The GATE Grind

GATE 2024 CS (CS2) – Question 44

Engineering Mathematics · Probability and Statistics · 2 marks · Multiple choice

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?

  1. $\bar{z}=\bar{x}\bar{y}$
  2. $\bar{z}\le\bar{x}\bar{y}$
  3. $\bar{z}\ge\bar{x}\bar{y}$
  4. $\bar{z}\le\bar{x}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (D) $\bar{z}\le\bar{x}$

Explanation

Since y ∈ [0,1], xy ≤ x pointwise, so E[xy] ≤ E[x]. Without independence, E[xy] can be above or below E[x]E[y], so A, B, C are not always true.