Exact false discovery rate under independent null p-values ($\mathrm{FDR}=\alpha m_0/m$)

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 $P_i\le \alpha r/m$ 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.