Let Y = Z^2, where Z = (X - mu) / sigma, and X is a normal random variable with mean mu and variance sigma^2. The variance of Y is
A. 1
B. 2
C. 3
D. 4
GATE 2025 · Engineering Mathematics · Variance · medium
Answer: B. 2
Identify Z as a standard normal: Since X has mean mu and variance sigma^2, Z = (X - mu)/sigma has mean (mu - mu)/sigma = 0 and variance sigma^2/sigma^2 = 1. Thus Z ~ N(0,1).
Compute E[Y] = E[Z^2]: For Z ~ N(0,1): E[Z] = 0 and Var(Z) = 1, so E[Z^2] = Var(Z) + (E[Z])^2 = 1 + 0 = 1.
Compute E[Y^2] = E[Z^4] using the 4th moment of N(0,1): For Z ~ N(0,1), the k-th raw moment is E[Z^k] = (k-1)!! for even k. For k = 4: E[Z^4] = 3!! = 3. So E[Y^2] = E[Z^4] = 3.