@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:-K9FXDR6F-N
  skos:prefLabel "loi de probabilité"@fr, "probability distribution"@en ;
  a skos:Concept ;
  skos:narrower psr:-NLD0HQRC-8 .

psr:-NLD0HQRC-8
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """In probability and statistics, the <b>logarithmic distribution</b> (also known as the <b>logarithmic series distribution</b> or the <b>log-series distribution</b>) is a discrete probability distribution derived from the Maclaurin series expansion  <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 -\\\\ln(1-p)=p+{\\rac {p^{2}}{2}}+{\\rac {p^{3}}{3}}+\\\\cdots .}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo>−<!-- − --></mo>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mn>1</mn>         <mo>−<!-- − --></mo>         <mi>p</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>p</mi>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <msup>               <mi>p</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msup>             <mn>2</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <msup>               <mi>p</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>3</mn>               </mrow>             </msup>             <mn>3</mn>           </mfrac>         </mrow>         <mo>+</mo>         <mo>⋯<!-- ⋯ --></mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle -\\\\ln(1-p)=p+{\\rac {p^{2}}{2}}+{\\rac {p^{3}}{3}}+\\\\cdots .}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/151b41df7c7713cf74722bfa195b99380a126dce" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.782ex; height:5.676ex;" alt="{\\\\displaystyle -\\\\ln(1-p)=p+{\\rac {p^{2}}{2}}+{\\rac {p^{3}}{3}}+\\\\cdots .}"></span></dd></dl> From this we obtain the identity  <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 \\\\sum _{k=1}^{\\\\infty }{\\rac {-1}{\\\\ln(1-p)}}\\\\;{\\rac {p^{k}}{k}}=1.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <munderover>           <mo>∑<!-- ∑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>k</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="normal">∞<!-- ∞ --></mi>           </mrow>         </munderover>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo>−<!-- − --></mo>               <mn>1</mn>             </mrow>             <mrow>               <mi>ln</mi>               <mo>⁡<!-- ⁡ --></mo>               <mo stretchy="false">(</mo>               <mn>1</mn>               <mo>−<!-- − --></mo>               <mi>p</mi>               <mo stretchy="false">)</mo>             </mrow>           </mfrac>         </mrow>         <mspace width="thickmathspace"></mspace>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <msup>               <mi>p</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>k</mi>               </mrow>             </msup>             <mi>k</mi>           </mfrac>         </mrow>         <mo>=</mo>         <mn>1.</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sum _{k=1}^{\\\\infty }{\\rac {-1}{\\\\ln(1-p)}}\\\\;{\\rac {p^{k}}{k}}=1.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2409883844df6705a2f66303c493d7116b29d85" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.146ex; height:6.843ex;" alt="{\\\\displaystyle \\\\sum _{k=1}^{\\\\infty }{\\rac {-1}{\\\\ln(1-p)}}\\\\;{\\rac {p^{k}}{k}}=1.}"></span></dd></dl> This leads directly to the probability mass function of a Log(<i>p</i>)-distributed random variable:  <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(k)={\\rac {-1}{\\\\ln(1-p)}}\\\\;{\\rac {p^{k}}{k}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>f</mi>         <mo stretchy="false">(</mo>         <mi>k</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo>−<!-- − --></mo>               <mn>1</mn>             </mrow>             <mrow>               <mi>ln</mi>               <mo>⁡<!-- ⁡ --></mo>               <mo stretchy="false">(</mo>               <mn>1</mn>               <mo>−<!-- − --></mo>               <mi>p</mi>               <mo stretchy="false">)</mo>             </mrow>           </mfrac>         </mrow>         <mspace width="thickmathspace"></mspace>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <msup>               <mi>p</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mi>k</mi>               </mrow>             </msup>             <mi>k</mi>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle f(k)={\\rac {-1}{\\\\ln(1-p)}}\\\\;{\\rac {p^{k}}{k}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f516a37604b70dfd0c891c11ea0939177e0b147d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.894ex; height:6.509ex;" alt="{\\\\displaystyle f(k)={\\rac {-1}{\\\\ln(1-p)}}\\\\;{\\rac {p^{k}}{k}}}"></span></dd></dl> for <i>k</i> ≥ 1, and where 0 &lt; <i>p</i> &lt; 1.  Because of the identity above, the distribution is properly normalized. The cumulative distribution 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(k)=1+{\\rac {\\\\mathrm {B} (p;k+1,0)}{\\\\ln(1-p)}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>F</mi>         <mo stretchy="false">(</mo>         <mi>k</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mn>1</mn>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mrow class="MJX-TeXAtom-ORD">                 <mi mathvariant="normal">B</mi>               </mrow>               <mo stretchy="false">(</mo>               <mi>p</mi>               <mo>;</mo>               <mi>k</mi>               <mo>+</mo>               <mn>1</mn>               <mo>,</mo>               <mn>0</mn>               <mo stretchy="false">)</mo>             </mrow>             <mrow>               <mi>ln</mi>               <mo>⁡<!-- ⁡ --></mo>               <mo stretchy="false">(</mo>               <mn>1</mn>               <mo>−<!-- − --></mo>               <mi>p</mi>               <mo stretchy="false">)</mo>             </mrow>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle F(k)=1+{\\rac {\\\\mathrm {B} (p;k+1,0)}{\\\\ln(1-p)}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c363e4b101fa17b65252c27727e9429e07a1357" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.767ex; height:6.509ex;" alt="{\\\\displaystyle F(k)=1+{\\rac {\\\\mathrm {B} (p;k+1,0)}{\\\\ln(1-p)}}}"></span></dd></dl> where <i>B</i> is the incomplete beta function. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Logarithmic_distribution">https://en.wikipedia.org/wiki/Logarithmic_distribution</a>)"""@en, """En probabilité et en statistiques, la loi logarithmique est une loi de probabilité discrète, dérivée du développement de Taylor de la fonction logarithme népérien. En anglais, cette loi est plutôt appelée logarithmic series distribution ou log-series distribution, pour éviter la confusion avec les lois dont les variables sont les logarithmes de variables suivant d'autres lois, comme la loi log-normale. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Loi_logarithmique">https://fr.wikipedia.org/wiki/Loi_logarithmique</a>)"""@fr ;
  skos:broader psr:-K9FXDR6F-N ;
  skos:prefLabel "loi logarithmique"@fr, "logarithmic distribution"@en ;
  skos:altLabel "log-series distribution"@en, "logarithmic series distribution"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Logarithmic_distribution>, <https://fr.wikipedia.org/wiki/Loi_logarithmique> ;
  a skos:Concept ;
  skos:inScheme psr: ;
  dc:created "2023-07-27"^^xsd:date .