More on Benjamini–Hochberg

(a) Show that if the true-null $p$-values are independent, exactly uniform, and independent of the false-null $p$-values, then $\mathrm{FDR}=m_0\alpha/m$ exactly. (b) Prove that BH rejects every hypothesis that Holm rejects. (c) When all nulls are true ($m_0=m$), show that FDR equals FWER for any procedure, so BH controls FWER in that case.