I will appreciate any help on this (simple?) question:
Let $r$ be a real number with $0<r<1$. Set $m(n)=r(r+1)...(r+n-1)/n! $, $m(0)=1, m(1)=r, m(2)=r(r+1)/2$, e.t.c.
What is the probability distribution of the random variable X with
$$E(X^n)=m(n),\ \ \ n=0,1,... ?$$
In particular, does X admits a density? Is it discrete?
Thanks in advance,
P.S. I know that the support of X is concentrated in the interval [0,1].