题目
3.设二维随机变量(X,Y)的联合概率密度为f(x,y)=}8xy, & 0le xle 1, 0le yle x,0, & 其他.求cov(X,Y),ρ_(XY).
3.设二维随机变量(X,Y)的联合概率密度为
$f(x,y)=\begin{cases}8xy, & 0\le x\le 1, 0\le y\le x,\\0, & 其他.\end{cases}$
求cov(X,Y),$ρ_{XY}$.
题目解答
答案
1. **计算期望**
$E(X) = \int_0^1 \int_0^x 8x^2y \, dy \, dx = \frac{4}{5}$，
$E(Y) = \int_0^1 \int_0^x 8xy^2 \, dy \, dx = \frac{8}{15}$，
$E(XY) = \int_0^1 \int_0^x 8x^2y^2 \, dy \, dx = \frac{4}{9}$。
2. **计算协方差**
$\text{cov}(X, Y) = E(XY) - E(X)E(Y) = \frac{4}{9} - \frac{4}{5} \times \frac{8}{15} = \frac{4}{225}$。
3. **计算方差**
$D(X) = E(X^2) - [E(X)]^2 = \frac{2}{3} - \left(\frac{4}{5}\right)^2 = \frac{2}{75}$，
$D(Y) = E(Y^2) - [E(Y)]^2 = \frac{1}{3} - \left(\frac{8}{15}\right)^2 = \frac{11}{225}$。
4. **计算相关系数**
$\rho_{XY} = \frac{\text{cov}(X, Y)}{\sqrt{D(X)}\sqrt{D(Y)}} = \frac{\frac{4}{225}}{\sqrt{\frac{2}{75}}\sqrt{\frac{11}{225}}} = \frac{2\sqrt{66}}{33}$。
**答案：**
$\boxed{\text{cov}(X, Y) = \frac{4}{225}, \quad \rho_{XY} = \frac{2\sqrt{66}}{33}}$
解析
步骤 1：计算期望 $E(X)$ 和 $E(Y)$
$E(X) = \int_0^1 \int_0^x 8x^2y \, dy \, dx = \int_0^1 8x^2 \left[\frac{y^2}{2}\right]_0^x \, dx = \int_0^1 4x^4 \, dx = \frac{4}{5}$
$E(Y) = \int_0^1 \int_0^x 8xy^2 \, dy \, dx = \int_0^1 8x \left[\frac{y^3}{3}\right]_0^x \, dx = \int_0^1 \frac{8}{3}x^4 \, dx = \frac{8}{15}$
步骤 2：计算 $E(XY)$
$E(XY) = \int_0^1 \int_0^x 8x^2y^2 \, dy \, dx = \int_0^1 8x^2 \left[\frac{y^3}{3}\right]_0^x \, dx = \int_0^1 \frac{8}{3}x^5 \, dx = \frac{4}{9}$
步骤 3：计算协方差 $\text{cov}(X, Y)$
$\text{cov}(X, Y) = E(XY) - E(X)E(Y) = \frac{4}{9} - \frac{4}{5} \times \frac{8}{15} = \frac{4}{225}$
步骤 4：计算方差 $D(X)$ 和 $D(Y)$
$D(X) = E(X^2) - [E(X)]^2 = \int_0^1 \int_0^x 8x^3y \, dy \, dx - \left(\frac{4}{5}\right)^2 = \int_0^1 4x^4 \, dx - \frac{16}{25} = \frac{2}{3} - \frac{16}{25} = \frac{2}{75}$
$D(Y) = E(Y^2) - [E(Y)]^2 = \int_0^1 \int_0^x 8xy^3 \, dy \, dx - \left(\frac{8}{15}\right)^2 = \int_0^1 \frac{8}{3}x^4 \, dx - \frac{64}{225} = \frac{1}{3} - \frac{64}{225} = \frac{11}{225}$
步骤 5：计算相关系数 $\rho_{XY}$
$\rho_{XY} = \frac{\text{cov}(X, Y)}{\sqrt{D(X)}\sqrt{D(Y)}} = \frac{\frac{4}{225}}{\sqrt{\frac{2}{75}}\sqrt{\frac{11}{225}}} = \frac{2\sqrt{66}}{33}$
$E(X) = \int_0^1 \int_0^x 8x^2y \, dy \, dx = \int_0^1 8x^2 \left[\frac{y^2}{2}\right]_0^x \, dx = \int_0^1 4x^4 \, dx = \frac{4}{5}$
$E(Y) = \int_0^1 \int_0^x 8xy^2 \, dy \, dx = \int_0^1 8x \left[\frac{y^3}{3}\right]_0^x \, dx = \int_0^1 \frac{8}{3}x^4 \, dx = \frac{8}{15}$
步骤 2：计算 $E(XY)$
$E(XY) = \int_0^1 \int_0^x 8x^2y^2 \, dy \, dx = \int_0^1 8x^2 \left[\frac{y^3}{3}\right]_0^x \, dx = \int_0^1 \frac{8}{3}x^5 \, dx = \frac{4}{9}$
步骤 3：计算协方差 $\text{cov}(X, Y)$
$\text{cov}(X, Y) = E(XY) - E(X)E(Y) = \frac{4}{9} - \frac{4}{5} \times \frac{8}{15} = \frac{4}{225}$
步骤 4：计算方差 $D(X)$ 和 $D(Y)$
$D(X) = E(X^2) - [E(X)]^2 = \int_0^1 \int_0^x 8x^3y \, dy \, dx - \left(\frac{4}{5}\right)^2 = \int_0^1 4x^4 \, dx - \frac{16}{25} = \frac{2}{3} - \frac{16}{25} = \frac{2}{75}$
$D(Y) = E(Y^2) - [E(Y)]^2 = \int_0^1 \int_0^x 8xy^3 \, dy \, dx - \left(\frac{8}{15}\right)^2 = \int_0^1 \frac{8}{3}x^4 \, dx - \frac{64}{225} = \frac{1}{3} - \frac{64}{225} = \frac{11}{225}$
步骤 5：计算相关系数 $\rho_{XY}$
$\rho_{XY} = \frac{\text{cov}(X, Y)}{\sqrt{D(X)}\sqrt{D(Y)}} = \frac{\frac{4}{225}}{\sqrt{\frac{2}{75}}\sqrt{\frac{11}{225}}} = \frac{2\sqrt{66}}{33}$