Tail-sum formula

For nonnegative $Y$, $E[Y]={\int }_0^{\infty }P(Y\ge t)dt$; for nonnegative integer $N$, $E[N]={\sum }_{k\ge 1}P(N\ge k)$.

For a nonnegative integer-valued variable, $X={\sum }_{m\ge 0}{1}_{\{X>m\}}$ and the tail-sum formula gives $\mathbb{E}X={\sum }_{m\ge 0}\mathbb{P}(X>m)$. For arbitrary positive real values, that integer-indexed sum equals $\mathbb{E}\lceil X\rceil$ instead. Two independent constants equal to $3/2$ demonstrate that the product of such tail sums need not equal the tail sum of their product: the two sides are four and three.