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<title>Fuzzy-Zufallsvariable</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Fuzzy-Zufallsvariable</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>Eine <b>Fuzzy-Zufallsvariable</b>, engl. <b>Fuzzy Random Variable</b>, ist eine (scharfe) <a href="Zufallsvariable" title="Zufallsvariable">Zufallsvariable</a>, die aber nur unscharf wahrgenommen werden kann. Fuzzy-Zufallsvariablen wurden erstmals 1978 von H. Kwakernaak<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> und 1987 ausführlicher von R. Kruse und K.D. Meyer<sup id="cite_ref-Kruse_2-0" class="reference"><a href="#cite_note-Kruse-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> behandelt. Man betrachte z. B. einen Schwarm von Maifliegen (das Beispiel stammt aus<sup id="cite_ref-Kruse_2-1" class="reference"><a href="#cite_note-Kruse-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>). Das aktuelle Alter einer zufällig herausgegriffenen Fliege ist eine (positiv reellwertige) <a href="Zufallsgr%C3%B6%C3%9Fe" class="mw-redirect" title="Zufallsgröße">Zufallsgröße</a>, allerdings ist diese nicht exakt beobachtbar, weil das genaue Geburtsdatum der Maifliege nicht bekannt ist. Man wird also das Alter nur unscharf z. B. als „jung“, „mittleres Alter“ oder „alt“ wahrnehmen können.

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

<p>Seien <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 (\Omega ,{\mathfrak {B}},P)}">

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<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathfrak {B}},P)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1866fac0c01537d409be47f414c90a5fcc86a9fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.355ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathfrak {B}},P)}" loading="lazy"></span> ein <a href="Wahrscheinlichkeitsraum" title="Wahrscheinlichkeitsraum">Wahrscheinlichkeitsraum</a> 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 S}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> eine Menge von <a href="Fuzzymenge" class="mw-redirect" title="Fuzzymenge">Zugehörigkeitsfunktionen</a> auf einer Grundmenge <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 U}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span>, häufig 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 U=\mathbb {R} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7144f67be80ffe9b4b8f4d09cde56282d86443f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U=\mathbb {R} }" loading="lazy"></span>. Eine <i>Fuzzy-Zufallsvariable</i> <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 Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> ist nun eine Abbildung <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 Y|\Omega \to S}">

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<annotation encoding="application/x-tex">{\displaystyle Y|\Omega \to S}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaf06064348f9f5a66d31a1e15dbc80261450ac1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.212ex; height:2.843ex;" alt="{\displaystyle Y|\Omega \to S}" loading="lazy"></span>, wobei an <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 S}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span> bzw. an die resultierenden Zugehörigkeitsfunktionen

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<annotation encoding="application/x-tex">{\displaystyle m_{Y(\omega )}(x),\quad \omega \in \Omega ,\quad x\in U}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/340d1ac13b6894ef62b370e2fa4458ff09456a34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.598ex; height:3.176ex;" alt="{\displaystyle m_{Y(\omega )}(x),\quad \omega \in \Omega ,\quad x\in U}" loading="lazy"></span></dd></dl>

<p>noch (hier unterdrückte) Forderungen gestellt werden, die die <a href="Messbare_Funktion" title="Messbare Funktion">Messbarkeit</a> 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 Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> sichern, siehe aber<sup id="cite_ref-Kruse_2-2" class="reference"><a href="#cite_note-Kruse-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. Sei weiter <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 O}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" loading="lazy"></span> die Menge aller scharfen Zufallsvariablen 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 U}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span>, hier <i>Menge der Originale</i> genannt. Eine wesentliche Rolle, insbesondere für die Grundlagen einer Statistik mit unscharfen Daten, spielt nun die Fuzzymenge <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 O_{Y}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01c352fed89dfddfd11bb1ca2136806320d8f947.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.259ex; height:2.509ex;" alt="{\displaystyle O_{Y}}" loading="lazy"></span> aller möglichen Originale der Fuzzy-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 Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span>. Diese Fuzzymenge hat die Zugehörigkeitsfunktion

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<annotation encoding="application/x-tex">{\displaystyle m_{O_{Y}}(Z)=\inf _{\omega \in \Omega }m_{Y(\omega )}(Z(\omega )),\quad Z\in O}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf0d051d8b642d8333ac6f1fb0a02d00093c69ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.213ex; height:4.176ex;" alt="{\displaystyle m_{O_{Y}}(Z)=\inf _{\omega \in \Omega }m_{Y(\omega )}(Z(\omega )),\quad Z\in O}" loading="lazy"></span>.</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle m_{O_{Y}}(Z)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1314c4ec3a045c32528f34f53a011a8147fee39f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.198ex; height:3.009ex;" alt="{\displaystyle m_{O_{Y}}(Z)}" loading="lazy"></span> beschreibt den Grad der <a href="Possibilit%C3%A4tstheorie" title="Possibilitätstheorie">Möglichkeit</a>, 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 Z}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> ein (nichtbeobachtbares) Original 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 Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> ist. Etwas salopper ausgedrückt: Welche (scharfen) Zufallsgrößen <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_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> sind möglich, wenn man nur das unscharfe <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 Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> beobachten kann. Auf obiges Beispiel mit dem Schwarm von Maifliegen bezogen: Welche (scharfen) Alterskonstellationen sind möglich, wenn man nur Zufallsvariablen mit den Werten "jung", "mittleres Alter" und "alt" beobachten kann.

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<div class="mw-heading mw-heading2"><h2 id="Erwartungswert_einer_Fuzzy-Zufallsvariablen">Erwartungswert einer Fuzzy-Zufallsvariablen</h2></div>

<p>Der <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</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 EY}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d42c02bf52fb8d93b2ef696dc59b5a04bcba413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.549ex; height:2.176ex;" alt="{\displaystyle EY}" loading="lazy"></span> einer Fuzzy-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 Y}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> ist die Fuzzymenge 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 U}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span>, die gemäß dem sogenannten <a href="Erweiterungsprinzip_in_der_Theorie_der_Fuzzymengen" title="Erweiterungsprinzip in der Theorie der Fuzzymengen">Erweiterungsprinzip </a><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> durch

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<annotation encoding="application/x-tex">{\displaystyle m_{EY}(x)=\sup _{Z\in O:EZ=x}m_{O_{Y}}(Z),\quad x\in U}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/697ff52448e3fe21694a99fd9f7e76fa25ea9bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.573ex; height:4.509ex;" alt="{\displaystyle m_{EY}(x)=\sup _{Z\in O:EZ=x}m_{O_{Y}}(Z),\quad x\in U}" loading="lazy"></span></dd></dl>

<p>erhalten wird. Dieser Erwartungswert stimmt mit dem Erwartungswert einer <a href="Zuf%C3%A4llige_Fuzzymenge" title="Zufällige Fuzzymenge">zufälligen Fuzzymenge</a> überein.<sup id="cite_ref-Kruse_2-3" class="reference"><a href="#cite_note-Kruse-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Im Spezialfall <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 U=\mathbb {R} }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7144f67be80ffe9b4b8f4d09cde56282d86443f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U=\mathbb {R} }" loading="lazy"></span> kann <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 EY}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d42c02bf52fb8d93b2ef696dc59b5a04bcba413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.549ex; height:2.176ex;" alt="{\displaystyle EY}" loading="lazy"></span> mittels der <a href="Fuzzymenge" class="mw-redirect" title="Fuzzymenge">Alpha-Schnitte</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 (EY)_{\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle (EY)_{\alpha }}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48077f203d03916b88b718f399f67b2fd03a4df3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.642ex; height:2.843ex;" alt="{\displaystyle (EY)_{\alpha }}" loading="lazy"></span> konstruiert werden durch die Intervalle

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<annotation encoding="application/x-tex">{\displaystyle (EY)_{\alpha }=[E\inf(Y(\omega ))_{\alpha },E\sup(Y(\omega ))_{\alpha }],\quad \alpha \in (0,1]}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d5c1be5ff283bbc2d06e2d6ad43911b277250b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.385ex; height:2.843ex;" alt="{\displaystyle (EY)_{\alpha }=[E\inf(Y(\omega ))_{\alpha },E\sup(Y(\omega ))_{\alpha }],\quad \alpha \in (0,1]}" loading="lazy"></span>.</dd></dl>

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

<p>Für Fuzzy-Zufallsvariablen kann auch eine <a href="Varianz_(Stochastik)" title="Varianz (Stochastik)">Varianz</a> definiert werden.<sup id="cite_ref-Kruse_2-4" class="reference"><a href="#cite_note-Kruse-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Sie ist eine Fuzzymenge. Im Unterschied dazu ist die Varianz einer zufälligen Fuzzymenge eine reelle Zahl.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</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">Kwakernaak, H. (1978).<i> Fuzzy random variables. Part I: Definitions and theorems</i>. Information Sciences 15, 1–15</span>

</li>

<li id="cite_note-Kruse-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Kruse_2-0">a</a></sup> <sup><a href="#cite_ref-Kruse_2-1">b</a></sup> <sup><a href="#cite_ref-Kruse_2-2">c</a></sup> <sup><a href="#cite_ref-Kruse_2-3">d</a></sup> <sup><a href="#cite_ref-Kruse_2-4">e</a></sup></span> <span class="reference-text">Kruse, R. and Meyer, K.D. (1987). <i>Statistics with Vague Data</i>. Reidel, Dordrecht</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">D. Dubois and H. Prade (1980) <i>Fuzzy Sets and Systems</i>. Academic Press, New York</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Körner, R. (1997).<i> On the variance of fuzzy random variables</i>. Fuzzy Sets and Systems 92, 83–93</span>

</li>

</ol>

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

<ul><li>I. Couso, D. Dubois and Sanchez, L. (2014).<i> Random Sets and Random Fuzzy Sets as Ill-Perceived Random Variables</i>. Springer</li>

<li>I. Couso and Dubois, D. (2009). <i>On the variability of the concept of variance for fuzzy random variables</i>. IEEE Trans. Fuzzy Syst. 17, 1070–1080</li>

<li>V. Krätschmer (2001). <i>A unified approach to fuzzy random variables</i>. Fuzzy Sets and Systems 123, 1–9</li></ul></div><div><div class="zim-footer">

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