@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-BL62N1W1-H
  skos:definition """En théorie des probabilités, les loi arc sinus est un ensemble de lois de probabilité à densité dont le support est un intervalle compact. Elles sont un cas particulier de la loi bêta. Les lois arc sinus sont des résultats des marches aléatoires linéaires (en dimension 1) modélisant le mouvement brownien. Plus précisément, elles modélisent le processus de Wiener. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Loi_arc_sinus">https://fr.wikipedia.org/wiki/Loi_arc_sinus</a>)"""@fr, """In probability theory, the <b>arcsine distribution</b> is the probability distribution whose cumulative distribution function involves the arcsine and the square root:  <dl><dd><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)={\\rac {2}{\\\\pi }}\\\\arcsin \\\\left({\\\\sqrt {x}}\\
ight)={\\rac {\\\\arcsin(2x-1)}{\\\\pi }}+{\\rac {1}{2}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>F</mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>2</mn>             <mi>π<!-- π --></mi>           </mfrac>         </mrow>         <mi>arcsin</mi>         <mo>⁡<!-- ⁡ --></mo>         <mrow>           <mo>(</mo>           <mrow class="MJX-TeXAtom-ORD">             <msqrt>               <mi>x</mi>             </msqrt>           </mrow>           <mo>)</mo>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>arcsin</mi>               <mo>⁡<!-- ⁡ --></mo>               <mo stretchy="false">(</mo>               <mn>2</mn>               <mi>x</mi>               <mo>−<!-- − --></mo>               <mn>1</mn>               <mo stretchy="false">)</mo>             </mrow>             <mi>π<!-- π --></mi>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>2</mn>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle F(x)={\\rac {2}{\\\\pi }}\\\\arcsin \\\\left({\\\\sqrt {x}}\\
ight)={\\rac {\\\\arcsin(2x-1)}{\\\\pi }}+{\\rac {1}{2}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545a661772241694be9ee2bc39f05870cd73cb97" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.61ex; height:5.676ex;" alt="{\\\\displaystyle F(x)={\\rac {2}{\\\\pi }}\\\\arcsin \\\\left({\\\\sqrt {x}}\\
ight)={\\rac {\\\\arcsin(2x-1)}{\\\\pi }}+{\\rac {1}{2}}}"></span></dd></dl> for 0 ≤ <i>x</i> ≤ 1, and whose probability density function is  <dl><dd><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)={\\rac {1}{\\\\pi {\\\\sqrt {x(1-x)}}}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mrow>               <mi>π<!-- π --></mi>               <mrow class="MJX-TeXAtom-ORD">                 <msqrt>                   <mi>x</mi>                   <mo stretchy="false">(</mo>                   <mn>1</mn>                   <mo>−<!-- − --></mo>                   <mi>x</mi>                   <mo stretchy="false">)</mo>                 </msqrt>               </mrow>             </mrow>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f(x)={\\rac {1}{\\\\pi {\\\\sqrt {x(1-x)}}}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500fbdb509236e85253d8bb0bf7087c022e8fe63" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.479ex; height:6.509ex;" alt="{\\\\displaystyle f(x)={\\rac {1}{\\\\pi {\\\\sqrt {x(1-x)}}}}}"></span></dd></dl> on (0, 1). The standard arcsine distribution is a special case of the beta distribution with <i>α</i> = <i>β</i> = 1/2. That is, if <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}">   <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\\\\displaystyle X}"></span> is an arcsine-distributed random variable, then <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\\\\sim {\\
m {Beta}}{\\igl (}{\\	frac {1}{2}},{\\	frac {1}{2}}{\\igr )}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>X</mi>         <mo>∼<!-- ∼ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">B</mi>             <mi mathvariant="normal">e</mi>             <mi mathvariant="normal">t</mi>             <mi mathvariant="normal">a</mi>           </mrow>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mrow class="MJX-TeXAtom-OPEN">             <mo maxsize="1.2em" minsize="1.2em">(</mo>           </mrow>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="false" scriptlevel="0">             <mfrac>               <mn>1</mn>               <mn>2</mn>             </mfrac>           </mstyle>         </mrow>         <mo>,</mo>         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="false" scriptlevel="0">             <mfrac>               <mn>1</mn>               <mn>2</mn>             </mfrac>           </mstyle>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mrow class="MJX-TeXAtom-CLOSE">             <mo maxsize="1.2em" minsize="1.2em">)</mo>           </mrow>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle X\\\\sim {\\
m {Beta}}{\\igl (}{\\	frac {1}{2}},{\\	frac {1}{2}}{\\igr )}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f202d20c0ad661e20efc12b7879fb32250179c22" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.303ex; height:3.509ex;" alt="{\\\\displaystyle X\\\\sim {\\
m {Beta}}{\\igl (}{\\	frac {1}{2}},{\\	frac {1}{2}}{\\igr )}}"></span>. By extension, the arcsine distribution is a special case of the Pearson type I distribution.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Arcsine_distribution">https://en.wikipedia.org/wiki/Arcsine_distribution</a>)"""@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Arcsine_distribution>, <https://fr.wikipedia.org/wiki/Loi_arc_sinus> ;
  dc:created "2023-07-28"^^xsd:date ;
  skos:prefLabel "loi arc sinus"@fr, "arcsine distribution"@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-K9FXDR6F-N ;
  dc:modified "2024-10-18"^^xsd:date .

psr:-K9FXDR6F-N
  skos:prefLabel "loi de probabilité"@fr, "probability distribution"@en ;
  a skos:Concept ;
  skos:narrower psr:-BL62N1W1-H .