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<title>Formel von Wald</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Formel von Wald</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Formel von Wald</b> oder <b>Waldsche Identität</b> ist in der <a href="Stochastik" title="Stochastik">Stochastik</a> eine <a href="Gleichung" title="Gleichung">Gleichung</a>, mit deren Hilfe der <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> von Summen von <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariablen</a> mit einer zufälligen Anzahl von Summanden berechnet werden kann. Sie wurde 1944 in einer Arbeit des Mathematikers <a href="Abraham_Wald" title="Abraham Wald">Abraham Wald</a> bewiesen.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Formulierung">Formulierung</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle (X_{n})_{n\geq 1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46a33659d18632a22501249c54c6be339cf08a4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.271ex; height:2.843ex;" alt="{\displaystyle (X_{n})_{n\geq 1}}" loading="lazy"></span> eine Folge <a href="Unabh%C3%A4ngig_und_identisch_verteilt" class="mw-redirect" title="Unabhängig und identisch verteilt">unabhängiger, identisch verteilter</a>, <a href="Zufallsvariable#Integrierbar_und_quasi-integrierbar" title="Zufallsvariable">integrierbarer</a> Zufallsvariablen und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> eine <span class="mwe-math-element mwe-math-element-inline"><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 {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"></span>-wertige Zufallsvariable mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (T)<\infty }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (T)<\infty }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef24b1307a8bcef538bb2faf21781111f056cfa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.451ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (T)<\infty }" loading="lazy"></span>, die von der Folge <span class="mwe-math-element mwe-math-element-inline"><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_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (X_{n})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3076fe5f4da5c004de5c8a5f99eb0f8b2989f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle (X_{n})}" loading="lazy"></span> <a href="Stochastisch_unabh%C3%A4ngige_Zufallsvariablen" title="Stochastisch unabhängige Zufallsvariablen">unabhängig</a> ist. Dann gilt<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>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} (T)\operatorname {E} (X_{1})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1959ae72ccd9f753672335bb6e21aebebfd40e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.904ex; height:7.509ex;" alt="{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} (T)\operatorname {E} (X_{1})}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beweis">Beweis</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> unabhängig von der Folge <span class="mwe-math-element mwe-math-element-inline"><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_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (X_{n})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3076fe5f4da5c004de5c8a5f99eb0f8b2989f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle (X_{n})}" loading="lazy"></span> ist, folgt durch <a href="Bedingter_Erwartungswert" title="Bedingter Erwartungswert">Bedingen</a> auf den Wert von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T=n\right)=\operatorname {E} \left(\sum _{k=1}^{n}X_{k}\right)=\sum _{k=1}^{n}\operatorname {E} (X_{k})=n\operatorname {E} (X_{1})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac681a32d2f967fd8750aa8ab00b70eef714a14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.7ex; height:7.509ex;" alt="{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T=n\right)=\operatorname {E} \left(\sum _{k=1}^{n}X_{k}\right)=\sum _{k=1}^{n}\operatorname {E} (X_{k})=n\operatorname {E} (X_{1})}" loading="lazy"></span>,</dd></dl>

<p>also

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T\right)=T\operatorname {E} (X_{1})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba3866aa04849ea04c65d230f4a388419154bdde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.085ex; height:7.509ex;" alt="{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T\right)=T\operatorname {E} (X_{1})}" loading="lazy"></span>.</dd></dl>

<p>Durch Anwenden des Erwartungswerts auf diese Gleichung erhält man schließlich

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} \left(\operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T\right)\right)=\operatorname {E} (T\operatorname {E} (X_{1}))=\operatorname {E} (T)\operatorname {E} (X_{1})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf73cbaab30bde594ba4acf0316cdee8ab0686e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.744ex; height:7.509ex;" alt="{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} \left(\operatorname {E} \left(\sum _{k=1}^{T}X_{k}\;{\Big |}\;T\right)\right)=\operatorname {E} (T\operatorname {E} (X_{1}))=\operatorname {E} (T)\operatorname {E} (X_{1})}" loading="lazy"></span>.</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> alle <span class="mwe-math-element mwe-math-element-inline"><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 {N} _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{0}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerung_auf_Stoppzeiten">Verallgemeinerung auf Stoppzeiten</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle (X_{n})_{n\geq 1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46a33659d18632a22501249c54c6be339cf08a4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.271ex; height:2.843ex;" alt="{\displaystyle (X_{n})_{n\geq 1}}" loading="lazy"></span> eine Folge identisch verteilter integrierbarer Zufallsvariablen, die an eine <a href="Filtrierung_(Wahrscheinlichkeitstheorie)" title="Filtrierung (Wahrscheinlichkeitstheorie)">Filtrierung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {F}}_{n})_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}_{n})_{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6db9767a4bde24fbe44fa07ecaa45c7933103e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.917ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {F}}_{n})_{n}}" loading="lazy"></span> <a href="Adaptierter_stochastischer_Prozess" title="Adaptierter stochastischer Prozess">adaptiert</a> ist, das heißt für alle <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"></span> ist <span class="mwe-math-element mwe-math-element-inline"><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_{n}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72a8564cedc659cf2f95ae68bc5de2f5207a3285.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{n}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aedfc87ef7b181d03cfd9b4d74662cdea74e5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.89ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{n}}" loading="lazy"></span>-messbar. Wenn <span class="mwe-math-element mwe-math-element-inline"><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_{n+1}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a5e2ebcd9ef3ffc3f45a5745884e0a46de672cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.243ex; height:2.509ex;" alt="{\displaystyle X_{n+1}}" loading="lazy"></span> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{n}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aedfc87ef7b181d03cfd9b4d74662cdea74e5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.89ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{n}}" loading="lazy"></span> unabhängig ist für alle <span class="mwe-math-element mwe-math-element-inline"><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\in \mathbb {N} }">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> eine integrierbare <a href="Stoppzeit" title="Stoppzeit">Stoppzeit</a> bezüglich <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {F}}_{n})_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}}_{n})_{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6db9767a4bde24fbe44fa07ecaa45c7933103e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.917ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {F}}_{n})_{n}}" loading="lazy"></span> ist, so gilt ebenfalls die Formel von Wald:<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>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} (T)\operatorname {E} (X_{1})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1959ae72ccd9f753672335bb6e21aebebfd40e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.904ex; height:7.509ex;" alt="{\displaystyle \operatorname {E} \left(\sum _{k=1}^{T}X_{k}\right)=\operatorname {E} (T)\operatorname {E} (X_{1})}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Verwandte_Konzepte">Verwandte Konzepte</h2></div>

<p>Ähnliche Aussagen über die Varianz von zusammengesetzten Verteilungen lassen sich mit der <a href="Blackwell-Girshick-Gleichung" title="Blackwell-Girshick-Gleichung">Blackwell-Girshick-Gleichung</a> treffen.

</p>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite style="font-style:italic">On Cumulative Sums of Random Variables</cite>. In: <cite style="font-style:italic">The Annals of Mathematical Statistics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>15</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>3</span>, 1944, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>283–296</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1214/aoms%2F1177731235">10.1214/aoms/1177731235</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Formel+von+Wald&rft.atitle=On+Cumulative+Sums+of+Random+Variables&rft.date=1944&rft.doi=10.1214%2Faoms%2F1177731235&rft.genre=journal&rft.issue=3&rft.jtitle=The+Annals+of+Mathematical+Statistics&rft.pages=283-296&rft.volume=15" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">David Meintrup, Stefan Schäffler: <i>Stochastik. Theorie und Anwendungen.</i> Springer, Berlin/Heidelberg 2005, ISBN 3-540-21676-6, S. 287.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a href="Heinz_Bauer_(Mathematiker)" title="Heinz Bauer (Mathematiker)">Heinz Bauer</a>: <i>Wahrscheinlichkeitstheorie.</i> 5. Auflage. De-Gruyter-Lehrbuch, Berlin 2002, ISBN 3-11-017236-4, Kapitel 17.</span>

</li>

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