Consider the two statements: S_1: There exist random variables X and Y such that (E[X*Y] - E[X]*E[Y])^2 > Var[X]*Var[Y] S_2: For all random variables X and Y, E[X^2*Y^2] >= E[X^2]*E[Y^2] Which one of the following choices is correct? A. Both S_1 and S_2 are true B. S_1 is true, but S_2 is false C. S_2 is true, but S_1 is false D. Neither S_1 nor S_2 is true

GATE 2021 · Engineering Mathematics · Expectation · medium

Answer: D. Neither S_1 nor S_2 is true

1. Evaluate S_1 using Cauchy-Schwarz: By the Cauchy-Schwarz inequality applied to random variables, (E[XY] - E[X]E[Y])^2 <= Var(X)*Var(Y) always holds, with equality only when X and Y are perfectly linearly dependent. Therefore S_1, which claims strict inequality in the opposite direction, is always FALSE.
2. Evaluate S_2 with a counterexample: For independent X,Y: E[X^2*Y^2] = E[X^2]*E[Y^2] (equality, not strict). For dependent X,Y where X^2 and Y^2 have negative covariance, E[X^2*Y^2] < E[X^2]*E[Y^2]. For example, let X and Y be such that when |X| is large |Y| is small and vice versa. Thus S_2 is FALSE in general.
3. Conclusion: Neither S_1 nor S_2 is true in general. The correct answer is D.