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GATE DA 2025 Q36: Variance of Y = Z² for a standardised normal Z

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Let Y=Z2Y = Z^2, Z=X−μσZ = \dfrac{X - \mu}{\sigma}, where XX is a normal random variable with mean μ\mu and variance σ2\sigma^2. The variance of YY is

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Answer: B (official GATE 2025 final answer key)

Concept used: Moments of the standard normal distribution: E[Z2]=1E[Z^2] = 1, E[Z4]=3E[Z^4] = 3 (so Z2Z^2 is chi-squared with 1 degree of freedom).

First step: ZZ is standard normal, so Var⁡(Y)=E[Z4]−(E[Z2])2\operatorname{Var}(Y) = E[Z^4] - (E[Z^2])^2. Use E[Z2]=1E[Z^2] = 1 and E[Z4]=3E[Z^4] = 3.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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