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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, σl=1, σ2=2. Then.

(a) Y is normal distributed with mean 0 and variance 1

(b) Y is normally distributed with mean 0 and variance 5

(c) Y has mean 0 and variance 5, but is NOT normally distributed

(d) Y has mean 0 and variance 1, but is NOT normally distributed

## 1 Answer

$var(X_1) = \sigma^2_1 = 1$

$var(X_2) = \sigma^2_2 = 4$

$\mu_1 = \mu_2 =1$

$X_1 \sim$ $N (\mu_1 , \sigma^2_1 )$

$X_2 \sim$ $N (\mu_2 , \sigma^2_2 )$

$Y = X_1 - X_2$

$E[X_1 - X_2] = E[X_1] - E[X_2] = \mu_1 - \mu_2 = 0$

$Var(X_1 - X_2) = Var(X_1) + Var(X_2) = 1 + 4 = 5$

$Y \sim$ $N (0 , 5 )$

Option B is correct

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