GATE CS 2021 Set 1 Q35 — Probability: Random Variables and Distributions

Statement $S_1$:

By the Cauchy-Schwarz inequality for random variables, for any two real-valued random variables $U$ and $V$, we have:

Letting $U = X - \mathbb{E}[X]$ and $V = Y - \mathbb{E}[Y]$, this inequality becomes:

Thus, the strict inequality in $S_1$ can never hold for any random variables. Therefore, $S_1$ is false.

Statement $S_2$:

By definition, the covariance is given by:

This is generally not equal to the expectation of the absolute product. For a simple counterexample, let $Y = -X$ where $X$ is a non-constant random variable with mean $0$ (e.g., $X \in \{-1, 1\}$ with equal probability):

• $\text{Cov}[X, Y] = \mathbb{E}[X(-X)] = -\mathbb{E}[X^2] = -1$
• $\mathbb{E}[|X - \mathbb{E}[X]|\;|Y - \mathbb{E}[Y]|] = \mathbb{E}[|X|\;|-X|] = \mathbb{E}[X^2] = 1$

Since $-1 \neq 1$, $S_2$ is false.

Consequently, both statements $S_1$ and $S_2$ are false.