@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr: a skos:ConceptScheme .
psr:-RPSD94K6-1
  skos:altLabel "cumulative distribution function"@en, "fonction de distribution cumulative"@fr ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Cumulative_distribution_function>, <https://fr.wikipedia.org/wiki/Fonction_de_r%C3%A9partition> ;
  skos:broader psr:-ZCKZW2CP-B, psr:-HTRLKD5K-S ;
  skos:prefLabel "distribution function"@en, "fonction de répartition"@fr ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:definition """En théorie des probabilités, la <b>fonction de répartition</b>, ou <b>fonction de distribution cumulative</b>, d'une variable aléatoire réelle <span class="texhtml mvar" style="font-style:italic;">X</span> est la fonction <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">F<sub><span class="texhtml mvar" style="font-style:italic;">X</span></sub></span></span> qui, à tout réel <span class="texhtml mvar" style="font-style:italic;">x</span>, associe la probabilité d’obtenir une valeur inférieure ou égale&nbsp;:
<br/>
<br/><center><span class="mwe-math-element"><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_{X}(x)=\\\\mathbb {P} (X\\\\leq x)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>F</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>X</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">P</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>X</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle F_{X}(x)=\\\\mathbb {P} (X\\\\leq x)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7ea7407411757ef5834b67afb44b0a57d3bfde" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.002ex; height:2.843ex;" alt="{\\\\displaystyle F_{X}(x)=\\\\mathbb {P} (X\\\\leq x)}"></span>.</center>
<br/>Cette fonction est caractéristique de la loi de probabilité de la variable aléatoire. Elle permet de calculer la probabilité de chaque intervalle semi-ouvert à gauche <span class="texhtml">]<i>a</i>, <i>b</i>]</span> où <span class="texhtml"><i>a</i> &lt; <i>b</i></span>, par
<br/>
<br/><center><span class="mwe-math-element"><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 {P} (X\\\\in ]a,b])=\\\\mathbb {P} (a<X\\\\leq b)=F_{X}(b)-F_{X}(a)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">P</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>X</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">P</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo>&lt;</mo>
<br/>        <mi>X</mi>
<br/>        <mo>≤<!-- ≤ --></mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mi>F</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>X</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <msub>
<br/>          <mi>F</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>X</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>a</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {P} (X\\\\in ]a,b])=\\\\mathbb {P} (a&lt;X\\\\leq b)=F_{X}(b)-F_{X}(a)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1116262237a0b52e83f8a5d5bba742202ad8bf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:46.731ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mathbb {P} (X\\\\in ]a,b])=\\\\mathbb {P} (a<X\\\\leq b)=F_{X}(b)-F_{X}(a)}"></span>.</center>
<br/>La <b>fonction de répartition</b> d'une mesure de probabilité <span class="mwe-math-element"><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 {P} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">P</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {P} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\\\\mathbb  P}"></span> définie sur la tribu borélienne <span class="mwe-math-element"><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 {B}}(\\\\mathbb {R} )}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathcal {B}}(\\\\mathbb {R} )}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b72c5154b8532f1f97a9d217a1ec867e934e772f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.031ex; height:2.843ex;" alt="{\\\\mathcal  B}(\\\\mathbb{R} )"></span> est la fonction <span class="texhtml mvar" style="font-style:italic;">F</span> qui à tout réel <span class="texhtml mvar" style="font-style:italic;">x</span> associe 
<br/>
<br/><center><span class="mwe-math-element"><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(x)=\\\\mathbb {P} (]-\\\\infty ,x]).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>F</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">P</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>        <mo>,</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle F(x)=\\\\mathbb {P} (]-\\\\infty ,x]).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d73ac113ede549a6298b0cbc694792dfdecb8bf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.676ex; height:2.843ex;" alt="F(x)={\\\\mathbb  P}(]-\\\\infty ,x])."> </center>
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Fonction_de_r%C3%A9partition">https://fr.wikipedia.org/wiki/Fonction_de_r%C3%A9partition</a>)"""@fr, """In probability theory and statistics, the <b>cumulative distribution function</b> (<b>CDF</b>) of a real-valued random variable <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>X</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle X}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span>, or just <b>distribution function</b> of <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>X</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle X}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span>, evaluated at <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span>, is the probability that <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>X</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle X}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> will take a value less than or equal to <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span>.
<br/>Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) <span class="mwe-math-element"><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\\\\colon \\\\mathbb {R} \\
ightarrow [0,1]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>F</mi>
<br/>        <mo>:<!-- : --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">]</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle F\\\\colon \\\\mathbb {R} \\
ightarrow [0,1]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c45b6faf38bb3fb300ab4678d3675afd172f56" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.719ex; height:2.843ex;" alt="{\\\\displaystyle F\\\\colon \\\\mathbb {R} \\
ightarrow [0,1]}"></span> satisfying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\lim _{x\\
ightarrow -\\\\infty }F(x)=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">lim</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>x</mi>
<br/>            <mo stretchy="false">→<!-- → --></mo>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mi>F</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lim _{x\\
ightarrow -\\\\infty }F(x)=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5624f61bba2a30c533cc18a8e50b11c65313590" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:15.033ex; height:4.009ex;" alt="{\\\\displaystyle \\\\lim _{x\\
ightarrow -\\\\infty }F(x)=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\lim _{x\\
ightarrow \\\\infty }F(x)=1}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">lim</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>x</mi>
<br/>            <mo stretchy="false">→<!-- → --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mi>F</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>1</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lim _{x\\
ightarrow \\\\infty }F(x)=1}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1653d3d823bf499f00c6de578df43b16da3db35f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:13.754ex; height:3.843ex;" alt="{\\\\displaystyle \\\\lim _{x\\
ightarrow \\\\infty }F(x)=1}"></span>.
<br/>In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to <span class="mwe-math-element"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span>. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Cumulative_distribution_function">https://en.wikipedia.org/wiki/Cumulative_distribution_function</a>)"""@en .

psr:-HTRLKD5K-S
  skos:prefLabel "statistique mathématique"@fr, "mathematical statistics"@en ;
  a skos:Concept ;
  skos:narrower psr:-RPSD94K6-1 .

psr:-ZCKZW2CP-B
  skos:prefLabel "probability theory"@en, "théorie des probabilités"@fr ;
  a skos:Concept ;
  skos:narrower psr:-RPSD94K6-1 .