Find the random variable given its moments.

Question

What is the random variable whose nth moment is $ \frac{c}{c+n}$ where $c$is positive.

Answer 1

Assume that $X$ is a random variable with a PDF supported on $\mathbb{R}^+$ and given by $f(x)$.
Assuming, for some constant $c>0$, $$ \mathbb{E}[X^n]=\int_{0}^{+\infty}x^n f(x)\,dx = \frac{c}{c+n} $$ we have $$ (\mathcal{L} f)(s)=\int_{0}^{+\infty}e^{-sx}f(x)\,dx=\sum_{n\geq 0}\frac{c}{n+c}\cdot\frac{s^n(-1)^n}{n!}=\frac{\Gamma(c+1)-c\,\Gamma(c,s)}{s^c} $$ and by applying the inverse Laplace transform to both sides we get $$ f(x) = c x^{c-1}\mathbb{1}_{(0,1)}(x) $$ which we could have guessed by inspection.