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href="#Techniques_of_fitting"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Techniques of fitting</span> </div> </a> <ul id="toc-Techniques_of_fitting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_of_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalization_of_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Generalization of distributions</span> </div> </a> <ul id="toc-Generalization_of_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inversion_of_skewness" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Inversion_of_skewness"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Inversion of skewness</span> </div> </a> <ul id="toc-Inversion_of_skewness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shifting_of_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Shifting_of_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Shifting of distributions</span> </div> </a> <ul id="toc-Shifting_of_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Composite_distributions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Composite_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Composite distributions</span> </div> </a> <ul id="toc-Composite_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uncertainty_of_prediction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uncertainty_of_prediction"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Uncertainty of prediction</span> </div> </a> <button aria-controls="toc-Uncertainty_of_prediction-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Uncertainty of prediction subsection</span> </button> <ul id="toc-Uncertainty_of_prediction-sublist" class="vector-toc-list"> <li id="toc-Variance_of_Bayesian_fitted_probability_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variance_of_Bayesian_fitted_probability_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Variance of Bayesian fitted probability functions</span> </div> </a> <ul id="toc-Variance_of_Bayesian_fitted_probability_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Goodness_of_fit" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Goodness_of_fit"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Goodness of fit</span> </div> </a> <ul id="toc-Goodness_of_fit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Histogram_and_density_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Histogram_and_density_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Histogram and density function</span> </div> </a> <ul id="toc-Histogram_and_density_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-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 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Distribution_fitting&redirect=no" class="mw-redirect" title="Distribution fitting">Distribution fitting</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical concept</div> <p> <b>Probability distribution fitting</b> or simply <b>distribution fitting</b> is the fitting of a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> to a series of data concerning the repeated measurement of a variable phenomenon. The aim of distribution fitting is to <a href="/wiki/Prediction" title="Prediction">predict</a> the <a href="/wiki/Probability" title="Probability">probability</a> or to <a href="/wiki/Forecasting" title="Forecasting">forecast</a> the <a href="/wiki/Frequency_(statistics)" title="Frequency (statistics)">frequency</a> of occurrence of the magnitude of the phenomenon in a certain interval. </p><p>There are many probability distributions (see <a href="/wiki/List_of_probability_distributions" title="List of probability distributions">list of probability distributions</a>) of which some can be fitted more closely to the observed frequency of the data than others, depending on the characteristics of the phenomenon and of the distribution. The distribution giving a close fit is supposed to lead to good predictions. In distribution fitting, therefore, one needs to select a distribution that suits the data well. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Selection_of_distribution">Selection of distribution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=1" title="Edit section: Selection of distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Normal_Distribution_PDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/220px-Normal_Distribution_PDF.svg.png" decoding="async" width="220" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/330px-Normal_Distribution_PDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/440px-Normal_Distribution_PDF.svg.png 2x" data-file-width="720" data-file-height="460" /></a><figcaption>Different shapes of the symmetrical normal distribution depending on mean <i>μ</i> and variance <i>σ</i><sup> 2</sup></figcaption></figure> <p>The selection of the appropriate distribution depends on the presence or absence of symmetry of the data set with respect to the <a href="/wiki/Central_tendency" title="Central tendency">central tendency</a>. </p><p><i>Symmetrical distributions</i> </p><p>When the data are symmetrically distributed around the mean while the frequency of occurrence of data farther away from the mean diminishes, one may for example select the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, the <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic distribution</a>, or the <a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution">Student's t-distribution</a>. The first two are very similar, while the last, with one degree of freedom, has "heavier tails" meaning that the values farther away from the mean occur relatively more often (i.e. the <a href="/wiki/Kurtosis" title="Kurtosis">kurtosis</a> is higher). The <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a> is also symmetric. </p><p><i>Skew distributions to the right</i> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Negative_and_positive_skew_diagrams_(English).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Negative_and_positive_skew_diagrams_%28English%29.svg/220px-Negative_and_positive_skew_diagrams_%28English%29.svg.png" decoding="async" width="220" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Negative_and_positive_skew_diagrams_%28English%29.svg/330px-Negative_and_positive_skew_diagrams_%28English%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Negative_and_positive_skew_diagrams_%28English%29.svg/440px-Negative_and_positive_skew_diagrams_%28English%29.svg.png 2x" data-file-width="446" data-file-height="159" /></a><figcaption>Skewness to left and right</figcaption></figure> <p>When the larger values tend to be farther away from the mean than the smaller values, one has a skew distribution to the right (i.e. there is positive <a href="/wiki/Skewness" title="Skewness">skewness</a>), one may for example select the <a href="/wiki/Lognormal_distribution" class="mw-redirect" title="Lognormal distribution">log-normal distribution</a> (i.e. the log values of the data are <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a>), the <a href="/wiki/Loglogistic_distribution" class="mw-redirect" title="Loglogistic distribution">log-logistic distribution</a> (i.e. the log values of the data follow a <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic distribution</a>), the <a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel distribution</a>, the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>, the <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a>, the <a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull distribution</a>, the <a href="/wiki/Burr_distribution" title="Burr distribution">Burr distribution</a>, or the <a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet distribution</a>. The last four distributions are bounded to the left. </p><p><i>Skew distributions to the left</i> </p><p>When the smaller values tend to be farther away from the mean than the larger values, one has a skew distribution to the left (i.e. there is negative skewness), one may for example select the <i>square-normal distribution</i> (i.e. the normal distribution applied to the square of the data values),<sup id="cite_ref-skew_1-0" class="reference"><a href="#cite_note-skew-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the inverted (mirrored) Gumbel distribution,<sup id="cite_ref-skew_1-1" class="reference"><a href="#cite_note-skew-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum distribution</a> (mirrored Burr distribution), or the <a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz distribution</a>, which is bounded to the left. </p> <div class="mw-heading mw-heading2"><h2 id="Techniques_of_fitting">Techniques of fitting</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=2" title="Edit section: Techniques of fitting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following techniques of distribution fitting exist:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ul><li><i>Parametric methods</i>, by which the <a href="/wiki/Parameter" title="Parameter">parameters</a> of the distribution are calculated from the data series.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The parametric methods are: <ul><li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/Maximum_spacing_estimation" title="Maximum spacing estimation">Maximum spacing estimation</a></li> <li>Method of <a href="/wiki/L-moment" title="L-moment">L-moments</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a> method<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul></li></ul> <dl><dd><dl><dd><table class="wikitable"> <tbody><tr> <td bgcolor="white"><i>For example, the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> (the</i> <i><a href="/wiki/Expected_value" title="Expected value">expectation</a>) can be estimated by the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">mean</a> of the data and the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span> (the <a href="/wiki/Variance" title="Variance">variance</a>) can be estimated from the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> of the data. The mean is found as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle m=\sum {X}/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle m=\sum {X}/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7beb4c5d224ab35994000036414a2adc7812cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.517ex; height:2.843ex;" alt="{\textstyle m=\sum {X}/n}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is the data value and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> the number of data, while the standard deviation is calculated as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s={\sqrt {{\frac {1}{n-1}}\sum {(X-m)^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mi>m</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s={\sqrt {{\frac {1}{n-1}}\sum {(X-m)^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e3288e5a09aa5d058d1a49c26aacaa5372a430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.388ex; height:4.843ex;" alt="{\textstyle s={\sqrt {{\frac {1}{n-1}}\sum {(X-m)^{2}}}}}"></span>. With these parameters many distributions, e.g. the normal distribution, are completely defined.</i> </td></tr></tbody></table></dd></dl></dd></dl> <figure typeof="mw:File/Thumb"><a href="/wiki/File:FitGumbelDistr.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/FitGumbelDistr.tif/lossless-page1-220px-FitGumbelDistr.tif.png" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/FitGumbelDistr.tif/lossless-page1-330px-FitGumbelDistr.tif.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/FitGumbelDistr.tif/lossless-page1-440px-FitGumbelDistr.tif.png 2x" data-file-width="623" data-file-height="465" /></a><figcaption>Cumulative Gumbel distribution fitted to maximum one-day October rainfalls in <a href="/wiki/Suriname" title="Suriname">Suriname</a> by the regression method with added <b><a href="/wiki/Confidence_band" class="mw-redirect" title="Confidence band">confidence band</a></b> using <a href="/wiki/CumFreq" title="CumFreq">cumfreq</a> </figcaption></figure> <ul><li><a href="/wiki/Plotting_position" class="mw-redirect" title="Plotting position">Plotting position</a> plus <a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a>, using a transformation of the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> so that a <a href="/wiki/Linear_relation" title="Linear relation">linear relation</a> is found between the <a href="/wiki/Cumulative_probability" class="mw-redirect" title="Cumulative probability">cumulative probability</a> and the values of the data, which may also need to be transformed, depending on the selected probability distribution. In this method the cumulative probability needs to be estimated by the <a href="/wiki/Plotting_position" class="mw-redirect" title="Plotting position">plotting position</a><sup id="cite_ref-gen_6-0" class="reference"><a href="#cite_note-gen-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd><dl><dd><table class="wikitable"> <tbody><tr> <td bgcolor="white">For example, the cumulative <a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel distribution</a> can be linearized to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=aX+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>a</mi> <mi>X</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=aX+b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15aad86ca14a646d56156b130debb5d815d0e383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.92ex; height:2.343ex;" alt="{\displaystyle Y=aX+b}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is the data variable and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=-\ln(-\ln P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=-\ln(-\ln P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95b2a658c48f11b8dcd36ec95168f2ea55b4c78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.083ex; height:2.843ex;" alt="{\displaystyle Y=-\ln(-\ln P)}"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> being the cumulative probability, i.e. the probability that the data value is less than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>. Thus, using the <a href="/wiki/Plotting_position" class="mw-redirect" title="Plotting position">plotting position</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, one finds the parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> from a linear regression of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, and the Gumbel distribution is fully defined. </td></tr></tbody></table></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Generalization_of_distributions">Generalization of distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=3" title="Edit section: Generalization of distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is customary to transform data logarithmically to fit symmetrical distributions (like the <a href="/wiki/Normal_distribution" title="Normal distribution">normal</a> and <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic</a>) to data obeying a distribution that is positively skewed (i.e. skew to the right, with <a href="/wiki/Mean" title="Mean">mean</a> > <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a>, and with a right hand tail that is longer than the left hand tail), see <a href="/wiki/Lognormal_distribution" class="mw-redirect" title="Lognormal distribution">lognormal distribution</a> and the <a href="/wiki/Loglogistic_distribution" class="mw-redirect" title="Loglogistic distribution">loglogistic distribution</a>. A similar effect can be achieved by taking the square root of the data. </p><p>To fit a symmetrical distribution to data obeying a negatively skewed distribution (i.e. skewed to the left, with <a href="/wiki/Mean" title="Mean">mean</a> < <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a>, and with a right hand tail this is shorter than the left hand tail) one could use the squared values of the data to accomplish the fit. </p><p>More generally one can raise the data to a power <i>p</i> in order to fit symmetrical distributions to data obeying a distribution of any skewness, whereby <i>p</i> < 1 when the skewness is positive and <i>p</i> > 1 when the skewness is negative. The optimal value of <i>p</i> is to be found by a <a href="/wiki/Numerical_method" title="Numerical method">numerical method</a>. The numerical method may consist of assuming a range of <i>p</i> values, then applying the distribution fitting procedure repeatedly for all the assumed <i>p</i> values, and finally selecting the value of <i>p</i> for which the sum of squares of deviations of calculated probabilities from measured frequencies (<a href="/wiki/Chi-squared_test" title="Chi-squared test">chi squared</a>) is minimum, as is done in <a href="/wiki/CumFreq" title="CumFreq">CumFreq</a>. </p><p>The generalization enhances the flexibility of probability distributions and increases their applicability in distribution fitting.<sup id="cite_ref-gen_6-1" class="reference"><a href="#cite_note-gen-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The versatility of generalization makes it possible, for example, to fit approximately normally distributed data sets to a large number of different probability distributions,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> while negatively skewed distributions can be fitted to square normal and mirrored Gumbel distributions.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Inversion_of_skewness">Inversion of skewness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=4" title="Edit section: Inversion of skewness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gumbel_distribution_and_Gumbel_mirrored.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Gumbel_distribution_and_Gumbel_mirrored.png/220px-Gumbel_distribution_and_Gumbel_mirrored.png" decoding="async" width="220" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Gumbel_distribution_and_Gumbel_mirrored.png/330px-Gumbel_distribution_and_Gumbel_mirrored.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Gumbel_distribution_and_Gumbel_mirrored.png/440px-Gumbel_distribution_and_Gumbel_mirrored.png 2x" data-file-width="467" data-file-height="199" /></a><figcaption>(A) Gumbel probability distribution skew to right and (B) Gumbel mirrored skew to left</figcaption></figure> <p>Skewed distributions can be inverted (or mirrored) by replacing in the mathematical expression of the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (F) by its complement: F'=1-F, obtaining the <a href="/wiki/Cumulative_distribution_function#Complementary_cumulative_distribution_function_(tail_distribution)" title="Cumulative distribution function">complementary distribution function</a> (also called <a href="/wiki/Survival_function" title="Survival function">survival function</a>) that gives a mirror image. In this manner, a distribution that is skewed to the right is transformed into a distribution that is skewed to the left and vice versa. </p> <dl><dd><dl><dd><table class="wikitable"> <tbody><tr> <td bgcolor="white"><i>Example</i>. The F-expression of the positively skewed <a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel distribution</a> is: F=exp[-exp{-(<i>X</i>-<i>u</i>)/0.78<i>s</i>}], where <i>u</i> is the <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a> (i.e. the value occurring most frequently) and <i>s</i> is the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>. The Gumbel distribution can be transformed using F'=1-exp[-exp{-(<i>x</i>-<i>u</i>)/0.78<i>s</i>}] . This transformation yields the inverse, mirrored, or complementary Gumbel distribution that may fit a data series obeying a negatively skewed distribution. </td></tr></tbody></table></dd></dl></dd></dl> <p>The technique of skewness inversion increases the number of probability distributions available for distribution fitting and enlarges the distribution fitting opportunities. </p> <div class="mw-heading mw-heading2"><h2 id="Shifting_of_distributions">Shifting of distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=5" title="Edit section: Shifting of distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some probability distributions, like the <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential</a>, do not support negative data values (<i>X</i>). Yet, when negative data are present, such distributions can still be used replacing <i>X</i> by <i>Y</i>=<i>X</i>-<i>Xm</i>, where <i>Xm</i> is the minimum value of <i>X</i>. This replacement represents a shift of the probability distribution in positive direction, i.e. to the right, because <i>Xm</i> is negative. After completing the distribution fitting of <i>Y</i>, the corresponding <i>X</i>-values are found from <i>X</i>=<i>Y</i>+<i>Xm</i>, which represents a back-shift of the distribution in negative direction, i.e. to the left.<br /> The technique of distribution shifting augments the chance to find a properly fitting probability distribution. </p> <div class="mw-heading mw-heading2"><h2 id="Composite_distributions">Composite distributions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=6" title="Edit section: Composite distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:SanLor.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/SanLor.jpg/220px-SanLor.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/SanLor.jpg/330px-SanLor.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/SanLor.jpg/440px-SanLor.jpg 2x" data-file-width="640" data-file-height="480" /></a><figcaption>Composite (discontinuous) distribution with confidence belt<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <p>The option exists to use two different probability distributions, one for the lower data range, and one for the higher like for example the <a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace distribution</a>. The ranges are separated by a break-point. The use of such composite (discontinuous) probability distributions can be opportune when the data of the phenomenon studied were obtained under two sets different conditions.<sup id="cite_ref-gen_6-2" class="reference"><a href="#cite_note-gen-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Uncertainty_of_prediction">Uncertainty of prediction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=7" title="Edit section: Uncertainty of prediction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:BinomialConfBelts.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/BinomialConfBelts.jpg/220px-BinomialConfBelts.jpg" decoding="async" width="220" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/BinomialConfBelts.jpg/330px-BinomialConfBelts.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/BinomialConfBelts.jpg/440px-BinomialConfBelts.jpg 2x" data-file-width="2364" data-file-height="2580" /></a><figcaption><small>Uncertainty analysis with confidence belts using the binomial distribution</small><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Predictions of occurrence based on fitted probability distributions are subject to <a href="/wiki/Uncertainty" title="Uncertainty">uncertainty</a>, which arises from the following conditions: </p> <ul><li>The true probability distribution of events may deviate from the fitted distribution, as the observed data series may not be totally representative of the real probability of occurrence of the phenomenon due to <a href="/wiki/Random_error" class="mw-redirect" title="Random error">random error</a></li> <li>The occurrence of events in another situation or in the future may deviate from the fitted distribution as this occurrence can also be subject to random error</li> <li>A change of environmental conditions may cause a change in the probability of occurrence of the phenomenon</li></ul> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:SampleFreqCurves.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/SampleFreqCurves.tif/lossy-page1-220px-SampleFreqCurves.tif.jpg" decoding="async" width="220" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/SampleFreqCurves.tif/lossy-page1-330px-SampleFreqCurves.tif.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/SampleFreqCurves.tif/lossy-page1-440px-SampleFreqCurves.tif.jpg 2x" data-file-width="690" data-file-height="562" /></a><figcaption>Variations of nine <i><a href="/wiki/Return_period" title="Return period">return period</a></i> curves of 50-year samples from a theoretical 1000 year record (base line), data from Benson<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>An estimate of the uncertainty in the first and second case can be obtained with the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial probability distribution</a> using for example the probability of exceedance <i>Pe</i> (i.e. the chance that the event <i>X</i> is larger than a reference value <i>Xr</i> of <i>X</i>) and the probability of non-exceedance <i>Pn</i> (i.e. the chance that the event <i>X</i> is smaller than or equal to the reference value <i>Xr</i>, this is also called <a href="/wiki/Cumulative_probability" class="mw-redirect" title="Cumulative probability">cumulative probability</a>). In this case there are only two possibilities: either there is exceedance or there is non-exceedance. This duality is the reason that the binomial distribution is applicable. </p><p>With the binomial distribution one can obtain a <a href="/wiki/Prediction_interval" title="Prediction interval">prediction interval</a>. Such an interval also estimates the risk of failure, i.e. the chance that the predicted event still remains outside the confidence interval. The confidence or risk analysis may include the <a href="/wiki/Return_period" title="Return period">return period</a> <i>T=1/Pe</i> as is done in <a href="/wiki/Hydrology" title="Hydrology">hydrology</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Variance_of_Bayesian_fitted_probability_functions"><a href="/wiki/Variance" title="Variance">Variance</a> of <a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian</a> fitted probability functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=8" title="Edit section: Variance of Bayesian fitted probability functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A Bayesian approach can be used for fitting a model <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x|\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x|\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6951d50bc9f0e043149cecd0c15c92bd6c1d9273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.622ex; height:2.843ex;" alt="{\displaystyle P(x|\theta )}"></span> having a prior distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/543ecac78af032493892f6608bff05542e961edb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.645ex; height:2.843ex;" alt="{\displaystyle P(\theta )}"></span> for the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>. When one has samples <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> that are independently drawn from the underlying distribution then one can derive the so-called posterior distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\theta |X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\theta |X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f089aa91a620a2dcbd00e3d9f5192f059384d371" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.272ex; height:2.843ex;" alt="{\displaystyle P(\theta |X)}"></span>. This posterior can be used to update the probability mass function for a new sample <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> given the observations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, one obtains </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\theta }(x|X):=\int d\theta \ P(x|\theta )\ P(\theta |X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo>∫</mo> <mi>d</mi> <mi>θ</mi> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\theta }(x|X):=\int d\theta \ P(x|\theta )\ P(\theta |X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bff8a6fc4d8ac9c24ae883cb283e1301f932afbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.949ex; height:5.676ex;" alt="{\displaystyle P_{\theta }(x|X):=\int d\theta \ P(x|\theta )\ P(\theta |X)}"></span>. </p><p>The variance of the newly obtained probability mass function can also be determined. The variance for a Bayesian probability mass function can be defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P_{\theta }(x|X)}^{2}:=\int d\theta \ \left[P(x|\theta )-P_{\theta }(x|X)\right]^{2}\ P(\theta |X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>:=</mo> <mo>∫</mo> <mi>d</mi> <mi>θ</mi> <mtext> </mtext> <msup> <mrow> <mo>[</mo> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P_{\theta }(x|X)}^{2}:=\int d\theta \ \left[P(x|\theta )-P_{\theta }(x|X)\right]^{2}\ P(\theta |X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de226ff5d1c090c1f59bcffe841e95cbeb02577a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.619ex; height:5.676ex;" alt="{\displaystyle \sigma _{P_{\theta }(x|X)}^{2}:=\int d\theta \ \left[P(x|\theta )-P_{\theta }(x|X)\right]^{2}\ P(\theta |X)}"></span>. </p><p>This expression for the variance can be substantially simplified (assuming independently drawn samples). Defining the "self probability mass function" as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\theta }(x|\left\{X,x\right\})=\int d\theta \ P(x|\theta )\ P(\theta |\left\{X,x\right\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>{</mo> <mrow> <mi>X</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>θ</mi> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>{</mo> <mrow> <mi>X</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\theta }(x|\left\{X,x\right\})=\int d\theta \ P(x|\theta )\ P(\theta |\left\{X,x\right\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe31dc404f6ede8b89c5585dbf030bfe2723960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.453ex; height:5.676ex;" alt="{\displaystyle P_{\theta }(x|\left\{X,x\right\})=\int d\theta \ P(x|\theta )\ P(\theta |\left\{X,x\right\})}"></span>, </p><p>one obtains for the variance<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P_{\theta }(x|X)}^{2}=P_{\theta }(x|X)\left[P_{\theta }(x|\left\{X,x\right\})-P_{\theta }(x|X)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>{</mo> <mrow> <mi>X</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P_{\theta }(x|X)}^{2}=P_{\theta }(x|X)\left[P_{\theta }(x|\left\{X,x\right\})-P_{\theta }(x|X)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d836ebe9321e14e2c6ffc959fa71ebac5f0a2c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:44.962ex; height:3.676ex;" alt="{\displaystyle \sigma _{P_{\theta }(x|X)}^{2}=P_{\theta }(x|X)\left[P_{\theta }(x|\left\{X,x\right\})-P_{\theta }(x|X)\right]}"></span>. </p><p> The expression for variance involves an additional fit that includes the sample <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> of interest.</p><figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:CumList.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/CumList.png/220px-CumList.png" decoding="async" width="220" height="393" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/CumList.png/330px-CumList.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f6/CumList.png 2x" data-file-width="363" data-file-height="649" /></a><figcaption>List of probability distributions ranked by goodness of fit<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:GEVdistrHistogr%2BDensity.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/GEVdistrHistogr%2BDensity.png/220px-GEVdistrHistogr%2BDensity.png" decoding="async" width="220" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/GEVdistrHistogr%2BDensity.png/330px-GEVdistrHistogr%2BDensity.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/GEVdistrHistogr%2BDensity.png/440px-GEVdistrHistogr%2BDensity.png 2x" data-file-width="623" data-file-height="441" /></a><figcaption>Histogram and probability density of a data set fitting the <a href="/wiki/GEV_distribution" class="mw-redirect" title="GEV distribution">GEV distribution</a> </figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Goodness_of_fit">Goodness of fit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=9" title="Edit section: Goodness of fit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By ranking the <a href="/wiki/Goodness_of_fit" title="Goodness of fit">goodness of fit</a> of various distributions one can get an impression of which distribution is acceptable and which is not. </p> <div class="mw-heading mw-heading2"><h2 id="Histogram_and_density_function">Histogram and density function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=10" title="Edit section: Histogram and density function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF) one can derive a <a href="/wiki/Histogram" title="Histogram">histogram</a> and the <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> (PDF). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Curve_fitting" title="Curve fitting">Curve fitting</a></li> <li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li> <li><a href="/wiki/Mixture_distribution" title="Mixture distribution">Mixture distribution</a></li> <li><a href="/wiki/Product_distribution" class="mw-redirect" title="Product distribution">Product distribution</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Probability_distribution_fitting&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-skew-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-skew_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-skew_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Left (negatively) skewed frequency histograms can be fitted to square Normal or mirrored Gumbel probability functions. On line: <a rel="nofollow" class="external autonumber" href="https://www.researchgate.net/publication/338633570_Left_negatively_skewed_frequency_histograms_can_be_fitted_to_square_Normal_or_mirrored_Gumbel_probability_functions">[1]</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><i>Frequency and Regression Analysis</i>. Chapter 6 in: H.P.Ritzema (ed., 1994), <i>Drainage Principles and Applications</i>, Publ. 16, pp. 175–224, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. <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 .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9070754339" title="Special:BookSources/9070754339">9070754339</a>. Free download from the webpage <a rel="nofollow" class="external autonumber" href="http://www.waterlog.info/articles.htm">[2]</a> under nr. 12, or directly as PDF : <a rel="nofollow" class="external autonumber" href="http://www.waterlog.info/pdf/freqtxt.pdf">[3]</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">H. Cramér, "Mathematical methods of statistics", Princeton Univ. Press (1946)</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHosking1990" class="citation journal cs1">Hosking, J.R.M. (1990). "L-moments: analysis and estimation of distributions using linear combinations of order statistics". <i>Journal of the Royal Statistical Society, Series B</i>. <b>52</b> (1): 105–124. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2345653">2345653</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B&rft.atitle=L-moments%3A+analysis+and+estimation+of+distributions+using+linear+combinations+of+order+statistics&rft.volume=52&rft.issue=1&rft.pages=105-124&rft.date=1990&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2345653%23id-name%3DJSTOR&rft.aulast=Hosking&rft.aufirst=J.R.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProbability+distribution+fitting" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="citeref_Aldrich1997" class="citation journal cs1">Aldrich, John (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1030037906">"R. A. Fisher and the making of maximum likelihood 1912–1922"</a>. <i>Statistical Science</i>. <b>12</b> (3): 162–176. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1030037906">10.1214/ss/1030037906</a></span>. <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=1617519">1617519</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=R.+A.+Fisher+and+the+making+of+maximum+likelihood+1912%E2%80%931922&rft.volume=12&rft.issue=3&rft.pages=162-176&rft.date=1997&rft_id=info%3Adoi%2F10.1214%2Fss%2F1030037906&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1617519%23id-name%3DMR&rft.aulast=Aldrich&rft.aufirst=John&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Fss%252F1030037906&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProbability+distribution+fitting" class="Z3988"></span></span> </li> <li id="cite_note-gen-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-gen_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-gen_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-gen_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Software for Generalized and Composite Probability Distributions. International Journal of Mathematical and Computational Methods, 4, 1-9 <a rel="nofollow" class="external autonumber" href="https://www.iaras.org/iaras/home/caijmcm/software-for-generalized-and-composite-probability-distributions">[4]</a> or <a rel="nofollow" class="external autonumber" href="https://www.waterlog.info/pdf/MathJournal.pdf">[5]</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Example of an approximately normally distributed data set to which a large number of different probability distributions can be fitted, <a rel="nofollow" class="external autonumber" href="https://www.waterlog.info/pdf/Multiple%20fit.pdf">[6]</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Left (negatively) skewed frequency histograms can be fitted to square normal or mirrored Gumbel probability functions. <a rel="nofollow" class="external autonumber" href="https://www.waterlog.info/pdf/LeftSkew.pdf">[7]</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.waterlog.info/composite.htm">Intro to composite probability distributions</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Frequency predictions and their binomial confidence limits. In: International Commission on Irrigation and Drainage, Special Technical Session: Economic Aspects of Flood Control and non-Structural Measures, Dubrovnik, Yugoslavia, 1988. <a rel="nofollow" class="external text" href="http://www.waterlog.info/pdf/binomial.pdf">On line</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Benson, M.A. 1960. Characteristics of frequency curves based on a theoretical 1000 year record. In: T.Dalrymple (Ed.), Flood frequency analysis. U.S. Geological Survey Water Supply Paper, 1543-A, pp. 51-71.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPijlmanLinnartz2023" class="citation journal cs1">Pijlman; Linnartz (2023). <a rel="nofollow" class="external text" href="https://sitb2023.ulb.be/proceedings/">"Variance of Likelihood of data"</a>. <i>SITB 2023 Proceedings</i>: 34.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SITB+2023+Proceedings&rft.atitle=Variance+of+Likelihood+of+data&rft.pages=34&rft.date=2023&rft.au=Pijlman&rft.au=Linnartz&rft_id=https%3A%2F%2Fsitb2023.ulb.be%2Fproceedings%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProbability+distribution+fitting" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.waterlog.info/cumfreq.htm">Software for probability distribution fitting</a></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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id="Distribution_fitting" style="font-size:114%;margin:0 4em">Distribution fitting</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview and methods</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Probability plot <ul><li><a href="/wiki/Normal_probability_plot" title="Normal probability plot">Normal probability plot</a></li> <li><a href="/wiki/P%E2%80%93P_plot" title="P–P plot">P–P plot</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li></ul></li> <li><a href="/wiki/Plotting_position" class="mw-redirect" title="Plotting position">Plotting position</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moment</a></li> <li><a class="mw-selflink selflink">Distribution fitting</a></li> <li><a href="/wiki/Cumulative_frequency_analysis" title="Cumulative frequency analysis">Cumulative frequency analysis</a></li></ul> </div></td></tr><tr><th scope="row" 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