(1) 设 X, Y 是两个随机变量，且 (Cov)(X, Y) = 1，D(X) = 3，D(Y) = 4，则 D(2X - Y) = ____.(2) 若 X, Y 为两个随机变量，已知 X sim N(1, 5^2)，Y sim E(4)，且 rho = 1，则 D(X + 2Y) = ____.
(1) 设 $X, Y$ 是两个随机变量，且 $\text{Cov}(X, Y) = 1$，$D(X) = 3$，$D(Y) = 4$，则 $D(2X - Y) = \_\_\_\_$.
(2) 若 $X, Y$ 为两个随机变量，已知 $X \sim N(1, 5^2)$，$Y \sim E(4)$，且 $\rho = 1$，则 $D(X + 2Y) = \_\_\_\_$.
题目解答
答案
(1) 由方差性质公式 $D(aX + bY) = a^2D(X) + b^2D(Y) + 2ab\text{Cov}(X, Y)$，代入 $a = 2$，$b = -1$，$D(X) = 3$，$D(Y) = 4$，$\text{Cov}(X, Y) = 1$，得
$D(2X - Y) = 4 \times 3 + 1 \times 4 + 2 \times 2 \times (-1) \times 1 = 12 + 4 - 4 = 12$
答案： $\boxed{12}$
(2) 由 $X \sim N(1, 5^2)$ 得 $D(X) = 25$，由 $Y \sim E(4)$ 得 $D(Y) = \frac{1}{16}$，相关系数 $\rho = 1$ 表示完全正相关，$\text{Cov}(X, Y) = \rho \sqrt{D(X)} \sqrt{D(Y)} = \frac{5}{4}$。代入方差公式，得
$D(X + 2Y) = D(X) + 4D(Y) + 4\text{Cov}(X, Y) = 25 + 4 \times \frac{1}{16} + 4 \times \frac{5}{4} = 25 + \frac{1}{4} + 5 = \frac{121}{4}$
答案： $\boxed{\frac{121}{4}}$（或 $30.25$）