The GATE Grind

GATE 2026 DA – Question 44

Probability and Statistics · Conditional expectation and variance · 2 marks · Multiple choice

Let $X$ and $Y$ be two independent random variables. $X$ follows $Bernoulli(p = 0.3)$ distribution and $Y$ follows $Normal(\mu = 0, \sigma^2 = 100)$ distribution.

Which of the following options is the variance of $(2X - 1)Y$?

  1. 100
  2. 90
  3. 49
  4. 21

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Show answer and explanation

Correct answer: (A) 100

Explanation

$2X - 1$ is $+1$ when $X = 1$ and $-1$ when $X = 0$, so $(2X - 1)^2 = 1$ always. Since $X$ and $Y$ are independent and $E[Y] = 0$, $E[(2X-1)Y] = E[2X - 1]\,E[Y] = 0$. So the variance is $E[(2X-1)^2Y^2] = E[(2X-1)^2]\,E[Y^2] = 1 \times 100 = 100$.