Let Z and W be random variables, not necessarily independent, that take real values in the interval [0, 1]. Let mu_Z = E[Z] and mu_W = E[W] be the mean values of Z and W respectively. Which one of the following statements is TRUE? A. Z <= mu_W B. Z <= mu_Z C. E[ZW] >= mu_Z * mu_W D. E[ZW] <= (mu_Z + mu_W) / 2

GATE 2024 · Engineering Mathematics · Statistics · medium

Answer: D. E[ZW] <= (mu_Z + mu_W) / 2 is always true for Z, W in [0,1].

1. Establish pointwise upper bound on ZW: Since 0 <= W <= 1 and 0 <= Z <= 1: ZW <= Z and ZW <= W. Therefore ZW <= min(Z, W).
2. Apply AM bound: min <= average: ZW <= min(Z,W) <= (Z+W)/2. So ZW <= (Z+W)/2 pointwise.
3. Take expectations: E[ZW] <= E[(Z+W)/2] = (E[Z] + E[W])/2 = (mu_Z + mu_W)/2. Statement D is proven.
4. Eliminate other options: A and B are false: a random variable can exceed its mean (take Z = 1, mu_Z = 0.5). C is false: if Z and W are independent uniform on [0,1], E[ZW] = 1/4 = mu_Z * mu_W (equality), but with negative dependence E[ZW] < mu_Z * mu_W.