= Integer-valued hypothesis in the tail-sum formula
{title2=$\sum_{m\ge0}\mathbb P(X>m)=\mathbb E\lceil X\rceil\quad(X\geq0)$}
For a nonnegative integer-valued variable, $X=\sum_{m\ge0}\mathbf1_{\{X>m\}}$ and the <tail-sum formula> gives $\mathbb E X=\sum_{m\ge0}\mathbb P(X>m)$. 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.
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