Discussion
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,
Answer: B
Y is normally distributed with mean 0 and variance 5.
Any linear combination of independent normal variables is itself normal. For Y = X1 - X2:
\[ \begin{aligned} E[Y] &= \mu_{1} - \mu_{2} = 1 - 1 = 0 \\ Var[Y] &= \sigma_{1}^{2} + \sigma_{2}^{2} = 1 + 4 = 5 \end{aligned} \]
Variances add even when the variables are subtracted, because Var(-X2) = Var(X2). Variance 1 comes from subtracting the variances (or the standard deviations), and the options saying Y is not normal deny the closure of the normal family under linear combination.
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