The problem points to connections between methods that control the FDP in the sense of (9.49) and methods that control its expected value, the FDR.

(i) Show, for any random variable X on [0, 1], we have E(X) − γ

1 − γ

≤ P{X > γ} ≤

E(X)

γ .

(ii) Apply the above to X = FDP to show that if a method controls the FDR at level q, then it controls the FDP in the sense P{FDP > γ} ≤ q/γ. Also, if the FDP is controlled in the sense of (9.49), then the FDR is controlled at level α(1 − γ) + γ, which is ≥ α but typically only slightly. (These crude arguments suggest that control of the FDP is perhaps more stringent than control of the FDR.)