GATE CS 2024 Set 2 Q.44: Probability and Statistics

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?

$\bar{z} = \bar{x}\bar{y}$

$\bar{z} \le \bar{x}\bar{y}$

$\bar{z} \ge \bar{x}\bar{y}$

$\bar{z} \le \bar{x}$

Official answer and solution

Official answer: D
From: Official GATE 2024 answer key

Solution

Use the pointwise bound $0 \le y \le 1$ and $x \ge 0$ to bound $z = xy \le x$ , then take expectations.

Since $x, y \in [0, 1]$ with probability $1$ , we have $x \ge 0$ and $y \le 1$ , which implies:

z = xy \le x \cdot 1 = x

By the monotonicity of expectation:

\bar{z} = E[z] = E[xy] \le E[x] = \bar{x}

Note that Options (A), (B), and (C) need not hold because $x$ and $y$ are not necessarily independent: if $y = x$ , $\bar{z} = E[x^2] > (E[x])^2 = \bar{x}\bar{y}$ (for non-constant $x$ ), whereas if $y = 1 - x$ , $\bar{z} = E[x(1-x)] < \bar{x}\bar{y}$ .

Final answer

Option (D)

Common trap

Assuming $E[xy] = E[x]E[y]$ when $x$ and $y$ are not independent.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 2 (CS2)
Question: 44 of 65
Section: Computer Science and Information Technology

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