= Exponential-logarithmic distribution
{title2=$f_{p,\beta}(x)=-\frac{\beta}{\log p}\frac{(1-p)e^{-\beta x}}{1-(1-p)e^{-\beta x}}$}
For $0<p<1$, $\beta>0$ and $x>0$, this <probability density function> has <cumulative distribution function> $F(x)=1-\log(1-(1-p)e^{-\beta x})/\log p$. Differentiating gives the density, while the endpoint limits are zero and one. Its <quantile function> is $F^{-1}(u)=\beta^{-1}[\log(1-p)-\log(1-p^{1-u})]$ for $0<u<1$, enabling <inverse transform sampling>.
Expanding the denominator as a <geometric series> gives
$$
f_{p,\beta}(x)=-\frac{\beta}{\log p}\sum_{k\geq1}(1-p)^ke^{-k\beta x}.
$$
Termwise integration, justified by the <monotone convergence theorem>, and the <gamma function> integral yield <raw moments> $\mathbb E[X^j]=-j!\operatorname{Li}_{j+1}(1-p)/(\beta^j\log p)$. Thus the <polylogarithm> expresses its <mean>, <variance> and higher <moments>. Here $p$ is the complement of the series parameter $1-p$; parameter conventions must be checked when comparing formulas.
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