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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Stichprobenfunktion</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>In der <a href="Statistik" title="Statistik">Statistik</a> fasst eine <b>Stichprobenfunktion</b>, auch <b>Stichprobenstatistik</b> oder schlicht <b>Statistik</b>, Informationen aus einer <a href="Stichprobe" title="Stichprobe">Stichprobe</a> in spezifischer Form als <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a> zusammen. Beispiele für Stichprobenfunktionen sind <a href="Sch%C3%A4tzfunktion" title="Schätzfunktion">Schätzfunktionen</a>, <a href="Pr%C3%BCfgr%C3%B6%C3%9Fe" class="mw-redirect" title="Prüfgröße">Prüfgrößen</a> (Teststatistik, Testgröße, Testfunktion) oder die Grenze eines <a href="Konfidenzintervall" title="Konfidenzintervall">Konfidenzintervalls</a>. Bekannte Stichprobenfunktionen sind das <a href="Stichprobenmittel" title="Stichprobenmittel">Stichprobenmittel</a>, die <a href="Stichprobenvarianz_(Sch%C3%A4tzfunktion)" title="Stichprobenvarianz (Schätzfunktion)">Stichprobenvarianz</a> sowie der <a href="Stichprobenmedian" title="Stichprobenmedian">Stichprobenmedian</a>. Die Wahrscheinlichkeitsverteilung einer Stichprobenfunktion heißt auch <a href="Stichprobenverteilung" title="Stichprobenverteilung">Stichprobenverteilung</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Die <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariablen</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_{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>
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<annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{n}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac794f5521dcce89913085a6d566e7cdb615dbb0.svg" 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}}" loading="lazy"></span> seien eine <a href="Zufallsstichprobe" title="Zufallsstichprobe">Stichprobe</a> des Umfangs <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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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
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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>, weiterhin sei
</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 g:\mathbb {R} ^{n}\to \mathbb {R} }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>:</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6032f72d4d3deedc8f4a9e49617e46b95bfc8e6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.242ex; height:2.676ex;" alt="{\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span></dd></dl>
<p>eine <a href="Messbare_Funktion" title="Messbare Funktion">messbare Funktion</a>. Dann heißt die Zufallsvariable
</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 G=g(X_{1},\ldots ,X_{n})}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>,</mo>
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<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle G=g(X_{1},\ldots ,X_{n})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44e2dbe6b22678879386efe4dbffd58f032ac843.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.15ex; height:2.843ex;" alt="{\displaystyle G=g(X_{1},\ldots ,X_{n})}" loading="lazy"></span></dd></dl>
<p>eine <i>Stichprobenfunktion</i>.
</p><p>Die Messbarkeit der Funktion <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 g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> garantiert, dass <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 G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> eine Zufallsvariable ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<p>In der Statistik und Wahrscheinlichkeitstheorie häufig verwendete Stichprobenfunktionen sind die Summenvariable <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 \sum _{i=1}^{n}X_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<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>
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</munderover>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}X_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c3dba4ae379a68304be2bb75ced40c0f1496bb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.466ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}X_{i}}" loading="lazy"></span>, die in diesem Zusammenhang auch <b>Stichprobensumme</b><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> heißt,
das <a href="Stichprobenmittel" title="Stichprobenmittel">Stichprobenmittel</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 {\frac {1}{n}}\sum _{i=1}^{n}X_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<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>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}X_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/245eef0f3e6d404d0a3fd32159719954c4616070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.084ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}X_{i}}" 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 {\frac {1}{n}}\sum _{i=1}^{n}X_{i}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<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>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}X_{i}^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2cad2e57b39a4279c3713e8d70d7654e5677977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.411ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}X_{i}^{2}}" 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 \max(X_{1},\dots ,X_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo movablelimits="true" form="prefix">max</mo>
<mo stretchy="false">(</mo>
<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>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \max(X_{1},\dots ,X_{n})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24f2bfc2be0a3e1e5ae5d60dec24409ab4c98dcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.435ex; height:2.843ex;" alt="{\displaystyle \max(X_{1},\dots ,X_{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 \min(X_{1},\dots ,X_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo movablelimits="true" form="prefix">min</mo>
<mo stretchy="false">(</mo>
<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>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \min(X_{1},\dots ,X_{n})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e5981d82bd3627a9ed9da1bbaf60391b56221d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.984ex; height:2.843ex;" alt="{\displaystyle \min(X_{1},\dots ,X_{n})}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Papula, Lothar: <cite style="font-style:italic">Mathematik für Ingenieure und Naturwissenschaftler</cite>. 7. Auflage. 3: <i>Vektoranalysis, Wahrscheinlichkeitsrechnung, Mathematische Statistik, Fehler- und Ausgleichsrechnung</i>. Springer Vieweg, Wiesbaden 2016, ISBN 978-3-658-11924-9.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Stichprobenfunktion&amp;rft.au=Papula%2C+Lothar&amp;rft.btitle=Mathematik+f%C3%BCr+Ingenieure+und+Naturwissenschaftler&amp;rft.date=2016&amp;rft.edition=7&amp;rft.genre=book&amp;rft.isbn=9783658119249&amp;rft.place=Wiesbaden&amp;rft.pub=Springer+Vieweg&amp;rft.volume=3%3A+Vektoranalysis%2C+Wahrscheinlichkeitsrechnung%2C+Mathematische+Statistik%2C+Fehler-+und+Ausgleichsrechnung" style="display:none">&nbsp;</span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="Horst_Rinne" title="Horst Rinne">Horst Rinne</a>: <cite style="font-style:italic">Taschenbuch der Statistik</cite>. 4. Auflage. Harri Deutsch, Frankfurt am Main 2008, ISBN 978-3-8171-1827-4, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>437</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Stichprobenfunktion&amp;rft.au=Horst+Rinne&amp;rft.btitle=Taschenbuch+der+Statistik&amp;rft.date=2008&amp;rft.edition=4&amp;rft.genre=book&amp;rft.isbn=9783817118274&amp;rft.pages=437&amp;rft.place=Frankfurt+am+Main&amp;rft.pub=Harri+Deutsch" style="display:none">&nbsp;</span></span>
</li>
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