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href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Метод_обратного_преобразования rootpage-Метод_обратного_преобразования skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Метод обратного преобразования</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">Материал из Википедии — свободной энциклопедии</div> <div id="contentSub"><div id="mw-content-subtitle"><div id="mw-fr-revision-messages"><div class="cdx-message mw-fr-message-box cdx-message--block cdx-message--notice mw-fr-basic mw-fr-draft-not-synced plainlinks noprint"><span class="cdx-message__icon"></span><div class="cdx-message__content">Текущая версия страницы пока <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D1%80%D0%BE%D0%B2%D0%B5%D1%80%D0%BA%D0%B0_%D1%81%D1%82%D0%B0%D1%82%D0%B5%D0%B9/%D0%9F%D0%BE%D1%8F%D1%81%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%BB%D1%8F_%D1%87%D0%B8%D1%82%D0%B0%D1%82%D0%B5%D0%BB%D0%B5%D0%B9" title="Википедия:Проверка статей/Пояснение для читателей">не проверялась</a> опытными участниками и может значительно отличаться от <a class="external text" href="https://ru.wikipedia.org/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&stable=1">версии, проверенной 17 апреля 2019 года</a>; проверки требуют <a class="external text" href="https://ru.wikipedia.org/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&oldid=90212403&diff=cur&diffonly=0">5 правок</a>.</div></div></div></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Перейти к навигации</a> <a class="mw-jump-link" href="#searchInput">Перейти к поиску</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ru" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Inversion_method2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Inversion_method2.svg/300px-Inversion_method2.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Inversion_method2.svg/450px-Inversion_method2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Inversion_method2.svg/600px-Inversion_method2.svg.png 2x" data-file-width="630" data-file-height="630" /></a><figcaption>Ме́тод обра́тного преобразова́ния.</figcaption></figure> <p><b>Ме́тод обра́тного преобразова́ния</b> (Преобразование <a href="/wiki/%D0%A1%D0%BC%D0%B8%D1%80%D0%BD%D0%BE%D0%B2,_%D0%9D%D0%B8%D0%BA%D0%BE%D0%BB%D0%B0%D0%B9_%D0%92%D0%B0%D1%81%D0%B8%D0%BB%D1%8C%D0%B5%D0%B2%D0%B8%D1%87_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA)" title="Смирнов, Николай Васильевич (математик)">Н. В. Смирнова</a>) — способ генерации <a href="/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Случайная величина">случайных величин</a> с заданной функцией распределения, путём модификации работы генератора равномерно распределённых чисел. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="ru" dir="ltr"><h2 id="mw-toc-heading">Содержание</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Описание_алгоритма"><span class="tocnumber">1</span> <span class="toctext">Описание алгоритма</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Строго_возрастающая_функция_распределения"><span class="tocnumber">1.1</span> <span class="toctext">Строго возрастающая функция распределения</span></a> <ul> <li class="toclevel-3 tocsection-3"><a href="#Пример"><span class="tocnumber">1.1.1</span> <span class="toctext">Пример</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-4"><a href="#Неубывающая_функция_распределения"><span class="tocnumber">1.2</span> <span class="toctext">Неубывающая функция распределения</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-5"><a href="#Замечания"><span class="tocnumber">2</span> <span class="toctext">Замечания</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Математическое_обоснование"><span class="tocnumber">3</span> <span class="toctext">Математическое обоснование</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#См._также"><span class="tocnumber">4</span> <span class="toctext">См. также</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Литература"><span class="tocnumber">5</span> <span class="toctext">Литература</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Описание_алгоритма"><span id=".D0.9E.D0.BF.D0.B8.D1.81.D0.B0.D0.BD.D0.B8.D0.B5_.D0.B0.D0.BB.D0.B3.D0.BE.D1.80.D0.B8.D1.82.D0.BC.D0.B0"></span>Описание алгоритма</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&veaction=edit&section=1" title="Редактировать раздел «Описание алгоритма»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&action=edit&section=1" title="Редактировать код раздела «Описание алгоритма»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Пусть <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 F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span> является <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Функция распределения">функцией произвольного распределения</a>. Покажем, как, имея <a href="/wiki/%D0%93%D0%B5%D0%BD%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80_%D0%BF%D1%81%D0%B5%D0%B2%D0%B4%D0%BE%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D1%85_%D1%87%D0%B8%D1%81%D0%B5%D0%BB" title="Генератор псевдослучайных чисел">генератор</a> <a href="/wiki/%D0%92%D1%8B%D0%B1%D0%BE%D1%80%D0%BA%D0%B0" title="Выборка">выборки</a> из <a href="/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D1%80%D1%8B%D0%B2%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D0%B2%D0%BD%D0%BE%D0%BC%D0%B5%D1%80%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="Непрерывное равномерное распределение">стандартного непрерывного равномерного распределения</a>, получить выборку из распределения, задаваемого функцией распределения <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 F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Строго_возрастающая_функция_распределения"><span id=".D0.A1.D1.82.D1.80.D0.BE.D0.B3.D0.BE_.D0.B2.D0.BE.D0.B7.D1.80.D0.B0.D1.81.D1.82.D0.B0.D1.8E.D1.89.D0.B0.D1.8F_.D1.84.D1.83.D0.BD.D0.BA.D1.86.D0.B8.D1.8F_.D1.80.D0.B0.D1.81.D0.BF.D1.80.D0.B5.D0.B4.D0.B5.D0.BB.D0.B5.D0.BD.D0.B8.D1.8F"></span>Строго возрастающая функция распределения</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&veaction=edit&section=2" title="Редактировать раздел «Строго возрастающая функция распределения»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&action=edit&section=2" title="Редактировать код раздела «Строго возрастающая функция распределения»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Если функция <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 F:\mathbb {R} \to [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:\mathbb {R} \to [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c8f53a3d4bfe23906df3cad3f66f98329c4344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.623ex; height:2.843ex;" alt="{\displaystyle F:\mathbb {R} \to [0,1]}"></span> <a href="/wiki/%D0%9C%D0%BE%D0%BD%D0%BE%D1%82%D0%BE%D0%BD%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Монотонная функция">строго возрастает</a> на всей <a href="/wiki/%D0%9E%D0%B1%D0%BB%D0%B0%D1%81%D1%82%D1%8C_%D0%BE%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" class="mw-redirect" title="Область определения">области определения</a>, то она <a href="/wiki/%D0%91%D0%B8%D0%B5%D0%BA%D1%86%D0%B8%D1%8F" title="Биекция">биективна</a>, а следовательно имеет <a href="/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Обратная функция">обратную функцию</a> <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 F^{-1}:[0,1]\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{-1}:[0,1]\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1f92c8282fd0e8eede86a688b35fcfe2d73880b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.029ex; height:3.176ex;" alt="{\displaystyle F^{-1}:[0,1]\to \mathbb {R} }"></span>. </p> <ul><li>Пусть <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 U_{1},\ldots ,U_{n}\sim U[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mi>U</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1},\ldots ,U_{n}\sim U[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7feb24e453110efff386c6a8d3588374bdbee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.159ex; height:2.843ex;" alt="{\displaystyle U_{1},\ldots ,U_{n}\sim U[0,1]}"></span> — выборка из стандартного непрерывного равномерного распределения.</li> <li>Тогда <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_{1},\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> <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},\ldots ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac794f5521dcce89913085a6d566e7cdb615dbb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{n}}"></span>, где <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_{i}=F^{-1}(U_{i}),\;i=1,\ldots ,n}"> <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> <mo>=</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}=F^{-1}(U_{i}),\;i=1,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d364e550e2b5143bc3afb3088df1664a76928458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.482ex; height:3.176ex;" alt="{\displaystyle X_{i}=F^{-1}(U_{i}),\;i=1,\ldots ,n}"></span>, — выборка из интересующего нас распределения.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Пример"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.80"></span>Пример</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&veaction=edit&section=3" title="Редактировать раздел «Пример»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&action=edit&section=3" title="Редактировать код раздела «Пример»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Пусть требуется сгенерировать выборку из <a href="/wiki/%D0%AD%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D0%B8%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="Экспоненциальное распределение">экспоненциального распределения</a> с параметром <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 \lambda >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda >0}"></span>. Функция этого распределения <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 F(x)=1-e^{-\lambda x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=1-e^{-\lambda x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6d5780de5cb8a64b0ebeadd01899ef5bb1e1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.474ex; height:3.176ex;" alt="{\displaystyle F(x)=1-e^{-\lambda x}}"></span> строго возрастает, и её обратная функция имеет вид <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 F^{-1}(x)=-{\frac {1}{\lambda }}\,\ln(1-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{-1}(x)=-{\frac {1}{\lambda }}\,\ln(1-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38b546fe31cd90c7fc7e45d4b704d16d9e76a9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.24ex; height:5.343ex;" alt="{\displaystyle F^{-1}(x)=-{\frac {1}{\lambda }}\,\ln(1-x)}"></span>. Таким образом, если <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 U_{1},\ldots ,U_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1},\ldots ,U_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdefcb9e1f2a3b58c4bb3925fa1d3ed76215b3e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.626ex; height:2.509ex;" alt="{\displaystyle U_{1},\ldots ,U_{n}}"></span> — выборка из стандартного непрерывного равномерного распределения, то <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_{1},\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> <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},\ldots ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac794f5521dcce89913085a6d566e7cdb615dbb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{n}}"></span>, где </p> <dl><dd><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_{i}=-{\frac {1}{\lambda }}\ln(1-U_{i}),\;i=1,\ldots ,n}"> <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> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}=-{\frac {1}{\lambda }}\ln(1-U_{i}),\;i=1,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a04605def7b3b1396fa13f374f036670da487b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.663ex; height:5.343ex;" alt="{\displaystyle X_{i}=-{\frac {1}{\lambda }}\ln(1-U_{i}),\;i=1,\ldots ,n}"></span></dd></dl> <p>— искомая выборка из экспоненциального распределения. </p> <div class="mw-heading mw-heading3"><h3 id="Неубывающая_функция_распределения"><span id=".D0.9D.D0.B5.D1.83.D0.B1.D1.8B.D0.B2.D0.B0.D1.8E.D1.89.D0.B0.D1.8F_.D1.84.D1.83.D0.BD.D0.BA.D1.86.D0.B8.D1.8F_.D1.80.D0.B0.D1.81.D0.BF.D1.80.D0.B5.D0.B4.D0.B5.D0.BB.D0.B5.D0.BD.D0.B8.D1.8F"></span>Неубывающая функция распределения</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&veaction=edit&section=4" title="Редактировать раздел «Неубывающая функция распределения»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&action=edit&section=4" title="Редактировать код раздела «Неубывающая функция распределения»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Если функция <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 F:\mathbb {R} \to [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:\mathbb {R} \to [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c8f53a3d4bfe23906df3cad3f66f98329c4344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.623ex; height:2.843ex;" alt="{\displaystyle F:\mathbb {R} \to [0,1]}"></span> лишь не убывает, то её обратная функция может не существовать. В таком случае необходимо модифицировать приведённый выше <a href="/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC" title="Алгоритм">алгоритм</a>. </p> <ul><li>Пусть <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 U_{1},\ldots ,U_{n}\sim U[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mi>U</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1},\ldots ,U_{n}\sim U[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7feb24e453110efff386c6a8d3588374bdbee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.159ex; height:2.843ex;" alt="{\displaystyle U_{1},\ldots ,U_{n}\sim U[0,1]}"></span> — выборка из стандартного непрерывного равномерного распределения.</li> <li>Тогда <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_{1},\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> <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},\ldots ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac794f5521dcce89913085a6d566e7cdb615dbb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{n}}"></span>, где <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_{i}=\inf\{x\mid F(x)=U_{i}\}=\min\{x\mid F(x)=U_{i}\},\;i=1,\ldots ,n}"> <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> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}=\inf\{x\mid F(x)=U_{i}\}=\min\{x\mid F(x)=U_{i}\},\;i=1,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd98712c4dc0d75f5bbd454c164c6e25fa7e8e71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.677ex; height:2.843ex;" alt="{\displaystyle X_{i}=\inf\{x\mid F(x)=U_{i}\}=\min\{x\mid F(x)=U_{i}\},\;i=1,\ldots ,n}"></span>, — выборка из интересующего нас распределения. Равенство точной нижней грани минимуму выполняется ввиду непрерывности функции распределения справа, что означает, что точная нижняя грань достигается.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Замечания"><span id=".D0.97.D0.B0.D0.BC.D0.B5.D1.87.D0.B0.D0.BD.D0.B8.D1.8F"></span>Замечания</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&veaction=edit&section=5" title="Редактировать раздел «Замечания»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&action=edit&section=5" title="Редактировать код раздела «Замечания»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Если <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 F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span> строго возрастает, то <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 F^{-1}(u)=\inf\{x\mid F(x)=u\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{-1}(u)=\inf\{x\mid F(x)=u\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9577f11b86c663e5ed893673e337251d32bddf9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.936ex; height:3.176ex;" alt="{\displaystyle F^{-1}(u)=\inf\{x\mid F(x)=u\}}"></span>. Таким образом, модифицированный алгоритм для произвольной функции распределения включает в себя отдельно разобранный случай строго возрастающей функции распределения.</li> <li>Несмотря на кажущуюся универсальность, данный алгоритм имеет серьёзные практические ограничения. Даже если функция распределения строго возрастает, вычислить её обратную не всегда просто, особенно если она не задана в виде <a href="/wiki/%D0%AD%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D1%8B%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Элементарные функции">элементарной функции</a>, как, например, в случае <a href="/wiki/%D0%9D%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="Нормальное распределение">нормального распределения</a>. В случае функции распределения общего вида чаще всего необходимо <a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8B%D0%B5_%D0%BC%D0%B5%D1%82%D0%BE%D0%B4%D1%8B" title="Численные методы">численно</a> находить <a href="/wiki/%D0%A2%D0%BE%D1%87%D0%BD%D0%B0%D1%8F_%D0%BD%D0%B8%D0%B6%D0%BD%D1%8F%D1%8F_%D0%B3%D1%80%D0%B0%D0%BD%D1%8C" class="mw-redirect" title="Точная нижняя грань">точную нижнюю грань</a>, что может быть очень трудоёмко.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Математическое_обоснование"><span id=".D0.9C.D0.B0.D1.82.D0.B5.D0.BC.D0.B0.D1.82.D0.B8.D1.87.D0.B5.D1.81.D0.BA.D0.BE.D0.B5_.D0.BE.D0.B1.D0.BE.D1.81.D0.BD.D0.BE.D0.B2.D0.B0.D0.BD.D0.B8.D0.B5"></span>Математическое обоснование</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&veaction=edit&section=6" title="Редактировать раздел «Математическое обоснование»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&action=edit&section=6" title="Редактировать код раздела «Математическое обоснование»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Generalized_inversion_method.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Generalized_inversion_method.svg/300px-Generalized_inversion_method.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Generalized_inversion_method.svg/450px-Generalized_inversion_method.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Generalized_inversion_method.svg/600px-Generalized_inversion_method.svg.png 2x" data-file-width="630" data-file-height="630" /></a><figcaption>Математическое обоснование.</figcaption></figure> <p>Пусть <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 U\sim U[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>∼</mo> <mi>U</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\sim U[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6e45472430f8b45e7bbb99bef71249034d30eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.316ex; height:2.843ex;" alt="{\displaystyle U\sim U[0,1]}"></span>, то есть <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 F_{U}(u)=u,\;u\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>u</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>u</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{U}(u)=u,\;u\in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d8cf0017f32f70ee9c897a0bba80ad5cd12f567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.056ex; height:2.843ex;" alt="{\displaystyle F_{U}(u)=u,\;u\in [0,1]}"></span>. Рассмотрим функцию распределения случайной величины <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=\inf\{x\mid F(x)=U\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\inf\{x\mid F(x)=U\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f8c45a7c91b039264b7ab9f8cac84d208e4c32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.082ex; height:2.843ex;" alt="{\displaystyle X=\inf\{x\mid F(x)=U\}}"></span>. </p> <dl><dd><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 \mathbb {P} (X\leq x)=\mathbb {P} (\inf\{x'\mid F(x')=U\}\leq x)=\mathbb {P} (U\leq F(x))=F_{U}(F(x))=F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>∣</mo> <mi>F</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo fence="false" stretchy="false">}</mo> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo>≤</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (X\leq x)=\mathbb {P} (\inf\{x'\mid F(x')=U\}\leq x)=\mathbb {P} (U\leq F(x))=F_{U}(F(x))=F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf934b1237e3cc5e6d6d1675721909893b290e1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.609ex; height:3.009ex;" alt="{\displaystyle \mathbb {P} (X\leq x)=\mathbb {P} (\inf\{x'\mid F(x')=U\}\leq x)=\mathbb {P} (U\leq F(x))=F_{U}(F(x))=F(x)}"></span>.</dd></dl> <p>То есть <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> имеет функцию распределения <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 F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; 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href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%B2%D1%8F%D0%B7%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B8/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F" rel="nofollow" title="Последние изменения в страницах, на которые ссылается эта страница [k]" accesskey="k"><span>Связанные правки</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BF%D0%B5%D1%86%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8B" title="Список служебных страниц [q]" accesskey="q"><span>Служебные страницы</span></a></li><li id="t-permalink" class="mw-list-item"><a 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href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:QrCode&url=https%3A%2F%2Fru.wikipedia.org%2Fwiki%2F%25D0%259C%25D0%25B5%25D1%2582%25D0%25BE%25D0%25B4_%25D0%25BE%25D0%25B1%25D1%2580%25D0%25B0%25D1%2582%25D0%25BD%25D0%25BE%25D0%25B3%25D0%25BE_%25D0%25BF%25D1%2580%25D0%25B5%25D0%25BE%25D0%25B1%25D1%2580%25D0%25B0%25D0%25B7%25D0%25BE%25D0%25B2%25D0%25B0%25D0%25BD%25D0%25B8%25D1%258F"><span>Скачать QR-код</span></a></li> </ul> </div> </nav> <nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label" > <h3 id="p-coll-print_export-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Печать/экспорт</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:DownloadAsPdf&page=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&action=show-download-screen" title="Скачать эту страницу как файл PDF"><span>Скачать как PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%BE%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D1%8F&printable=yes" title="Версия этой страницы для печати [p]" accesskey="p"><span>Версия для печати</span></a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">В других проектах</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1377019" title="Ссылка на связанный элемент репозитория данных [g]" accesskey="g"><span>Элемент Викиданных</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">На других языках</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mostreig_de_transformaci%C3%B3_inversa" title="Mostreig de transformació inversa — каталанский" lang="ca" hreflang="ca" data-title="Mostreig de transformació inversa" data-language-autonym="Català" data-language-local-name="каталанский" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Inversionsmethode" title="Inversionsmethode — немецкий" lang="de" hreflang="de" data-title="Inversionsmethode" data-language-autonym="Deutsch" data-language-local-name="немецкий" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Inverse_transform_sampling" title="Inverse transform sampling — английский" lang="en" hreflang="en" data-title="Inverse transform sampling" data-language-autonym="English" data-language-local-name="английский" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/M%C3%A9todo_de_la_transformada_inversa" title="Método de la transformada inversa — испанский" lang="es" hreflang="es" data-title="Método de la transformada inversa" data-language-autonym="Español" data-language-local-name="испанский" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D9%88%D8%B4_%D8%AA%D8%A8%D8%AF%DB%8C%D9%84_%D9%85%D8%B9%DA%A9%D9%88%D8%B3" title="روش تبدیل معکوس — персидский" lang="fa" hreflang="fa" data-title="روش تبدیل معکوس" data-language-autonym="فارسی" data-language-local-name="персидский" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9thode_de_la_transform%C3%A9e_inverse" title="Méthode de la transformée inverse — французский" lang="fr" hreflang="fr" data-title="Méthode de la transformée inverse" data-language-autonym="Français" data-language-local-name="французский" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%92%D7%99%D7%9E%D7%94_%D7%9E%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%94%D7%95%D7%A4%D7%9B%D7%99%D7%AA" title="דגימה מהעתקה הופכית — иврит" lang="he" hreflang="he" data-title="דגימה מהעתקה הופכית" data-language-autonym="עברית" data-language-local-name="иврит" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Metodo_dell%27inversione" title="Metodo dell'inversione — итальянский" lang="it" hreflang="it" data-title="Metodo dell'inversione" data-language-autonym="Italiano" data-language-local-name="итальянский" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%80%86%E9%96%A2%E6%95%B0%E6%B3%95" title="逆関数法 — японский" lang="ja" hreflang="ja" data-title="逆関数法" data-language-autonym="日本語" data-language-local-name="японский" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/M%C3%A9todo_da_transforma%C3%A7%C3%A3o_inversa" title="Método da transformação inversa — португальский" lang="pt" hreflang="pt" data-title="Método da transformação inversa" data-language-autonym="Português" data-language-local-name="португальский" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Inverse_transform_sampling_method" title="Inverse transform sampling method — сунданский" lang="su" hreflang="su" data-title="Inverse transform sampling method" data-language-autonym="Sunda" data-language-local-name="сунданский" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link 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