Let $x$ and $y$ be random variables, not necessarily independent, that take real values in the interval $[0,1]$. Let $z = xy$ and let the mean values of $x,y,z$ be $\bar{x},\bar{y},\bar{z}$, respectively. Which one of the following statements is TRUE?

Since $x \in [0, 1]$ and $y \in [0, 1]$, we have:

Taking the expectation (mean) on both sides of this inequality:

Hence, statement (D) is always TRUE.

Regarding the other options:

• $\bar{z} = E[xy] = E[x]E[y] + \operatorname{Cov}(x, y) = \bar{x}\bar{y} + \operatorname{Cov}(x, y)$.

• Depending on whether the covariance $\operatorname{Cov}(x, y)$ is zero, positive, or negative, $\bar{z}$ can be equal to, greater than, or less than $\bar{x}\bar{y}$. Thus, none of (A), (B), or (C) is guaranteed to hold for arbitrary random variables.