Logarithmic Distribution

A Continuous Distribution for a variate with probability function

$\begin{displaymath} P(x)={\log x\over b(\log b-1)-a(\log a-1)} \end{displaymath}$

and distribution function

$\begin{displaymath} D(x)={a(1-\log a)-x(1-\log x)\over a(1-\log a)-b(1-\log b)}. \end{displaymath}$

The Mean is

$\begin{displaymath} \mu={a^2(1-2\log a)-b^2(1-2\log b)\over 4[a(1-\log a)-b(1-\log b)]}, \end{displaymath}$

but higher order moments are rather messy.