Consider the two statements. $$\begin{aligned} S_1 :& \quad \text{There exist random variables } X \text{ and } Y \text{ such that} \\ & \quad \left(\mathbb{E}\left[(X - \mathbb{E}(X)) (Y - \mathbb{E}(Y))\right]\right)^2 > \mathrm{Var}[X] \, \mathrm{Var}[Y] \\ S_2 :& \quad \text{For all random variables } X \text{ and } Y, \\ & \quad \mathrm{Cov}[X,Y] = \mathbb{E}\left[|X - \mathbb{E}[X]| \, |Y - \mathbb{E}[Y]|\right] \end{aligned}$$ Which one of the following choices is correct?

Let us evaluate both statements:

1. Statement $S_1$:

By the Cauchy-Schwarz inequality for random variables:

Equivalently, the squared correlation coefficient satisfies $\rho^2_{X,Y} \le 1$. Therefore, there do not exist any random variables such that $(\mathrm{Cov}(X, Y))^2 > \mathrm{Var}[X] \, \mathrm{Var}[Y]$. Hence, $S_1$ is false.

2. Statement $S_2$:

The definition of covariance between two random variables $X$ and $Y$ is:

Since covariance can be negative (for example, if $Y = -X$), whereas $\mathbb{E}\left[|X - \mathbb{E}[X]| \, |Y - \mathbb{E}[Y]|\right]$ is always non-negative, the equality does not hold in general. Hence, $S_2$ is false.

Since both statements are false, option (D) is the correct choice.