GermanWikipedia - /page/entry.mu

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

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<p>Das <b>gaußsche Fehlerintegral</b> (nach <a href="Carl_Friedrich_Gau%C3%9F" title="Carl Friedrich Gauß">Carl Friedrich Gauß</a>) ist die <a href="Verteilungsfunktion" title="Verteilungsfunktion">Verteilungsfunktion</a> der <a href="Standardnormalverteilung" class="mw-redirect" title="Standardnormalverteilung">Standardnormalverteilung</a>. Es wird häufig 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 \Phi }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"></span> bezeichnet und ist das Integral 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 -\infty }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"></span> bis <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 z}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> über die <a href="Dichtefunktion" title="Dichtefunktion">Dichtefunktion</a> der Normalverteilung 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 \mu =0}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3753282c0ad2ea1e7d63f39425efd13c37da3169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}" 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 \sigma =1}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma =1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f759e9b01b4c117d116da9f6d0e635b2247ee502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma =1}" loading="lazy"></span>. Da die gesamte Fläche unterhalb der Dichtekurve (auch <i>Gauß-Glocke</i> genannt) gleich 1 ist, ist der Wert des Fehlerintegrals 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 z\rightarrow \infty }">

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<annotation encoding="application/x-tex">{\displaystyle z\rightarrow \infty }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26cef5dbce37d8b390eb867322e85d8e1b4145de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.026ex; height:1.843ex;" alt="{\displaystyle z\rightarrow \infty }" loading="lazy"></span> ebenfalls 1 (siehe Abschnitt <a href="#Normierung">Normierung</a>).

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

<p>Das Fehlerintegral ist durch

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<annotation encoding="application/x-tex">{\displaystyle \Phi (z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17f9eddf6e9ed932eb024ecb08ee22169c606f0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.816ex; height:6.343ex;" alt="{\displaystyle \Phi (z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t}" loading="lazy"></span></dd></dl>

<p>definiert.

</p><p>Lässt man das Integral erst bei <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 0}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> statt bei <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 -\infty }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"></span> beginnen, so spricht man 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 \Phi _{0}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32f33de1903508a18f10c5fc11d788de19e043dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Phi _{0}}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{0}(z)={\frac {1}{\sqrt {2\pi }}}\int _{0}^{z}e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t=\Phi (z)-{\tfrac {1}{2}}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92e4461dec393e4fe69e947abaadc68c04ad1856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.709ex; height:6.343ex;" alt="{\displaystyle \Phi _{0}(z)={\frac {1}{\sqrt {2\pi }}}\int _{0}^{z}e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t=\Phi (z)-{\tfrac {1}{2}}.}" loading="lazy"></span><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></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Zusammenhang_mit_der_gaußschen_Fehlerfunktion"><span id="Zusammenhang_mit_der_gau.C3.9Fschen_Fehlerfunktion"></span>Zusammenhang mit der gaußschen Fehlerfunktion</h2></div>

<p>Durch die Substitution <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={\tfrac {t}{\sqrt {2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle x={\tfrac {t}{\sqrt {2}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0cb95aa165ba470411752a0d7e0a1c907d8a6d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.455ex; height:4.176ex;" alt="{\displaystyle x={\tfrac {t}{\sqrt {2}}}}" loading="lazy"></span> in den oben genannten Formeln und durch passende Umformungen lässt sich aus <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 \Phi }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"></span> bzw. <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 \Phi _{0}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32f33de1903508a18f10c5fc11d788de19e043dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Phi _{0}}" loading="lazy"></span> die <a href="Fehlerfunktion" title="Fehlerfunktion">Fehlerfunktion</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (z)=2\Phi ({\sqrt {2}}\,z)-1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f791f524d72a7c554c217d727d314ee074623e54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.878ex; height:3.176ex;" alt="{\displaystyle \operatorname {erf} (z)=2\Phi ({\sqrt {2}}\,z)-1}" loading="lazy"></span></dd></dl>

<p>bzw.

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (z)=2\Phi _{0}({\sqrt {2}}\,z)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7fe7bdbf8afc9ed51904acd1abfd9362045cf381.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.929ex; height:3.176ex;" alt="{\displaystyle \operatorname {erf} (z)=2\Phi _{0}({\sqrt {2}}\,z)}" loading="lazy"></span></dd></dl>

<p>herleiten.

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

<p>Das Fehlerintegral <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 \Phi (z)}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fbffdd4f2cf1c6e8daaddfddd00677b16f12809.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \Phi (z)}" loading="lazy"></span> gibt die Wahrscheinlichkeit an, dass eine standardnormalverteilte <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariable</a> einen Wert kleiner oder gleich <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 z}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> annimmt. Umgekehrt kann auch die Wahrscheinlichkeit für einen Wert größer oder gleich <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 z}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> ermittelt werden, indem man <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 \Phi (\infty )-\Phi (z)=1-\Phi (z)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi (\infty )-\Phi (z)=1-\Phi (z)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f848cfae57574333ce8a4cfebe8eda61fb7ba630.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.904ex; height:2.843ex;" alt="{\displaystyle \Phi (\infty )-\Phi (z)=1-\Phi (z)}" loading="lazy"></span> bildet.

</p><p>Als elektrotechnisches Beispiel sei ein gaußverteiltes Störrauschen der Streuung <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 \sigma =1{,}25\,\mathrm {V} }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma =1{,}25\,\mathrm {V} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/688fce9869fc744e2471a2773b10f11b49e3b38e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.693ex; height:2.509ex;" alt="{\displaystyle \sigma =1{,}25\,\mathrm {V} }" loading="lazy"></span> angenommen, das einem Übertragungskanal überlagert ist. Dieser Kanal arbeite fehlerfrei, solange die Störungen im Bereich −5 V ... +5 V liegen. Es klärt sich nun schnell die Frage, wie wahrscheinlich eine fehlerhafte Übertragung ist:

</p><p>Wahrscheinlichkeit für einen Rauschwert nicht größer als -5 V:

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<annotation encoding="application/x-tex">{\displaystyle p_{1}=\Phi (-5\,\mathrm {V} /\sigma )=\Phi (-4)=0{,}317\cdot 10^{-4}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bbaed0db678a63045087a475fbeaf23d82da62a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:41.427ex; height:3.176ex;" alt="{\displaystyle p_{1}=\Phi (-5\,\mathrm {V} /\sigma )=\Phi (-4)=0{,}317\cdot 10^{-4}.}" loading="lazy"></span></dd></dl>

<p>Wahrscheinlichkeit für einen Rauschwert mindestens gleich +5 V:

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<annotation encoding="application/x-tex">{\displaystyle p_{2}=\Phi (\infty )-\Phi (5\,\mathrm {V} /\sigma )=1-\Phi (4)=0{,}317\cdot 10^{-4}.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e5e3c8552072f0b300deb05867bedf4389efb00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:50.465ex; height:3.176ex;" alt="{\displaystyle p_{2}=\Phi (\infty )-\Phi (5\,\mathrm {V} /\sigma )=1-\Phi (4)=0{,}317\cdot 10^{-4}.}" loading="lazy"></span></dd></dl>

<p>Die Gesamtwahrscheinlichkeit für einen Übertragungsfehler ergibt sich dann aus <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=p_{1}+p_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle p=p_{1}+p_{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d1e529ac4d83aadbc1b623af5121d3b865f6834.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.645ex; height:2.343ex;" alt="{\displaystyle p=p_{1}+p_{2}}" loading="lazy"></span>

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

<p>Um die Normiertheit <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 \Phi (\infty )=1}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi (\infty )=1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48fd1d7bfcde2cfac571ca9fb7c0a4181bf7d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.072ex; height:2.843ex;" alt="{\displaystyle \Phi (\infty )=1}" loading="lazy"></span> nachzuweisen, berechnen wir

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<annotation encoding="application/x-tex">{\displaystyle I:=\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9869d9e02783a1d31189cc67aa90416146e2e4cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.053ex; height:6.009ex;" alt="{\displaystyle I:=\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t.}" loading="lazy"></span></dd></dl>

<p>Auch wenn keine <a href="Stammfunktion" title="Stammfunktion">Stammfunktion</a> des Integranden als <a href="Elementare_Funktion" title="Elementare Funktion">elementare Funktion</a> ausdrückbar ist, gibt es trotzdem mehr als ein halbes Dutzend Lösungswege, seinen Wert zu bestimmen, angefangen bei ersten Näherungen <a href="Abraham_de_Moivre" title="Abraham de Moivre">De Moivres</a> aus dem Jahr 1733 über die Arbeiten von <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a> und <a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a> aus der Zeit um 1800 bis hin zu einem gänzlich neuen Lösungsansatz S. P. Evesons aus dem Jahr 2005.<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> Einer der entscheidenden Tricks für seine Berechnung (angeblich von Poisson<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>) ist es, auf eine höhere Dimension auszuweichen und das resultierende 2D-Integrationsgebiet anders zu parametrisieren:

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I^{2}&=\left(\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}x^{2}}\mathrm {d} x\right)\left(\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}y^{2}}\mathrm {d} y\right)\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}x^{2}}e^{-{\frac {1}{2}}y^{2}}\mathrm {d} x\,\mathrm {d} y\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}\left(x^{2}+y^{2}\right)}\mathrm {d} x\,\mathrm {d} y.\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa4f278f32806043423fa0b0488d9272fb939405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:39.707ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}I^{2}&=\left(\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}x^{2}}\mathrm {d} x\right)\left(\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}y^{2}}\mathrm {d} y\right)\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}x^{2}}e^{-{\frac {1}{2}}y^{2}}\mathrm {d} x\,\mathrm {d} y\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}\left(x^{2}+y^{2}\right)}\mathrm {d} x\,\mathrm {d} y.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Grundlage für die erste Umformung ist die <a href="Lineare_Abbildung" title="Lineare Abbildung">Linearität</a> des Integrals.

</p><p>Statt längs kartesischer Koordinaten wird über <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 {R} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"></span> nun längs <a href="Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a> integriert, was der Substitution <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=r\cos \varphi ,y=r\sin \varphi }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x=r\cos \varphi ,y=r\sin \varphi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5aeb37d32f8dbbc296fabbe17b28f53fa0a29a33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.369ex; height:2.676ex;" alt="{\displaystyle x=r\cos \varphi ,y=r\sin \varphi }" loading="lazy"></span> und daraus <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 r^{2}=x^{2}+y^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle r^{2}=x^{2}+y^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdb2a5f9d5d6102af609d3730471b017b2ac7ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.64ex; height:3.009ex;" alt="{\displaystyle r^{2}=x^{2}+y^{2}}" loading="lazy"></span> entspricht, und man erhält schließlich mit dem <a href="Transformationssatz" title="Transformationssatz">Transformationssatz</a>

</p>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I^{2}&=\int _{0}^{\infty }\int _{0}^{2\pi }e^{-{\frac {1}{2}}r^{2}}r\,\mathrm {d} \varphi \,\mathrm {d} r\\&=2\pi \int _{0}^{\infty }e^{-{\frac {1}{2}}r^{2}}r\,\mathrm {d} r\\&=-2\pi \left[e^{-{\frac {1}{2}}r^{2}}\right]_{r=0}^{\infty }\\&=2\pi .\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff854f439ee112e947bd1cbd39c38eaebcee24b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:28.342ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}I^{2}&=\int _{0}^{\infty }\int _{0}^{2\pi }e^{-{\frac {1}{2}}r^{2}}r\,\mathrm {d} \varphi \,\mathrm {d} r\\&=2\pi \int _{0}^{\infty }e^{-{\frac {1}{2}}r^{2}}r\,\mathrm {d} r\\&=-2\pi \left[e^{-{\frac {1}{2}}r^{2}}\right]_{r=0}^{\infty }\\&=2\pi .\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Damit erhalten wir:

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<annotation encoding="application/x-tex">{\displaystyle \lim _{z\to \infty }\Phi (z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t={\frac {1}{\sqrt {2\pi }}}I=1.}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/430f510fba3a8bb5e0efae02613e145b2ddfe706.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:44.703ex; height:6.343ex;" alt="{\displaystyle \lim _{z\to \infty }\Phi (z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t={\frac {1}{\sqrt {2\pi }}}I=1.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Tabelle_Standardnormalverteilung" class="mw-redirect" title="Tabelle Standardnormalverteilung">Tabelle Standardnormalverteilung</a></li></ul>

<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"><a href="Ilja_Nikolajewitsch_Bronstein" class="mw-redirect" title="Ilja Nikolajewitsch Bronstein">Ilja Nikolajewitsch Bronstein</a>, <a href="Konstantin_Adolfowitsch_Semendjajew" title="Konstantin Adolfowitsch Semendjajew">Konstantin Adolfowitsch Semendjajew</a>: <cite style="font-style:italic"><a href="Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a></cite>. 5. Auflage. Verlag Harri Deutsch, 2000, ISBN 3-8171-2005-2, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>477</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:Fehlerintegral&rft.au=Ilja+Nikolajewitsch+Bronstein%2C+Konstantin+Adolfowitsch+Semendjajew&rft.btitle=Taschenbuch+der+Mathematik&rft.date=2000&rft.edition=5.&rft.genre=book&rft.isbn=3817120052&rft.pages=477&rft.pub=Verlag+Harri+Deutsch" 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"><a rel="nofollow" class="external text" href="https://www.york.ac.uk/depts/maths/histstat/normal_history.pdf">Peter M. Lee: <i>The probability integral</i>; University of York, Department of Mathematics, 2011</a>, zuletzt abgerufen am 14. Mai 2016.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://ems.press/content/serial-article-files/4007">Denis Bell: <i>Poisson’s remarkable calculation - a method or a trick?</i>; Elemente der Mathematik 65, 2010</a> (PDF; 248 kB)</span>

</li>

</ol></div><div><div class="zim-footer">

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