= Exact false discovery rate under independent null p-values
{title2=$\operatorname{FDR}=\alpha m_0/m$}
For the <Benjamini-Hochberg procedure>, replace a true-null <p-value> $P_i$ by zero and call the new rejection count $R^{(i)}$. The <Benjamini-Hochberg leave-one-out identity> identifies rejection with $R=r$ with the event $P_i\leq\alpha r/m$ and $R^{(i)}=r$. If $P_i$ is uniform and independent of all the other <p-values>, each possible rejection count contributes $\alpha\mathbb P(R^{(i)}=r)/m$ to its expected false-discovery fraction. Summation gives $\alpha/m$ per true null and the displayed exact <false discovery rate>. Super-uniform independent nulls yield the corresponding upper bound.
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