题目

6.设Xsim E(0.1)，Ysim N(0,1)，rho_(XY)=0.5，求D(X+2Y).

6.设$X\sim E(0.1)$，$Y\sim N(0,1)$，$\rho_{XY}=0.5$，求$D(X+2Y)$.

题目解答

答案

已知 $X \sim Exp(0.1)$，$Y \sim N(0,1)$，$\rho_{XY} = 0.5$。计算方差： - $D(X) = \frac{1}{0.1^2} = 100$ - $D(Y) = 1$ - $Cov(X,Y) = \rho_{XY} \sqrt{D(X)} \sqrt{D(Y)} = 0.5 \times 10 \times 1 = 5$ 利用方差性质： \[ D(X+2Y) = D(X) + 4D(Y) + 4Cov(X,Y) = 100 + 4 + 20 = 124 \] **答案：** $\boxed{124}$