GATE CS 2024 Set 2 — Question 44

We are given that $x, y \in [0, 1]$.
We want to compare $\bar{z} = E[xy]$ with $\bar{x} = E[x]$.
Since $y \in [0, 1]$, we have $y \leq 1$.
Since $x \in [0, 1]$, $x$ is non-negative ($x \geq 0$).
Multiplying the inequality $y \leq 1$ by $x$ (which is non-negative) preserves the inequality:
$xy \leq x$
Taking the expectation on both sides (expectation is a linear and monotonic operator):
$E[xy] \leq E[x]$
Substituting the given notation:
$\bar{z} \leq \bar{x}$Why other options are incorrect:
(A) $\bar{z} = \bar{x}\bar{y}$ is true only if $x$ and $y$ are uncorrelated (e.g., independent). The problem states they are "not necessarily independent".
(B) & (C) The relationship between $E[xy]$ and $E[x]E[y]$ depends on the covariance $Cov(x,y) = E[xy] - E[x]E[y]$. If $x,y$ are positively correlated, $E[xy] > E[x]E[y]$. If negatively correlated, $E[xy] < E[x]E[y]$. Since the correlation is unknown, neither inequality holds universally.