Factorial-moment inversion
= Factorial-moment inversion
{title2=$\Pr(X=k)=\frac1{k!}\sum_{r\ge0}\frac{(-1)^r\mathbb E(X)_{k+r}}{r!}$}
For a nonnegative integer-valued <random variable>, the displayed inversion is valid when $\mathbb E(X)_s s^k/s!\to0$. Indeed the gap between consecutive <Jordan-Bonferroni exact-occurrence inequalities> is $\mathbb E(X)_{k+m+1}/[k!(m+1)!]$, which tends to zero under this hypothesis. No exchange of an infinite signed sum with an <expectation> is needed.