= Jordan-Bonferroni exact-occurrence inequalities
{c}
{title2=$T_{2s+1}\le\Pr(N=k)\le T_{2s}$}
For the number $N$ of occurring events, write $S_j=\mathbb E\binom Nj$ and $T_m=\sum_{r=0}^m(-1)^r\binom{k+r}kS_{k+r}$. Then odd truncations bound $\Pr(N=k)$ below and even truncations above. For each integer $N>k$, the truncated binomial sum equals $(-1)^m\binom Nk\binom{N-k-1}m$ when $m<N-k$, and is zero otherwise. Its sign proves the bounds on taking <expectations>. This is the exact-count version of the <Bonferroni inequalities>.
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