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GATE DA 2026 Q44: Variance of (2X − 1)Y with X Bernoulli and Y normal

Question

Let XX and YY be two independent random variables. XX follows Bernoulli(p=0.3)Bernoulli(p = 0.3) distribution and YY follows Normal(μ=0,σ2=100)Normal(\mu = 0, \sigma^2 = 100) distribution.

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

Try it: multiple choice: exactly one correct option

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Answer: A (official GATE 2026 final answer key)

Concept used: For independent variables, E[(2X−1)2Y2]=E[(2X−1)2] E[Y2]E[(2X-1)^2 Y^2] = E[(2X-1)^2]\,E[Y^2].

First step: 2X−12X - 1 is always ±1\pm 1, so (2X−1)2=1(2X-1)^2 = 1; and E[(2X−1)Y]=0E[(2X - 1)Y] = 0 because E[Y]=0E[Y] = 0.

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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