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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Binomial-Prozess</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>Als <b>Binomial-Prozesse</b> bezeichnet man eine spezielle Klasse von <a href="Punktprozess" title="Punktprozess">Punktprozessen</a> in der Theorie der <a href="Stochastischer_Prozess" title="Stochastischer Prozess">stochastischen Prozesse</a>, einem Teilgebiet der <a href="Wahrscheinlichkeitstheorie" title="Wahrscheinlichkeitstheorie">Wahrscheinlichkeitstheorie</a>. Binomial-Prozesse sind den <a href="Poisson-Prozess" title="Poisson-Prozess">Poisson-Prozessen</a> ähnlich, jedoch ist die Anzahl der Ereignisse pro Intervall <a href="Binomialverteilung" title="Binomialverteilung">binomialverteilt</a> und nicht <a href="Poisson-Verteilung" title="Poisson-Verteilung">Poisson-verteilt</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Gegeben sei eine ganze Zahl <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}">
<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="./_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> und eine <a href="Wahrscheinlichkeitsverteilung" class="mw-redirect" title="Wahrscheinlichkeitsverteilung">Wahrscheinlichkeitsverteilung</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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" 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}" loading="lazy"></span> auf einem <a href="Messraum_(Mathematik)" title="Messraum (Mathematik)">Messraum</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 (X,{\mathcal {B}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {B}})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac297ca9d2e879833a75ffdf0751993c507a77cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.367ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {B}})}" loading="lazy"></span> sowie <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}">
<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="./_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> <a href="Unabh%C3%A4ngig_und_identisch_verteilte_Zufallsvariablen" title="Unabhängig und identisch verteilte Zufallsvariablen">unabhängig, identisch und gemäß <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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" 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}" loading="lazy"></span> verteilte</a> Zufallsvariablen <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_{1},X_{2},\dots ,X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots ,X_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/197911bbfe9fe9540571f88763a0064df10686ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots ,X_{n}}" loading="lazy"></span>. Es gilt also <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_{i}\sim P}">
<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>
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{i}\sim P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c75f6c0f04cf425f1b55e08ab7c39ce470436ab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.568ex; height:2.509ex;" alt="{\displaystyle X_{i}\sim P}" loading="lazy"></span> 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 i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>. Des Weiteren bezeichne <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 \delta _{x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{x}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d31468dba98fabc9542edf69cae6d3362a3ac3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.205ex; height:2.676ex;" alt="{\displaystyle \delta _{x}}" loading="lazy"></span> das <a href="Dirac-Ma%C3%9F" class="mw-redirect" title="Dirac-Maß">Dirac-Maß</a> auf dem Punkt <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}">
<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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"></span>, also
</p>
<dl><dd><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 \delta _{x}(A):={\begin{cases}1&{\text{ falls }}x\in A\ \\0&{\text{ sonst }}\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext> falls </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>A</mi>
<mtext> </mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext> sonst </mtext>
</mrow>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{x}(A):={\begin{cases}1&{\text{ falls }}x\in A\ \\0&{\text{ sonst }}\end{cases}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a75e39c9f02d7da3a0b49e6cee2dede40dce564c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.222ex; height:6.176ex;" alt="{\displaystyle \delta _{x}(A):={\begin{cases}1&{\text{ falls }}x\in A\ \\0&{\text{ sonst }}\end{cases}}}" loading="lazy"></span></dd></dl>
<p>für <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 A\in {\mathcal {B}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {B}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09dde20ecba5f4b4a079bb31e1e8148f8aba7a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.127ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {B}}}" loading="lazy"></span>.
</p><p>Dann heißt das durch
</p>
<dl><dd><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 \zeta :=\sum _{i=1}^{n}\delta _{X_{i}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ζ<!-- ζ --></mi>
<mo>:=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \zeta :=\sum _{i=1}^{n}\delta _{X_{i}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da0b018820a47365d6417281481318d0e10281d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.833ex; height:6.843ex;" alt="{\displaystyle \zeta :=\sum _{i=1}^{n}\delta _{X_{i}}}" loading="lazy"></span></dd></dl>
<p>definierte <a href="Zuf%C3%A4lliges_Ma%C3%9F" title="Zufälliges Maß">zufällige Maß</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 \zeta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ζ<!-- ζ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"></span> auf <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,{\mathcal {B}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {B}})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac297ca9d2e879833a75ffdf0751993c507a77cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.367ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {B}})}" loading="lazy"></span> ein Binomial-Prozess.
</p>
<div class="mw-heading mw-heading2"><h2 id="Bemerkung">Bemerkung</h2></div>
<p>Für jede <a href="Messbare_Menge" class="mw-redirect" title="Messbare Menge">messbare Menge</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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> gilt per Definition
</p>
<dl><dd><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 \zeta (A)=\#\{i\mid X_{i}\in A\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ζ<!-- ζ --></mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">#<!-- # --></mi>
<mo fence="false" stretchy="false">{</mo>
<mi>i</mi>
<mo>∣<!-- ∣ --></mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>A</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \zeta (A)=\#\{i\mid X_{i}\in A\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69f11ced0f66dbed323391583eb1ece5cd888909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.054ex; height:2.843ex;" alt="{\displaystyle \zeta (A)=\#\{i\mid X_{i}\in A\}}" loading="lazy"></span></dd></dl>
<p>Hierbei bezeichnet <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 \#M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">#<!-- # --></mi>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \#M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17997301e5d9699f82cbcaa50de2a0994db250b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.378ex; height:2.509ex;" alt="{\displaystyle \#M}" loading="lazy"></span> die <a href="M%C3%A4chtigkeit_(Mathematik)" title="Mächtigkeit (Mathematik)">Mächtigkeit</a> der Menge <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 M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>, also die Anzahl ihrer Elemente. Der Prozess zählt somit, wie viele der <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}">
<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="./_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> <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariablen</a> Werte in der Menge <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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> annehmen. Somit ist für jede messbare Menge <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 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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> die Zufallsvariable <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 \zeta (A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ζ<!-- ζ --></mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \zeta (A)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/517e2c7a662b5865e50c0287d567a75cfd72a456.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.648ex; height:2.843ex;" alt="{\displaystyle \zeta (A)}" loading="lazy"></span> immer binomialverteilt mit Parametern <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}">
<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="./_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> 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 P(A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f264d19e21604793c6dc54f8044df454db82744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}" loading="lazy"></span>, es gilt also
</p>
<dl><dd><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 \zeta (A)\sim \operatorname {Bin} _{n,P(A)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ζ<!-- ζ --></mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>∼<!-- ∼ --></mo>
<msub>
<mi>Bin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>,</mo>
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \zeta (A)\sim \operatorname {Bin} _{n,P(A)}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a78df8e8335a58c4c10adc937e040f92c080d19b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.753ex; height:3.176ex;" alt="{\displaystyle \zeta (A)\sim \operatorname {Bin} _{n,P(A)}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Zugehöriger_Sprungprozess"><span id="Zugeh.C3.B6riger_Sprungprozess"></span>Zugehöriger Sprungprozess</h3></div>
<p>Im reellen Fall, also für <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,{\mathcal {B}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {B}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/822c66c3ef162f76c264f1d81af2da348decf117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.017ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {B}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}" loading="lazy"></span> ist der zum Punktprozess gehörende <a href="Sprungprozess" title="Sprungprozess">Sprungprozess</a> gegeben durch
</p>
<dl><dd><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_{t}:=\zeta ((-\infty ,t])=\#\{i\mid X_{i}\leq t\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>:=</mo>
<mi>ζ<!-- ζ --></mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">#<!-- # --></mi>
<mo fence="false" stretchy="false">{</mo>
<mi>i</mi>
<mo>∣<!-- ∣ --></mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mi>t</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{t}:=\zeta ((-\infty ,t])=\#\{i\mid X_{i}\leq t\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d080a25686bd5af05da6c6561d00a408c702d09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.718ex; height:2.843ex;" alt="{\displaystyle X_{t}:=\zeta ((-\infty ,t])=\#\{i\mid X_{i}\leq t\}}" loading="lazy"></span>.</dd></dl>
<p>Er gibt an, wie viele der Zufallsvariablen Werte kleinergleich <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> annehmen.
</p>
<div class="mw-heading mw-heading3"><h3 id="Laplace-Transformierte">Laplace-Transformierte</h3></div>
<p>Die <a href="Laplace-Transformation" title="Laplace-Transformation">Laplace-Transformation</a> eines Binomial-Prozesses ist gegeben durch
</p>
<dl><dd><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 {L}}_{P,n}(f)=\left[\int \exp(-f(x))\mathrm {P} (dx)\right]^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>P</mi>
<mo>,</mo>
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mrow>
<mo>[</mo>
<mrow>
<mo>∫<!-- ∫ --></mo>
<mi>exp</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">P</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>d</mi>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{P,n}(f)=\left[\int \exp(-f(x))\mathrm {P} (dx)\right]^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83b07eff25e85fe781764d51273e3930676a6cd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.479ex; height:6.176ex;" alt="{\displaystyle {\mathcal {L}}_{P,n}(f)=\left[\int \exp(-f(x))\mathrm {P} (dx)\right]^{n}}" loading="lazy"></span></dd></dl>
<p>für alle messbaren positiven Funktionen <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 f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Intensitätsmaß"><span id="Intensit.C3.A4tsma.C3.9F"></span>Intensitätsmaß</h3></div>
<p>Das <a href="Intensit%C3%A4tsma%C3%9F" title="Intensitätsmaß">Intensitätsmaß</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 \operatorname {E\zeta } }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mi mathvariant="normal">E</mi>
<mi>ζ<!-- ζ --></mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E\zeta } }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b12f8871f33bb0e1b76a0a80bb1cd1f939e8d4fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.678ex; height:2.509ex;" alt="{\displaystyle \operatorname {E\zeta } }" loading="lazy"></span> eines Binomial-Prozesses <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 \zeta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ζ<!-- ζ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"></span> ist gegeben durch
</p>
<dl><dd><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\zeta } =nP}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mi mathvariant="normal">E</mi>
<mi>ζ<!-- ζ --></mi>
</mrow>
<mo>=</mo>
<mi>n</mi>
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E\zeta } =nP}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a69ae3b38b1fa7b519f989f3d8332a259dc72d68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.917ex; height:2.509ex;" alt="{\displaystyle \operatorname {E\zeta } =nP}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerungen">Verallgemeinerungen</h2></div>
<p>Eine Verallgemeinerung der Binomial-Prozesse sind <a href="Gemischter_Binomial-Prozess" title="Gemischter Binomial-Prozess">gemischte Binomial-Prozesse</a>. Dabei wird die bei Binomial-Prozessen deterministische Anzahl der Zufallsvariablen <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}">
<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="./_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> durch eine Zufallsvariable ersetzt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Olav Kallenberg: <cite style="font-style:italic">Random Measures, Theory and Applications</cite>. Springer, Switzerland 2017, <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/10.1007/978-3-319-41598-7">10.1007/978-3-319-41598-7</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Binomial-Prozess&rft.au=Olav+Kallenberg&rft.btitle=Random+Measures%2C+Theory+and+Applications&rft.date=2017&rft.doi=10.1007%2F978-3-319-41598-7&rft.genre=book&rft.place=Switzerland&rft.pub=Springer" style="display:none"> </span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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