Project ForgeEngineering Mathematics
Calculus
GATE 2018
Medium
Let X1, X2 be two independent normal random variables with means μ1, μ2 and standard deviations σ1, σ2, respectively. Consider Y = X1 − X2; μ1 = μ2 = 1, σ1 = 1, σ2 = 2. Then,
- AY is normally distributed with mean 0 and variance 1
- Y is normally distributed with mean 0 and variance 5
- CY has mean 0 and variance 5, but is NOT normally distributed
- DY has mean 0 and variance 1, but is NOT normally distributed
Explanation
A linear combination of independent normal random variables is itself normal. Mean = μ1−μ2 = 1−1 = 0. Variance of a difference of independent variables adds: Var(Y) = σ1²+σ2² = 1+4 = 5. So Y ~ N(0, 5).
Source: GATE ME 2018 Session 1 (Set 1) (2018)
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