Question 45

Analyze statement $S_1$: By definition, $\text{Cov}[X, Y] = E[(X - E[X])(Y - E[Y])]$. The Pearson correlation coefficient $\rho$ is defined as:

By the Cauchy-Schwarz inequality, $|\rho| \le 1$ for all random variables with non-zero variances, which implies $(\text{Cov}[X, Y])^2 \le \text{Var}[X]\text{Var}[Y]$. The strict inequality $(\text{Cov}[X, Y])^2 > \text{Var}[X]\text{Var}[Y]$ is mathematically impossible. Hence, $S_1$ is false.

Analyze statement $S_2$: The definition of covariance is $\text{Cov}[X, Y] = E[(X - E[X])(Y - E[Y])]$, without absolute values. The absolute-value expression $E[|X - E[X]||Y - E[Y]|]$ is always non-negative, whereas covariance can be negative (when $X$ and $Y$ are negatively correlated). Hence, $S_2$ is false.

Because both $S_1$ and $S_2$ are false, option D is correct.