The GATE Grind

GATE 2026 EC – Question 38

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

Let $X$, $N$, $Y$ and $Z$ be random variables. The variables $X$ and $N$ are independent of each other. $X$ is uniformly distributed between -1 and 1; $N$ follows Normal distribution with zero mean and unity variance. $Y$ and $Z$ are defined as, $Y=X+N$ and $Z=X^2+N$. Which of the following pairs represents the values of correlation between $X$ and $Y$ and that between $X$ and $Z$?

  1. 1/3 and 0
  2. 1/3 and 1/9
  3. 1/3 and 1/3
  4. 1 and 0

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Show answer and explanation

Correct answer: (A) 1/3 and 0

Explanation

$E[XY]=E[X^2]+E[X]E[N]=1/3$. $E[XZ]=E[X^3]+E[X]E[N]=0$ because $X$ is symmetric about 0. So the correlations are 1/3 and 0.