Expected value of a factorial number

Question

If $X$ ~ $Pois(\lambda) $, what is the expected value of $X!$? And what would be the expected value of $2^X$?

Answer 1

$n!$ is strictly increasing. Hence $X!=n!$ iff $X=n$. Thus $X!$ takes the values $0!,1!,2!,...$ with probabilities $e^{-\lambda},e^{-\lambda}\frac \lambda {1!},e^{-\lambda}\frac {(\lambda)^{2}} {2!},...$. Hence $EX=\frac {e^{-\lambda}} {1-\lambda}$ if $0<\lambda <1$ and $\infty$ otherwise. $E2^{X}=\sum_0 ^{\infty} e^{-\lambda} \frac {(2\lambda)^{n}} {n!}=e^{\lambda}$