= Factorial of a Poisson random variable
{title2=$\mathbb E[X!]=e^{-\lambda}/(1-\lambda),\quad0<\lambda<1$}
If $X$ has a <Poisson distribution> with parameter $\lambda>0$, then the ordinary <factorial> $X!$ has finite <expectation> exactly when $\lambda<1$. Indeed, its nonnegative expectation <series> is $e^{-\lambda}\sum_{k\geq0}\lambda^k$, since $k!$ cancels the denominator of the <probability mass function>. The <geometric series> gives the formula and proves divergence for $\lambda\geq1$. This differs from a falling-factorial <moment> of fixed order, which is finite for every finite <Poisson distribution> parameter.
Back to article page
