Is there a closed form for $E(Y^n)$, where $Y$ is a random variable with a gamma distribution with parameters $\alpha$, $\beta$?



Expected value of a gamma-distributed random variable to the n-th power?
If the shape parameter is $\alpha$ and the scale parameter $\beta$, then $E(Y^r) = \beta^r \Gamma(\alpha + r)/\Gamma(\alpha)$ for real $r > 0$.


