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GATE 2025 DA – Question 36

Probability and Statistics · Random variables, discrete and continuous distributions · 2 marks · Multiple choice

Let $Y = Z^2$, $Z = \frac{X - \mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is

  1. 1
  2. 2
  3. 3
  4. 4

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Correct answer: (B) 2

Explanation

$Z$ is a standard normal variable, with $E[Z^2] = 1$ and $E[Z^4] = 3$. So $\text{Var}(Y) = E[Z^4] - (E[Z^2])^2 = 3 - 1 = 2$. ($Y$ is a chi-square variable with 1 degree of freedom, whose variance is 2.)