@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:-R9K39R4D-P
  skos:prefLabel "fraction continue"@fr, "continued fraction"@en ;
  a skos:Concept ;
  skos:narrower psr:-HVMHT7QM-W .

psr:-K9FXDR6F-N
  skos:prefLabel "loi de probabilité"@fr, "probability distribution"@en ;
  a skos:Concept ;
  skos:narrower psr:-HVMHT7QM-W .

psr:-HVMHT7QM-W
  skos:prefLabel "Gauss-Kuzmin distribution"@en, "loi de Gauss-Kuzmin"@fr ;
  skos:broader psr:-K9FXDR6F-N, psr:-R9K39R4D-P ;
  a skos:Concept ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """In mathematics, the <b>Gauss–Kuzmin distribution</b> is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1). The distribution is named after Carl Friedrich Gauss, who derived it around 1800, and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929. It is given by the probability mass function  <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 p(k)=-\\\\log _{2}\\\\left(1-{\\rac {1}{(1+k)^{2}}}\\
ight)~.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>p</mi>         <mo stretchy="false">(</mo>         <mi>k</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mo>−<!-- − --></mo>         <msub>           <mi>log</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>⁡<!-- ⁡ --></mo>         <mrow>           <mo>(</mo>           <mrow>             <mn>1</mn>             <mo>−<!-- − --></mo>             <mrow class="MJX-TeXAtom-ORD">               <mfrac>                 <mn>1</mn>                 <mrow>                   <mo stretchy="false">(</mo>                   <mn>1</mn>                   <mo>+</mo>                   <mi>k</mi>                   <msup>                     <mo stretchy="false">)</mo>                     <mrow class="MJX-TeXAtom-ORD">                       <mn>2</mn>                     </mrow>                   </msup>                 </mrow>               </mfrac>             </mrow>           </mrow>           <mo>)</mo>         </mrow>         <mtext> </mtext>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p(k)=-\\\\log _{2}\\\\left(1-{\\rac {1}{(1+k)^{2}}}\\
ight)~.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87e96f84788661275bc34ed13ed06b96ec5e7ef9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:31.164ex; height:6.343ex;" alt="{\\\\displaystyle p(k)=-\\\\log _{2}\\\\left(1-{\\rac {1}{(1+k)^{2}}}\\
ight)~.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution">https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution</a>)"""@en, """En théorie des probabilités, la <b>loi de Gauss-Kuzmin</b> est une loi de probabilité discrète à support infini qui apparaît comme loi de probabilité asymptotique des coefficients dans le développement en fraction continue d'une variable aléatoire uniforme sur <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 ]0,1[}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">]</mo>         <mn>0</mn>         <mo>,</mo>         <mn>1</mn>         <mo stretchy="false">[</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ]0,1[}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6a83a50a400fb17f0c9abe6e674c6526a7b0e1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\\\\displaystyle ]0,1[}"></span></span>. Le nom provient de Carl Friedrich Gauss qui considéra cette loi en 1800, et de Rodion Kuzmin qui donna une borne pour la vitesse de convergence en 1929 par l'intermédiaire de la fonction de masse :  <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 p(k):=\\\\mathbb {P} (X=k)=-\\\\log _{2}\\\\left(1-{\\rac {1}{(1+k)^{2}}}\\
ight)~.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>p</mi>         <mo stretchy="false">(</mo>         <mi>k</mi>         <mo stretchy="false">)</mo>         <mo>:=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mi mathvariant="double-struck">P</mi>         </mrow>         <mo stretchy="false">(</mo>         <mi>X</mi>         <mo>=</mo>         <mi>k</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mo>−<!-- − --></mo>         <msub>           <mi>log</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msub>         <mo>⁡<!-- ⁡ --></mo>         <mrow>           <mo>(</mo>           <mrow>             <mn>1</mn>             <mo>−<!-- − --></mo>             <mrow class="MJX-TeXAtom-ORD">               <mfrac>                 <mn>1</mn>                 <mrow>                   <mo stretchy="false">(</mo>                   <mn>1</mn>                   <mo>+</mo>                   <mi>k</mi>                   <msup>                     <mo stretchy="false">)</mo>                     <mrow class="MJX-TeXAtom-ORD">                       <mn>2</mn>                     </mrow>                   </msup>                 </mrow>               </mfrac>             </mrow>           </mrow>           <mo>)</mo>         </mrow>         <mtext> </mtext>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle p(k):=\\\\mathbb {P} (X=k)=-\\\\log _{2}\\\\left(1-{\\rac {1}{(1+k)^{2}}}\\
ight)~.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6ef156ccec4947308ece29cb2e5d778309a163" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:44.429ex; height:6.343ex;" alt="{\\\\displaystyle p(k):=\\\\mathbb {P} (X=k)=-\\\\log _{2}\\\\left(1-{\\rac {1}{(1+k)^{2}}}\\
ight)~.}"> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Loi_de_Gauss-Kuzmin">https://fr.wikipedia.org/wiki/Loi_de_Gauss-Kuzmin</a>)"""@fr ;
  dc:created "2023-07-27"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Loi_de_Gauss-Kuzmin>, <https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution> ;
  skos:inScheme psr: .