x_{i}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd89d33becbd9cf42a69f4e8b77f73cc128c9a6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.631ex; height:6.843ex;" alt="{\displaystyle L(\mu ,\sigma )=\prod _{i=1}^{n}{\frac {1}{x_{i}}}\varphi _{\mu ,\sigma }(\ln x_{i}),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> is the density function of the normal distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.966ex; height:3.176ex;" alt="{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}">. Therefore, the log-likelihood function is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (\mu ,\sigma \mid x_{1},x_{2},\ldots ,x_{n})=-\sum _{i}\ln x_{i}+\ell _{N}(\mu ,\sigma \mid \ln x_{1},\ln x_{2},\dots ,\ln x_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>∣</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>σ</mi> <mo>∣</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (\mu ,\sigma \mid x_{1},x_{2},\ldots ,x_{n})=-\sum _{i}\ln x_{i}+\ell _{N}(\mu ,\sigma \mid \ln x_{1},\ln x_{2},\dots ,\ln x_{n}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be1592bd473fc8b85aa0fb02e1ec788ef67fb8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:69.669ex; height:5.509ex;" alt="{\displaystyle \ell (\mu ,\sigma \mid x_{1},x_{2},\ldots ,x_{n})=-\sum _{i}\ln x_{i}+\ell _{N}(\mu ,\sigma \mid \ln x_{1},\ln x_{2},\dots ,\ln x_{n}).}"> Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3adc0cc904b2fcad44c27d9c5509f1f14ddadd37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.661ex; height:2.509ex;" alt="{\displaystyle \ell _{N}}">, reach their maximum with the same <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">. Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x_{1},\ln x_{2},\dots ,\ln x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x_{1},\ln x_{2},\dots ,\ln x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1a01da51f796f5c0a60c6b2a382584fce33d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.412ex; height:2.843ex;" alt="{\displaystyle \ln x_{1},\ln x_{2},\dots ,\ln x_{n})}">, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad {\widehat {\sigma }}^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad {\widehat {\sigma }}^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82932d2a009062ea383213d66c441debd4f623a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.299ex; height:6.176ex;" alt="{\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad {\widehat {\sigma }}^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n}}.}"> For finite n, the estimator for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is unbiased, but the one for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> is biased. As for the normal distribution, an unbiased estimator for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> can be obtained by replacing the denominator n by n−1 in the equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\sigma }}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\sigma }}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afe69bbdc03070ef019f43fe451b847d2acf1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle {\widehat {\sigma }}^{2}}">. When the individual values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},\ldots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\ldots ,x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8694289524164f895d6665f163e14c4dc5ec648d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\ldots ,x_{n}}"> are not available, but the sample's mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"> and <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> s is, then the <a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a> can be used. The corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dd294aa33c0865f58e2b1bdaf44ebe911dbf93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.857ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X]}"> and variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} [X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79297a808478243e9aab0b27dd1ab583c0f877d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.091ex; height:2.843ex;" alt="{\displaystyle \operatorname {Var} [X]}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\ln \left({\frac {\bar {x}}{\sqrt {1+{\widehat {\sigma }}^{2}/{\bar {x}}^{2}}}}\right),\qquad \sigma ^{2}=\ln \left(1+{{\widehat {\sigma }}^{2}}/{\bar {x}}^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\ln \left({\frac {\bar {x}}{\sqrt {1+{\widehat {\sigma }}^{2}/{\bar {x}}^{2}}}}\right),\qquad \sigma ^{2}=\ln \left(1+{{\widehat {\sigma }}^{2}}/{\bar {x}}^{2}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1aa40db73816305447d6f4cbd2ae635e928490b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:50.057ex; height:10.509ex;" alt="{\displaystyle \mu =\ln \left({\frac {\bar {x}}{\sqrt {1+{\widehat {\sigma }}^{2}/{\bar {x}}^{2}}}}\right),\qquad \sigma ^{2}=\ln \left(1+{{\widehat {\sigma }}^{2}}/{\bar {x}}^{2}\right).}"> <div class="mw-heading mw-heading3"><h3 id="Interval_estimates">Interval estimates</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=24" title="Edit section: Interval estimates">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Reference_range#Log-normal_distribution" title="Reference range">Reference range § Log-normal distribution</a></div> The most efficient way to obtain <a href="/wiki/Interval_estimate" class="mw-redirect" title="Interval estimate">interval estimates</a> when analyzing log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate. <div class="mw-heading mw-heading4"><h4 id="Prediction_intervals">Prediction intervals</h4>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=25" title="Edit section: Prediction intervals">edit</a>]</div> A basic example is given by <a href="/wiki/Prediction_interval" title="Prediction interval">prediction intervals</a>: For the normal distribution, the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mu -\sigma ,\mu +\sigma ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>μ</mi> <mo>−</mo> <mi>σ</mi> <mo>,</mo> <mi>μ</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mu -\sigma ,\mu +\sigma ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c87cf54494c57f8aa41a35e60cf1f4ba837fa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.471ex; height:2.843ex;" alt="{\displaystyle [\mu -\sigma ,\mu +\sigma ]}"> contains approximately two thirds (68%) of the probability (or of a large sample), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mu -2\sigma ,\mu +2\sigma ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>μ</mi> <mo>−</mo> <mn>2</mn> <mi>σ</mi> <mo>,</mo> <mi>μ</mi> <mo>+</mo> <mn>2</mn> <mi>σ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mu -2\sigma ,\mu +2\sigma ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb2f1b03c720b317b0fcf7e012a9bba1a3f418e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.796ex; height:2.843ex;" alt="{\displaystyle [\mu -2\sigma ,\mu +2\sigma ]}"> contain 95%. Therefore, for a log-normal distribution, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mu ^{*}/\sigma ^{*},\mu ^{*}\cdot \sigma ^{*}]=[\mu ^{*}{}^{\times }\!\!/\sigma ^{*}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>,</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⋅</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mu ^{*}/\sigma ^{*},\mu ^{*}\cdot \sigma ^{*}]=[\mu ^{*}{}^{\times }\!\!/\sigma ^{*}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90bfe33d3e1ec78fd21196427394f5f4fe5e1836" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.982ex; height:2.843ex;" alt="{\displaystyle [\mu ^{*}/\sigma ^{*},\mu ^{*}\cdot \sigma ^{*}]=[\mu ^{*}{}^{\times }\!\!/\sigma ^{*}]}"> contains 2/3, and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mu ^{*}/(\sigma ^{*})^{2},\mu ^{*}\cdot (\sigma ^{*})^{2}]=[\mu ^{*}{}^{\times }\!\!/(\sigma ^{*})^{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mu ^{*}/(\sigma ^{*})^{2},\mu ^{*}\cdot (\sigma ^{*})^{2}]=[\mu ^{*}{}^{\times }\!\!/(\sigma ^{*})^{2}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721c476ec6cdb74bed626ea73e2e5f44bff32d84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.573ex; height:3.176ex;" alt="{\displaystyle [\mu ^{*}/(\sigma ^{*})^{2},\mu ^{*}\cdot (\sigma ^{*})^{2}]=[\mu ^{*}{}^{\times }\!\!/(\sigma ^{*})^{2}]}"> contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals. <div class="mw-heading mw-heading4"><h4 id="Confidence_interval_for_eμ">Confidence interval for eμ</h4>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=26" title="Edit section: Confidence interval for eμ">edit</a>]</div> Using the principle, note that a <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\widehat {\mu }}\pm q\cdot {\widehat {\mathop {se} }}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>±</mo> <mi>q</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-OP"> <mi>s</mi> <mi>e</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\widehat {\mu }}\pm q\cdot {\widehat {\mathop {se} }}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38626a249b1d579a2af15d2d64ec382789448e60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.618ex; height:2.843ex;" alt="{\displaystyle [{\widehat {\mu }}\pm q\cdot {\widehat {\mathop {se} }}]}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathop {se} ={\widehat {\sigma }}/{\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <mi>s</mi> <mi>e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathop {se} ={\widehat {\sigma }}/{\sqrt {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39ef64cb23f0d77b6fb0969746c53ddee69251e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.095ex; height:3.009ex;" alt="{\displaystyle \mathop {se} ={\widehat {\sigma }}/{\sqrt {n}}}"> is the standard error and q is the 97.5% quantile of a <a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">t distribution</a> with n-1 degrees of freedom. Back-transformation leads to a confidence interval for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{*}=e^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{*}=e^{\mu }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd58ff2956fb3e609960cfabceba23464fa4e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.861ex; height:2.843ex;" alt="{\displaystyle \mu ^{*}=e^{\mu }}"> (the median), is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\widehat {\mu }}^{*}{}^{\times }\!\!/(\operatorname {sem} ^{*})^{q}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>sem</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\widehat {\mu }}^{*}{}^{\times }\!\!/(\operatorname {sem} ^{*})^{q}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9c1579089d540825002f6a247b9991d2d87936" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.395ex; height:3.009ex;" alt="{\displaystyle [{\widehat {\mu }}^{*}{}^{\times }\!\!/(\operatorname {sem} ^{*})^{q}]}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sem} ^{*}=({\widehat {\sigma }}^{*})^{1/{\sqrt {n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sem</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sem} ^{*}=({\widehat {\sigma }}^{*})^{1/{\sqrt {n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a701413dde6b94e4af350d605ada06fe43746f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.462ex; height:3.343ex;" alt="{\displaystyle \operatorname {sem} ^{*}=({\widehat {\sigma }}^{*})^{1/{\sqrt {n}}}}"> <div class="mw-heading mw-heading4"><h4 id="Confidence_interval_for_E(X)">Confidence interval for E(X)</h4>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=27" title="Edit section: Confidence interval for E(X)">edit</a>]</div> The literature discusses several options for calculating the <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> (the mean of the log-normal distribution). These include <a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">bootstrap</a> as well as various other methods.<a href="#cite_note-Olsson2005-42">[42]</a><a href="#cite_note-43">[43]</a> The Cox Method<a href="#cite_note-46">[a]</a> proposes to plug-in the estimators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad S^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad S^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n-1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b60f16bcc6af7c9890f41a642fb9e308ff3902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.844ex; height:6.343ex;" alt="{\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad S^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n-1}}}"> and use them to construct <a href="/wiki/Confidence_interval#Approximate_confidence_intervals" title="Confidence interval">approximate confidence intervals</a> in the following way: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CI(E(X)):e^{\left({\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>±</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CI(E(X)):e^{\left({\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a23f01628ceedf1417bf4a0641f7d4c7c32f421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.031ex; height:6.343ex;" alt="{\displaystyle CI(E(X)):e^{\left({\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}\right)}}"> <style data-mw-deduplicate="TemplateStyles:r1214851843">.mw-parser-output .hidden-begin{box-sizing:border-box;width:100%;padding:5px;border:none;font-size:95%}.mw-parser-output .hidden-title{font-weight:bold;line-height:1.6;text-align:left}.mw-parser-output .hidden-content{text-align:left}@media all and (max-width:500px){.mw-parser-output .hidden-begin{width:auto!important;clear:none!important;float:none!important}}</style><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px #aaa solid; width:100%"><div class="hidden-title skin-nightmode-reset-color" style="text-align:center;">[Proof]</div><div class="hidden-content mw-collapsible-content" style=""> We know that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(X)=e^{\mu +{\frac {\sigma ^{2}}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X)=e^{\mu +{\frac {\sigma ^{2}}{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7917c0fe50edf7f3171e97eaa81d75b62497fc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.649ex; height:4.509ex;" alt="{\displaystyle E(X)=e^{\mu +{\frac {\sigma ^{2}}{2}}}}">. Also, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mu }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0771ade78a4446e1f7bf6900e637b788c0cbad92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.019ex; width:1.43ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mu }}}"> is a normal distribution with parameters: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mu }}\sim N\left(\mu ,{\frac {\sigma ^{2}}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>∼</mo> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mu }}\sim N\left(\mu ,{\frac {\sigma ^{2}}{n}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb13b0072a9338474bbebbcbe5cbb11b98fa4906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.039ex; height:6.343ex;" alt="{\displaystyle {\widehat {\mu }}\sim N\left(\mu ,{\frac {\sigma ^{2}}{n}}\right)}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6401d5d0155afb1406770d1eb80badce4e08ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2}}"> has a <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a>, which is <a href="/wiki/Chi-squared_distribution#Related_distributions" title="Chi-squared distribution">approximately</a> normally distributed (via <a href="/wiki/Central_limit_theorem" title="Central limit theorem">CLT</a>), with <a href="/wiki/Variance#Distribution_of_the_sample_variance" title="Variance">parameters</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}{\dot {\sim }}N\left(\sigma ^{2},{\frac {2\sigma ^{4}}{n-1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∼</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}{\dot {\sim }}N\left(\sigma ^{2},{\frac {2\sigma ^{4}}{n-1}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44807e8d1228094c0f41e2ef1b3754330633f679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.908ex; height:6.343ex;" alt="{\displaystyle S^{2}{\dot {\sim }}N\left(\sigma ^{2},{\frac {2\sigma ^{4}}{n-1}}\right)}">. Hence, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {S^{2}}{2}}{\dot {\sim }}N\left({\frac {\sigma ^{2}}{2}},{\frac {\sigma ^{4}}{2(n-1)}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∼</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S^{2}}{2}}{\dot {\sim }}N\left({\frac {\sigma ^{2}}{2}},{\frac {\sigma ^{4}}{2(n-1)}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ae48f303627dc19faa4e74de2055222cb1248f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.552ex; height:6.509ex;" alt="{\displaystyle {\frac {S^{2}}{2}}{\dot {\sim }}N\left({\frac {\sigma ^{2}}{2}},{\frac {\sigma ^{4}}{2(n-1)}}\right)}">. Since the sample mean and variance are independent, and the sum of normally distributed variables is <a href="/wiki/Normal_distribution#Operations_on_two_independent_normal_variables" title="Normal distribution">also normal</a>, we get that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mu }}+{\frac {S^{2}}{2}}{\dot {\sim }}N\left(\mu +{\frac {\sigma ^{2}}{2}},{\frac {\sigma ^{2}}{n}}+{\frac {\sigma ^{4}}{2(n-1)}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∼</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mu }}+{\frac {S^{2}}{2}}{\dot {\sim }}N\left(\mu +{\frac {\sigma ^{2}}{2}},{\frac {\sigma ^{2}}{n}}+{\frac {\sigma ^{4}}{2(n-1)}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10e135b52517bd17e86d118b39cf20973b684ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.108ex; height:6.509ex;" alt="{\displaystyle {\widehat {\mu }}+{\frac {S^{2}}{2}}{\dot {\sim }}N\left(\mu +{\frac {\sigma ^{2}}{2}},{\frac {\sigma ^{2}}{n}}+{\frac {\sigma ^{4}}{2(n-1)}}\right)}"> Based on the above, standard <a href="/wiki/Normal_distribution#Confidence_intervals" title="Normal distribution">confidence intervals</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu +{\frac {\sigma ^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu +{\frac {\sigma ^{2}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c468ee63827ea0bdb9cbdff994b86d518d9c089e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.463ex; height:5.676ex;" alt="{\displaystyle \mu +{\frac {\sigma ^{2}}{2}}}"> can be constructed (using a <a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivotal quantity</a>) as: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>±</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a60440e4b8718f85198b0bbe4f8e665bb1002c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.38ex; height:7.676ex;" alt="{\displaystyle {\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}}"> And since confidence intervals are preserved for monotonic transformations, we get that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CI\left(E(X)=e^{\mu +{\frac {\sigma ^{2}}{2}}}\right):e^{\left({\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>I</mi> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>±</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CI\left(E(X)=e^{\mu +{\frac {\sigma ^{2}}{2}}}\right):e^{\left({\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38b73486827299574fc27eaef6ad9216bc95dc51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.115ex; height:8.009ex;" alt="{\displaystyle CI\left(E(X)=e^{\mu +{\frac {\sigma ^{2}}{2}}}\right):e^{\left({\hat {\mu }}+{\frac {S^{2}}{2}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S^{2}}{n}}+{\frac {S^{4}}{2(n-1)}}}}\right)}}"> As desired. </div></div> Olsson 2005, proposed a "modified Cox method" by replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1-{\frac {\alpha }{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1-{\frac {\alpha }{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e1be72c4b09dd940afa11ecbabb56e22a39e3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.104ex; height:3.176ex;" alt="{\displaystyle z_{1-{\frac {\alpha }{2}}}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n-1,1-{\frac {\alpha }{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n-1,1-{\frac {\alpha }{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ac2e5f824e32c2b912bc7a712aaed9dc3f9f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.407ex; height:3.509ex;" alt="{\displaystyle t_{n-1,1-{\frac {\alpha }{2}}}}">, which seemed to provide better coverage results for small sample sizes.<a href="#cite_note-Olsson2005-42">[42]</a>: Section 3.4  <div class="mw-heading mw-heading4"><h4 id="Confidence_interval_for_comparing_two_log_normals">Confidence interval for comparing two log normals</h4>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=28" title="Edit section: Confidence interval for comparing two log normals">edit</a>]</div> Comparing two log-normal distributions can often be of interest, for example, from a treatment and control group (e.g., in an <a href="/wiki/A/B_testing" title="A/B testing">A/B test</a>). We have samples from two independent log-normal distributions with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{1},\sigma _{1}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{1},\sigma _{1}^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ad62720670862eadc8aa0d2daf86577f290fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.684ex; height:3.176ex;" alt="{\displaystyle (\mu _{1},\sigma _{1}^{2})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mu _{2},\sigma _{2}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{2},\sigma _{2}^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f40887314ee18d7c9b1c1aa28fdc562fe44344a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.684ex; height:3.176ex;" alt="{\displaystyle (\mu _{2},\sigma _{2}^{2})}">, with sample sizes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee784b70e772f55ede5e6e0bdc929994bff63413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/840e456e3058bc0be28e5cf653b170cdbfcc3be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{2}}"> respectively. Comparing the medians of the two can easily be done by taking the log from each and then constructing straightforward confidence intervals and transforming it back to the exponential scale. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CI(e^{\mu _{1}-\mu _{2}}):e^{\left({\hat {\mu }}_{1}-{\hat {\mu }}_{2}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n}}+{\frac {S_{2}^{2}}{n}}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>I</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>±</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>n</mi> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CI(e^{\mu _{1}-\mu _{2}}):e^{\left({\hat {\mu }}_{1}-{\hat {\mu }}_{2}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n}}+{\frac {S_{2}^{2}}{n}}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e030f1442f96c3fa88109adacf5e90f61f65e124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.585ex; height:6.843ex;" alt="{\displaystyle CI(e^{\mu _{1}-\mu _{2}}):e^{\left({\hat {\mu }}_{1}-{\hat {\mu }}_{2}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n}}+{\frac {S_{2}^{2}}{n}}}}\right)}}"> These CI are what's often used in epidemiology for calculation the CI for <a href="/wiki/Relative-risk" class="mw-redirect" title="Relative-risk">relative-risk</a> and <a href="/wiki/Odds-ratio" class="mw-redirect" title="Odds-ratio">odds-ratio</a>.<a href="#cite_note-47">[46]</a> The way it is done there is that we have two approximately Normal distributions (e.g., p1 and p2, for RR), and we wish to calculate their ratio.<a href="#cite_note-48">[b]</a> However, the ratio of the expectations (means) of the two samples might also be of interest, while requiring more work to develop. The ratio of their means is: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/884cde97b34e74c0e90690c4efbe8107d0b94a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:37.654ex; height:10.176ex;" alt="{\displaystyle {\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2})}}"> Plugin in the estimators to each of these parameters yields also a log normal distribution, which means that the Cox Method, discussed above, could similarly be used for this use-case: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CI\left({\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}\right):e^{\left(({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>I</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>±</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CI\left({\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}\right):e^{\left(({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e69b8c8e362ee91838e818599a9b52d9c3588d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:75.055ex; height:10.843ex;" alt="{\displaystyle CI\left({\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}\right):e^{\left(({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}}"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214851843"><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px #aaa solid; width:100%"><div class="hidden-title skin-nightmode-reset-color" style="text-align:center;">[Proof]</div><div class="hidden-content mw-collapsible-content" style=""> To construct a confidence interval for this ratio, we first note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mu }}_{1}-{\hat {\mu }}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}_{1}-{\hat {\mu }}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb9aae532f442268ce170189c8e41abfaf1ec06c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.752ex; height:2.676ex;" alt="{\displaystyle {\hat {\mu }}_{1}-{\hat {\mu }}_{2}}"> follows a normal distribution, and that both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/538341b0169fdfc5e11c0009affc03c17efb70d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.576ex; height:3.176ex;" alt="{\displaystyle S_{1}^{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162588281d3e68c8754882166f8566153ebcda2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.576ex; height:3.176ex;" alt="{\displaystyle S_{2}^{2}}"> has a <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a>, which is <a href="/wiki/Chi-squared_distribution#Related_distributions" title="Chi-squared distribution">approximately</a> normally distributed (via <a href="/wiki/Central_limit_theorem" title="Central limit theorem">CLT</a>, with the relevant <a href="/wiki/Variance#Distribution_of_the_sample_variance" title="Variance">parameters</a>). This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\sim N\left((\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2}),{\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\sim N\left((\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2}),{\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4280b8f3c263109055553a584d4664c19c0e272b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:94.086ex; height:7.509ex;" alt="{\displaystyle ({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\sim N\left((\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2}),{\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}"> Based on the above, standard <a href="/wiki/Normal_distribution#Confidence_intervals" title="Normal distribution">confidence intervals</a> can be constructed (using a <a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivotal quantity</a>) as: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>±</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae980936522cab77d3b1d1933d2ea3811698121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.524ex; height:7.509ex;" alt="{\displaystyle ({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}}"> And since confidence intervals are preserved for monotonic transformations, we get that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CI\left({\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}\right):e^{\left(({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>I</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>±</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CI\left({\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}\right):e^{\left(({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e69b8c8e362ee91838e818599a9b52d9c3588d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:75.055ex; height:10.843ex;" alt="{\displaystyle CI\left({\frac {E(X_{1})}{E(X_{2})}}={\frac {e^{\mu _{1}+{\frac {\sigma _{1}^{2}}{2}}}}{e^{\mu _{2}+{\frac {\sigma _{2}^{2}}{2}}}}}\right):e^{\left(({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\frac {1}{2}}S_{1}^{2}-{\frac {1}{2}}S_{2}^{2})\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}}"> As desired. </div></div> It's worth noting that naively using the <a href="/wiki/Maximum_likelihood_estimation" title="Maximum likelihood estimation">MLE</a> in the ratio of the two expectations to create a <a href="/wiki/Ratio_estimator" title="Ratio estimator">ratio estimator</a> will lead to a <a href="/wiki/Consistency_(statistics)" title="Consistency (statistics)">consistent</a>, yet biased, point-estimation (we use the fact that the estimator of the ratio is a log normal distribution)<a href="#cite_note-49">[c]</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left[{\frac {{\widehat {E}}(X_{1})}{{\widehat {E}}(X_{2})}}\right]=E\left[e^{({\widehat {\mu }}_{1}-{\widehat {\mu }}_{2})+{\frac {1}{2}}(S_{1}^{2}-S_{2}^{2})}\right]=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2})+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>E</mi> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[{\frac {{\widehat {E}}(X_{1})}{{\widehat {E}}(X_{2})}}\right]=E\left[e^{({\widehat {\mu }}_{1}-{\widehat {\mu }}_{2})+{\frac {1}{2}}(S_{1}^{2}-S_{2}^{2})}\right]=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2})+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d691e4ea97cb627ead3fe0137ba424367fc29fa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:82.01ex; height:8.676ex;" alt="{\displaystyle E\left[{\frac {{\widehat {E}}(X_{1})}{{\widehat {E}}(X_{2})}}\right]=E\left[e^{({\widehat {\mu }}_{1}-{\widehat {\mu }}_{2})+{\frac {1}{2}}(S_{1}^{2}-S_{2}^{2})}\right]=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}(\sigma _{1}^{2}-\sigma _{2}^{2})+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}}"> <div class="mw-heading mw-heading3"><h3 id="Extremal_principle_of_entropy_to_fix_the_free_parameter_σ">Extremal principle of entropy to fix the free parameter σ</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=29" title="Edit section: Extremal principle of entropy to fix the free parameter σ">edit</a>]</div> In applications, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> is a parameter to be determined. For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that<a href="#cite_note-bai-50">[47]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\frac {1}{\sqrt {6}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>6</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {1}{\sqrt {6}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d815a4e3d3df3348876abf6751ca317d9421e330" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:8.363ex; height:6.176ex;" alt="{\displaystyle \sigma ={\frac {1}{\sqrt {6}}}}"> This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution.<a href="#cite_note-bai-50">[47]</a> This relationship is determined by the base of natural logarithm, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=2.718\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>2.718</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=2.718\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7518451f7860ee0d559926ff47f678db0de1a9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.589ex; height:2.176ex;" alt="{\displaystyle e=2.718\ldots }">, and exhibits some geometrical similarity to the minimal surface energy principle. These scaling relations are useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.). For example, the log-normal function with such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> fits well with the size of secondarily produced droplets during droplet impact <a href="#cite_note-wu-51">[48]</a> and the spreading of an epidemic disease.<a href="#cite_note-Wang-52">[49]</a> The value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma =1{\big /}{\sqrt {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma =1{\big /}{\sqrt {6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fefa55e5870b428d15027ad1db2d97f6feae414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.033ex; height:3.343ex;" alt="{\textstyle \sigma =1{\big /}{\sqrt {6}}}"> is used to provide a probabilistic solution for the Drake equation.<a href="#cite_note-Bloetscher-53">[50]</a> <div class="mw-heading mw-heading2"><h2 id="Heavy-tailness_of_the_Log-Normal">Heavy-tailness of the Log-Normal</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=30" title="Edit section: Heavy-tailness of the Log-Normal">edit</a>]</div> Whether a Log-Normal can be considered or not a true heavy-tail distribution is still debated. The main reason is that its variance is always finite, differently from what happen with certain Pareto distributions, for instance. However a recent study has shown how it is possible to create a Log-Normal distribution with infinite variance using Robinson Non-Standard Analysis.<a href="#cite_note-54">[51]</a> <div class="mw-heading mw-heading2"><h2 id="Occurrence_and_applications">Occurrence and applications</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=31" title="Edit section: Occurrence and applications">edit</a>]</div> The log-normal distribution is important in the description of natural phenomena. Many natural growth processes are driven by the accumulation of many small percentage changes which become additive on a log scale. Under appropriate regularity conditions, the distribution of the resulting accumulated changes will be increasingly well approximated by a log-normal, as noted in the section above on "<a href="#Multiplicative_Central_Limit_Theorem">Multiplicative Central Limit Theorem</a>". This is also known as <a href="/wiki/Gibrat%27s_law" title="Gibrat's law">Gibrat's law</a>, after Robert Gibrat (1904–1980) who formulated it for companies.<a href="#cite_note-55">[52]</a> If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if this assumption is not true, the size distributions at any age of things that grow over time tends to be log-normal.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Consequently, <a href="/wiki/Reference_ranges" class="mw-redirect" title="Reference ranges">reference ranges</a> for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases. Specific examples are given in the following subsections.<a href="#cite_note-:0-56">[53]</a> contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas.<a href="#cite_note-:4-57">[54]</a> is a review article on log-normal distributions in neuroscience, with annotated bibliography. <div class="mw-heading mw-heading3"><h3 id="Human_behavior">Human behavior</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=32" title="Edit section: Human behavior">edit</a>]</div> <ul><li>The length of comments posted in Internet discussion forums follows a log-normal distribution.<a href="#cite_note-:3-58">[55]</a></li> <li>Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.<a href="#cite_note-59">[56]</a></li> <li>The length of <a href="/wiki/Chess" title="Chess">chess</a> games tends to follow a log-normal distribution.<a href="#cite_note-60">[57]</a></li> <li>Onset durations of acoustic comparison stimuli that are matched to a standard stimulus follow a log-normal distribution.<a href="#cite_note-Acoustic_Stimuli_Revisited_2016-18">[18]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Biology_and_medicine">Biology and medicine</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=33" title="Edit section: Biology and medicine">edit</a>]</div> <ul><li>Measures of size of living tissue (length, skin area, weight).<a href="#cite_note-61">[58]</a></li> <li>Incubation period of diseases.<a href="#cite_note-62">[59]</a></li> <li>Diameters of banana leaf spots, powdery mildew on barley.<a href="#cite_note-:0-56">[53]</a></li> <li>For highly communicable epidemics, such as SARS in 2003, if public intervention control policies are involved, the number of hospitalized cases is shown to satisfy the log-normal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of <a href="/wiki/Entropy_production" title="Entropy production">entropy production</a>.<a href="#cite_note-63">[60]</a></li> <li>The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</li> <li>The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.</li> <li>The <a href="/wiki/Pacific_Biosciences" title="Pacific Biosciences">PacBio</a> sequencing read length follows a log-normal distribution.<a href="#cite_note-64">[61]</a></li> <li>Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).<a href="#cite_note-65">[62]</a></li> <li>Several <a href="/wiki/Pharmacokinetics" title="Pharmacokinetics">pharmacokinetic</a> variables, such as <a href="/wiki/Cmax_(pharmacology)" title="Cmax (pharmacology)">Cmax</a>, <a href="/wiki/Biological_half-life" title="Biological half-life">elimination half-life</a> and the <a href="/wiki/Elimination_rate_constant" title="Elimination rate constant">elimination rate constant</a>.<a href="#cite_note-66">[63]</a></li> <li>In neuroscience, the distribution of firing rates across a population of neurons is often approximately log-normal. This has been first observed in the cortex and striatum <a href="#cite_note-67">[64]</a> and later in hippocampus and entorhinal cortex,<a href="#cite_note-68">[65]</a> and elsewhere in the brain.<a href="#cite_note-:4-57">[54]</a><a href="#cite_note-69">[66]</a> Also, intrinsic gain distributions and synaptic weight distributions appear to be log-normal<a href="#cite_note-70">[67]</a> as well.</li> <li>Neuron densities in the cerebral cortex, due to the noisy cell division process during neurodevelopment.<a href="#cite_note-71">[68]</a></li> <li>In operating-rooms management, the distribution of <a href="/wiki/Predictive_methods_for_surgery_duration" title="Predictive methods for surgery duration">surgery duration</a>.</li> <li>In the size of avalanches of fractures in the cytoskeleton of living cells, showing log-normal distributions, with significantly higher size in cancer cells than healthy ones.<a href="#cite_note-72">[69]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Chemistry">Chemistry</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=34" title="Edit section: Chemistry">edit</a>]</div> <ul><li><a href="/wiki/Particle_size_distribution" class="mw-redirect" title="Particle size distribution">Particle size distributions</a> and <a href="/wiki/Molar_mass_distribution" title="Molar mass distribution">molar mass distributions</a>.</li> <li>The concentration of rare elements in minerals.<a href="#cite_note-73">[70]</a></li> <li>Diameters of crystals in ice cream, oil drops in mayonnaise, pores in cocoa press cake.<a href="#cite_note-:0-56">[53]</a></li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:FitLogNormDistr.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/FitLogNormDistr.tif/lossy-page1-220px-FitLogNormDistr.tif.jpg" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/FitLogNormDistr.tif/lossy-page1-330px-FitLogNormDistr.tif.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/FitLogNormDistr.tif/lossy-page1-440px-FitLogNormDistr.tif.jpg 2x" data-file-width="623" data-file-height="465" /></a><figcaption>Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see <a href="/wiki/Distribution_fitting" class="mw-redirect" title="Distribution fitting">distribution fitting</a> </figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Hydrology">Hydrology</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=35" title="Edit section: Hydrology">edit</a>]</div> <ul><li>In <a href="/wiki/Hydrology" title="Hydrology">hydrology</a>, the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.<a href="#cite_note-74">[71]</a></li></ul> <dl><dd><dl><dd>The image on the right, made with <a href="/wiki/CumFreq" title="CumFreq">CumFreq</a>, illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% <a href="/wiki/Confidence_belt" class="mw-redirect" title="Confidence belt">confidence belt</a> based on the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>.<a href="#cite_note-75">[72]</a></dd></dl></dd></dl> <dl><dd><dl><dd>The rainfall data are represented by <a href="/wiki/Plotting_position" class="mw-redirect" title="Plotting position">plotting positions</a> as part of a <a href="/wiki/Cumulative_frequency_analysis" title="Cumulative frequency analysis">cumulative frequency analysis</a>.</dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Social_sciences_and_demographics">Social sciences and demographics</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=36" title="Edit section: Social sciences and demographics">edit</a>]</div> <ul><li>In <a href="/wiki/Economics" title="Economics">economics</a>, there is evidence that the <a href="/wiki/Income" title="Income">income</a> of 97%–99% of the population is distributed log-normally.<a href="#cite_note-76">[73]</a> (The distribution of higher-income individuals follows a <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a>).<a href="#cite_note-77">[74]</a></li> <li>If an income distribution follows a log-normal distribution with standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">, then the <a href="/wiki/Gini_coefficient" title="Gini coefficient">Gini coefficient</a>, commonly use to evaluate income inequality, can be computed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\operatorname {erf} \left({\frac {\sigma }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>erf</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\operatorname {erf} \left({\frac {\sigma }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/050f58ceb781781fce54a6d12e8ea1389089e27f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.522ex; height:4.843ex;" alt="{\displaystyle G=\operatorname {erf} \left({\frac {\sigma }{2}}\right)}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>erf</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8ee8591a5cc9057c9c646d1d07b848c1d68e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.153ex; width:2.809ex; height:2.176ex;" alt="{\displaystyle \operatorname {erf} }"> is the <a href="/wiki/Error_function" title="Error function">error function</a>, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=2\Phi \left({\frac {\sigma }{\sqrt {2}}}\right)-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>2</mn> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=2\Phi \left({\frac {\sigma }{\sqrt {2}}}\right)-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2d9606af752a9a180d98dade729c0d7e2633df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.511ex; height:6.509ex;" alt="{\displaystyle G=2\Phi \left({\frac {\sigma }{\sqrt {2}}}\right)-1}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="{\displaystyle \Phi (x)}"> is the cumulative distribution function of a standard normal distribution.</li> <li>In <a href="/wiki/Finance" title="Finance">finance</a>, in particular the <a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes model</a>, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal<a href="#cite_note-78">[75]</a> (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a> have argued <a href="#cite_note-79">[76]</a> that <a href="/wiki/L%C3%A9vy_skew_alpha-stable_distribution" class="mw-redirect" title="Lévy skew alpha-stable distribution">log-Lévy distributions</a>, which possesses <a href="/wiki/Heavy_tails" class="mw-redirect" title="Heavy tails">heavy tails</a> would be a more appropriate model, in particular for the analysis for <a href="/wiki/Stock_market_crash" title="Stock market crash">stock market crashes</a>. Indeed, stock price distributions typically exhibit a <a href="/wiki/Fat_tail" class="mw-redirect" title="Fat tail">fat tail</a>.<a href="#cite_note-80">[77]</a> The fat tailed distribution of changes during stock market crashes invalidate the assumptions of the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>.</li> <li>In <a href="/wiki/Scientometrics" title="Scientometrics">scientometrics</a>, the number of citations to journal articles and patents follows a discrete log-normal distribution.<a href="#cite_note-81">[78]</a><a href="#cite_note-82">[79]</a></li> <li><a href="/wiki/Historical_urban_community_sizes" title="Historical urban community sizes">City sizes</a> (population) satisfy Gibrat's Law.<a href="#cite_note-83">[80]</a> The growth process of city sizes is proportionate and invariant with respect to size. From the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> therefore, the log of city size is normally distributed.</li> <li>The number of sexual partners appears to be best described by a log-normal distribution.<a href="#cite_note-84">[81]</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Technology">Technology</h3>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=37" title="Edit section: Technology">edit</a>]</div> <ul><li>In <a href="/wiki/Reliability_(statistics)" title="Reliability (statistics)">reliability</a> analysis, the log-normal distribution is often used to model times to repair a maintainable system.<a href="#cite_note-85">[82]</a></li> <li>In <a href="/wiki/Wireless_communication" class="mw-redirect" title="Wireless communication">wireless communication</a>, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution."<a href="#cite_note-86">[83]</a> Also, the random obstruction of radio signals due to large buildings and hills, called <a href="/wiki/Fading" title="Fading">shadowing</a>, is often modeled as a log-normal distribution.</li> <li>Particle size distributions produced by comminution with random impacts, such as in <a href="/wiki/Ball_mill" title="Ball mill">ball milling</a>.<a href="#cite_note-87">[84]</a></li> <li>The <a href="/wiki/File_size" title="File size">file size</a> distribution of publicly available audio and video data files (<a href="/wiki/MIME_types" class="mw-redirect" title="MIME types">MIME types</a>) follows a log-normal distribution over five <a href="/wiki/Orders_of_magnitude" class="mw-redirect" title="Orders of magnitude">orders of magnitude</a>.<a href="#cite_note-88">[85]</a></li> <li>File sizes of 140 million files on personal computers running the Windows OS, collected in 1999.<a href="#cite_note-89">[86]</a><a href="#cite_note-:3-58">[55]</a></li> <li>Sizes of text-based emails (1990s) and multimedia-based emails (2000s).<a href="#cite_note-:3-58">[55]</a></li> <li>In computer networks and <a href="/wiki/Internet_traffic" title="Internet traffic">Internet traffic</a> analysis, log-normal is shown as a good statistical model to represent the amount of traffic per unit time. This has been shown by applying a robust statistical approach on a large groups of real Internet traces. In this context, the log-normal distribution has shown a good performance in two main use cases: (1) predicting the proportion of time traffic will exceed a given level (for service level agreement or link capacity estimation) i.e. link dimensioning based on bandwidth provisioning and (2) predicting 95th percentile pricing.<a href="#cite_note-90">[87]</a></li> <li>in <a href="/wiki/Physical_test" title="Physical test">physical testing</a> when the test produces a time-to-failure of an item under specified conditions, the data is often best analyzed using a lognormal distribution.<a href="#cite_note-91">[88]</a><a href="#cite_note-92">[89]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=38" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">Heavy-tailed distribution</a></li> <li><a href="/wiki/Log-distance_path_loss_model" title="Log-distance path loss model">Log-distance path loss model</a></li> <li><a href="/wiki/Modified_lognormal_power-law_distribution" title="Modified lognormal power-law distribution">Modified lognormal power-law distribution</a></li> <li><a href="/wiki/Fading" title="Fading">Fading</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=39" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-46"><a href="#cite_ref-46">^</a> The Cox Method was quoted as “personal communication” in Land, 1971,<a href="#cite_note-44">[44]</a> and was also given in CitationZhou and Gao (1997)<a href="#cite_note-45">[45]</a> and Olsson 2005<a href="#cite_note-Olsson2005-42">[42]</a>: Section 3.3  </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> The issue is that we don't know how to do it directly, so we take their logs, and then use the <a href="/wiki/Delta_method" title="Delta method">delta method</a> to say that their logs is itself (approximately) normal. This trick allows us to pretend that their exp was log normal, and use that approximation to build the CI. Notice that in the RR case, the median and the mean in the base distribution (i.e., before taking the log), is actually identical (since they are originally normal, and not log normal). E.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}_{1}{\dot {\sim }}N(p_{1},p_{1}(1-p1)/n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∼</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}_{1}{\dot {\sim }}N(p_{1},p_{1}(1-p1)/n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b14657ce864277fc63815a7090e07894135db08b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:24.367ex; height:2.843ex;" alt="{\displaystyle {\hat {p}}_{1}{\dot {\sim }}N(p_{1},p_{1}(1-p1)/n)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle log({\hat {p}}_{1}){\dot {\sim }}N(log(p_{1}),(1-p1)/(p_{1}*n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∼</mo> <mo>˙</mo> </mover> </mrow> </mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∗</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle log({\hat {p}}_{1}){\dot {\sim }}N(log(p_{1}),(1-p1)/(p_{1}*n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f153b8b496f417ace56a7600b861123b5527b812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.774ex; height:2.843ex;" alt="{\displaystyle log({\hat {p}}_{1}){\dot {\sim }}N(log(p_{1}),(1-p1)/(p_{1}*n))}"> Hence, building a CI based on the log and than back-transform will give us <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CI(p_{1}):e^{log({\hat {p}}_{1})\pm (1-{\hat {p}}_{1})/({\hat {p}}_{1}*n))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>±</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∗</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CI(p_{1}):e^{log({\hat {p}}_{1})\pm (1-{\hat {p}}_{1})/({\hat {p}}_{1}*n))}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb97ee4422c2b668b3b7026759d9257508ddfa9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.336ex; height:3.509ex;" alt="{\displaystyle CI(p_{1}):e^{log({\hat {p}}_{1})\pm (1-{\hat {p}}_{1})/({\hat {p}}_{1}*n))}}">. So while we expect the CI to be for the median, in this case, it's actually also for the mean in the original distribution. i.e., if the original <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39907b12d79c0fcf7638de78a9e5fab428551ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:2.503ex; height:2.676ex;" alt="{\displaystyle {\hat {p}}_{1}}"> was log-normal, we'd expect that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[{\hat {p}}_{1}]=e^{log(p_{1})+1/2*(1-p1)/(p_{1}*n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>∗</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∗</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[{\hat {p}}_{1}]=e^{log(p_{1})+1/2*(1-p1)/(p_{1}*n)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3f7318a7ef62498010b80b435b5793839fd85a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.075ex; height:3.343ex;" alt="{\displaystyle E[{\hat {p}}_{1}]=e^{log(p_{1})+1/2*(1-p1)/(p_{1}*n)}}">. But in practice, we KNOW that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[{\hat {p}}_{1}]=e^{log(p_{1})}=p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[{\hat {p}}_{1}]=e^{log(p_{1})}=p_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea7b95a1e85e8f3a89f40a021349b461859f70e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.234ex; height:3.343ex;" alt="{\displaystyle E[{\hat {p}}_{1}]=e^{log(p_{1})}=p_{1}}">. Hence, the approximation we have is in the second step (of the delta method), but the CI are actually for the expectation (not just the median). This is because we are starting from a base distribution that is normal, and then using another approximation after the log again to normal. This means that a big approximation part of the CI is from the delta method. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> The bias can be partially minimized by using: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\left[{\frac {E(X_{1})}{E(X_{2})}}\right]}}=\left[{\frac {{\widehat {E}}(X_{1})}{{\widehat {E}}(X_{2})}}\right]{\frac {2}{\widehat {\left({\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}}}=\left[e^{({\widehat {\mu }}_{1}-{\widehat {\mu }}_{2})+{\frac {1}{2}}(S_{1}^{2}-S_{2}^{2})}\right]{\frac {2}{{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>^</mo> </mover> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </msup> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\left[{\frac {E(X_{1})}{E(X_{2})}}\right]}}=\left[{\frac {{\widehat {E}}(X_{1})}{{\widehat {E}}(X_{2})}}\right]{\frac {2}{\widehat {\left({\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}}}=\left[e^{({\widehat {\mu }}_{1}-{\widehat {\mu }}_{2})+{\frac {1}{2}}(S_{1}^{2}-S_{2}^{2})}\right]{\frac {2}{{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c22f1c8adacddf8c56c5a1757c0fe498308472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:106.471ex; height:11.509ex;" alt="{\displaystyle {\widehat {\left[{\frac {E(X_{1})}{E(X_{2})}}\right]}}=\left[{\frac {{\widehat {E}}(X_{1})}{{\widehat {E}}(X_{2})}}\right]{\frac {2}{\widehat {\left({\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}+{\frac {\sigma _{1}^{4}}{2(n_{1}-1)}}+{\frac {\sigma _{2}^{4}}{2(n_{2}-1)}}\right)}}}=\left[e^{({\widehat {\mu }}_{1}-{\widehat {\mu }}_{2})+{\frac {1}{2}}(S_{1}^{2}-S_{2}^{2})}\right]{\frac {2}{{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}}"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=40" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-norton-1"><a href="#cite_ref-norton_1-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output 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ACM SIGMETRICS Performance Evaluation Review. 27 (1): 59–70. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F301464.301480">10.1145/301464.301480</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0163-5999">0163-5999</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGMETRICS+Performance+Evaluation+Review&rft.atitle=A+large-scale+study+of+file-system+contents&rft.volume=27&rft.issue=1&rft.pages=59-70&rft.date=1999-05-01&rft_id=info%3Adoi%2F10.1145%2F301464.301480&rft.issn=0163-5999&rft.aulast=Douceur&rft.aufirst=John+R.&rft.au=Bolosky%2C+William+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F301464.301480&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALog-normal+distribution" class="Z3988"> </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlamsarParisisCleggZakhleniuk2019" class="citation arxiv cs1">Alamsar, Mohammed; Parisis, George; Clegg, Richard; Zakhleniuk, Nickolay (2019). "On the Distribution of Traffic Volumes in the Internet and its Implications". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1902.03853">1902.03853</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.NI">cs.NI</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=On+the+Distribution+of+Traffic+Volumes+in+the+Internet+and+its+Implications&rft.date=2019&rft_id=info%3Aarxiv%2F1902.03853&rft.aulast=Alamsar&rft.aufirst=Mohammed&rft.au=Parisis%2C+George&rft.au=Clegg%2C+Richard&rft.au=Zakhleniuk%2C+Nickolay&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALog-normal+distribution" class="Z3988"> </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> ASTM D3654, Standard Test Method for Shear Adhesion on Pressure-Sensitive Tapesw </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> ASTM D4577, Standard Test Method for Compression Resistance of a container Under Constant Load>\ </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=41" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrowShimizu1988" class="citation cs2">Crow, Edwin L.; Shimizu, Kunio, eds. (1988), Lognormal Distributions, Theory and Applications, Statistics: Textbooks and Monographs, vol. 88, New York: Marcel Dekker, Inc., pp. xvi+387, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-7803-3" title="Special:BookSources/978-0-8247-7803-3"><bdi>978-0-8247-7803-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0939191">0939191</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0644.62014">0644.62014</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lognormal+Distributions%2C+Theory+and+Applications&rft.place=New+York&rft.series=Statistics%3A+Textbooks+and+Monographs&rft.pages=xvi%2B387&rft.pub=Marcel+Dekker%2C+Inc.&rft.date=1988&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0644.62014%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0939191%23id-name%3DMR&rft.isbn=978-0-8247-7803-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALog-normal+distribution" class="Z3988"></li> <li>Aitchison, J. and Brown, J.A.C. (1957) The Lognormal Distribution, Cambridge University Press.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLimpertStahelAbbt2001" class="citation journal cs1">Limpert, E; Stahel, W; Abbt, M (2001). <a rel="nofollow" class="external text" href="https://doi.org/10.1641%2F0006-3568%282001%29051%5B0341%3ALNDATS%5D2.0.CO%3B2">"Lognormal distributions across the sciences: keys and clues"</a>. BioScience. 51 (5): 341–352. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1641%2F0006-3568%282001%29051%5B0341%3ALNDATS%5D2.0.CO%3B2">10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BioScience&rft.atitle=Lognormal+distributions+across+the+sciences%3A+keys+and+clues&rft.volume=51&rft.issue=5&rft.pages=341-352&rft.date=2001&rft_id=info%3Adoi%2F10.1641%2F0006-3568%282001%29051%5B0341%3ALNDATS%5D2.0.CO%3B2&rft.aulast=Limpert&rft.aufirst=E&rft.au=Stahel%2C+W&rft.au=Abbt%2C+M&rft_id=https%3A%2F%2Fdoi.org%2F10.1641%252F0006-3568%25282001%2529051%255B0341%253ALNDATS%255D2.0.CO%253B2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALog-normal+distribution" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolgate1989" class="citation journal cs1">Holgate, P. (1989). "The lognormal characteristic function". Communications in Statistics - Theory and Methods. 18 (12): 4539–4548. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610928908830173">10.1080/03610928908830173</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Statistics+-+Theory+and+Methods&rft.atitle=The+lognormal+characteristic+function&rft.volume=18&rft.issue=12&rft.pages=4539-4548&rft.date=1989&rft_id=info%3Adoi%2F10.1080%2F03610928908830173&rft.aulast=Holgate&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALog-normal+distribution" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrooksCorsonDonal1994" class="citation journal cs1">Brooks, Robert; Corson, Jon; <a href="/wiki/Jimbo_Wales" class="mw-redirect" title="Jimbo Wales">Donal, Wales</a> (1994). "The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion". Advances in Futures and Options Research. 7. <a href="/wiki/SSRN_(identifier)" class="mw-redirect" title="SSRN (identifier)">SSRN</a> <a rel="nofollow" class="external text" href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5735">5735</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Futures+and+Options+Research&rft.atitle=The+Pricing+of+Index+Options+When+the+Underlying+Assets+All+Follow+a+Lognormal+Diffusion&rft.volume=7&rft.date=1994&rft_id=https%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D5735%23id-name%3DSSRN&rft.aulast=Brooks&rft.aufirst=Robert&rft.au=Corson%2C+Jon&rft.au=Donal%2C+Wales&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALog-normal+distribution" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Log-normal_distribution&action=edit&section=42" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output 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talk:Probability distributions"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Probability_distributions_(list)" style="font-size:114%;margin:0 4em"><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distributions</a> (<a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Discrete univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">with finite support</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford</a></li> <li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li> <li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">Beta-binomial</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial</a></li> <li><a href="/wiki/Categorical_distribution" title="Categorical distribution">Categorical</a></li> <li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric</a> <ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">Negative</a></li></ul></li> <li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li> <li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li> <li><a href="/wiki/Soliton_distribution" title="Soliton distribution">Soliton</a></li> <li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">Discrete uniform</a></li> <li><a href="/wiki/Zipf%27s_law" title="Zipf's law">Zipf</a></li> <li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="Zipf–Mandelbrot law">Zipf–Mandelbrot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">Beta negative binomial</a></li> <li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li> <li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="Conway–Maxwell–Poisson distribution">Conway–Maxwell–Poisson</a></li> <li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">Discrete phase-type</a></li> <li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li> <li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">Extended negative binomial</a></li> <li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="Flory–Schulz distribution">Flory–Schulz</a></li> <li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="Gauss–Kuzmin distribution">Gauss–Kuzmin</a></li> <li><a href="/wiki/Geometric_distribution" title="Geometric distribution">Geometric</a></li> <li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">Logarithmic</a></li> <li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">Mixed Poisson</a></li> <li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">Negative binomial</a></li> <li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li> <li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">Parabolic fractal</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li> <li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li> <li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon</a></li> <li><a href="/wiki/Zeta_distribution" title="Zeta distribution">Zeta</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">Arcsine</a></li> <li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li> <li><a href="/wiki/Balding%E2%80%93Nichols_model" title="Balding–Nichols model">Balding–Nichols</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li> <li><a href="/wiki/Beta_distribution" title="Beta distribution">Beta</a> <ul><li><a href="/wiki/Generalized_beta_distribution" title="Generalized beta distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">Beta rectangular</a></li> <li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">Continuous Bernoulli</a></li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall</a></li> <li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li> <li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">Logit-normal</a></li> <li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">Noncentral beta</a></li> <li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li> <li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">Raised cosine</a></li> <li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">Reciprocal</a></li> <li><a href="/wiki/Triangular_distribution" title="Triangular distribution">Triangular</a></li> <li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li> <li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">Uniform</a></li> <li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a semi-infinite interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li> <li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li> <li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li> <li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">Beta prime</a></li> <li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li> <li><a href="/wiki/Chi_distribution" title="Chi distribution">Chi</a></li> <li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">Chi-squared</a> <ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">Noncentral</a></li> <li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">Inverse</a> <ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">Scaled</a></li></ul></li></ul></li> <li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li> <li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li> <li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a> <ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">Hyper</a></li></ul></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential</a> <ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">Hyperexponential</a></li> <li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">Hypoexponential</a></li> <li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">Logarithmic</a></li></ul></li> <li><a href="/wiki/F-distribution" title="F-distribution">F</a> <ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">Noncentral</a></li></ul></li> <li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">Folded normal</a></li> <li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li> <li><a href="/wiki/Gamma_distribution" title="Gamma distribution">Gamma</a> <ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">Generalized</a></li> <li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li> <li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a> <ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">Shifted</a></li></ul></li> <li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">Half-logistic</a></li> <li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">Half-normal</a></li> <li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling's T-squared distribution">Hotelling's T-squared</a></li> <li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">Inverse Gaussian</a> <ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov</a></li> <li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li> <li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">Log-Cauchy</a></li> <li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">Log-Laplace</a></li> <li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">Log-logistic</a></li> <li><a class="mw-selflink selflink">Log-normal</a></li> <li><a href="/wiki/Log-t_distribution" title="Log-t distribution">Log-t</a></li> <li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li> <li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">Matrix-exponential</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="Maxwell–Jüttner distribution">Maxwell–Jüttner</a></li> <li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li> <li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li> <li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">Phase-type</a></li> <li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li> <li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li> <li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic Breit–Wigner distribution">Relativistic Breit–Wigner</a></li> <li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li> <li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">Truncated normal</a></li> <li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li> <li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a> <ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">Discrete</a></li></ul></li> <li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks's lambda distribution">Wilks's lambda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on the whole real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li> <li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">Exponential power</a></li> <li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher's z-distribution">Fisher's z</a></li> <li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li> <li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian q</a></li> <li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">Generalized normal</a></li> <li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">Generalized hyperbolic</a></li> <li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">Geometric stable</a></li> <li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li> <li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li> <li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">Hyperbolic secant</a></li> <li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson's SU-distribution">Johnson's SU</a></li> <li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li> <li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a> <ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">Asymmetric</a></li></ul></li> <li><a href="/wiki/Logistic_distribution" title="Logistic distribution">Logistic</a></li> <li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">Noncentral t</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal (Gaussian)</a></li> <li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">Normal-inverse Gaussian</a></li> <li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">Skew normal</a></li> <li><a href="/wiki/Slash_distribution" title="Slash distribution">Slash</a></li> <li><a href="/wiki/Stable_distribution" title="Stable distribution">Stable</a></li> <li><a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t</a></li> <li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="Tracy–Widom distribution">Tracy–Widom</a></li> <li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">Variance-gamma</a></li> <li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">Generalized chi-squared</a></li> <li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">Generalized extreme value</a></li> <li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">Generalized Pareto</a></li> <li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="Marchenko–Pastur distribution">Marchenko–Pastur</a></li> <li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis κ-exponential</a></li> <li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis κ-Gamma</a></li> <li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis κ-Weibull</a></li> <li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis κ-Logistic</a></li> <li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis κ-Erlang</a></li> <li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution">q-exponential</a></li> <li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution">q-Gaussian</a></li> <li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution">q-Weibull</a></li> <li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">Shifted log-logistic</a></li> <li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous- discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate (joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Discrete: </li> <li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens's sampling formula">Ewens</a></li> <li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">Multinomial</a> <ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li> <li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">Negative</a></li></ul></li> <li>Continuous: </li> <li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a> <ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">Generalized</a></li></ul></li> <li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">Multivariate Laplace</a></li> <li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Multivariate normal</a></li> <li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">Multivariate stable</a></li> <li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">Multivariate t</a></li> <li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">Normal-gamma</a> <ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></li> <li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li> <li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal</a></li> <li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">Matrix t</a></li> <li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">Matrix gamma</a> <ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">Inverse</a></li></ul></li> <li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a> <ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">Normal</a></li> <li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">Inverse</a></li> <li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">Normal-inverse</a></li> <li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">Complex</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></dt> <dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd> <dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">Univariate von Mises</a></dd> <dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal</a></dd> <dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">Wrapped Cauchy</a></dd> <dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">Wrapped exponential</a></dd> <dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">Wrapped asymmetric Laplace</a></dd> <dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">Wrapped Lévy</a></dd> <dt>Bivariate (spherical)</dt> <dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd> <dt>Bivariate (toroidal)</dt> <dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">Bivariate von Mises</a></dd> <dt>Multivariate</dt> <dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von Mises–Fisher distribution">von Mises–Fisher</a></dd> <dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt>Degenerate</dt> <dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd> <dt>Singular</dt> <dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li> <li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">Compound Poisson</a></li> <li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential</a></li> <li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">Natural exponential</a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale</a></li> <li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">Maximum entropy</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture</a></li> <li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li> <li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">Wrapped</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /> <a href="https://commons.wikimedia.org/wiki/Category:Probability_distributions" class="extiw" title="commons:Category:Probability distributions">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <a href="https://www.wikidata.org/wiki/Q826116#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></th><td class="navbox-list-with-group navbox-list navbox-odd" 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Log-normal distribution - Wikipedia