题目
9.设随机变量(X,Y)的概率密度为-|||-f(x,y)= ) 12(y)^2,0leqslant yleqslant xleqslant 1 0, yleqslant xleqslant 1).
题目解答
答案
解析
步骤 1：计算E(X)
E(X) = $\int_{0}^{1}\int_{0}^{x}x\cdot12y^{2}dydx$
步骤 2：计算E(Y)
E(Y) = $\int_{0}^{1}\int_{0}^{x}y\cdot12y^{2}dydx$
步骤 3：计算E(XY)
E(XY) = $\int_{0}^{1}\int_{0}^{x}xy\cdot12y^{2}dydx$
步骤 4：计算 $E({X}^{2}+{Y}^{2})$
$E({X}^{2}+{Y}^{2})$ = $\int_{0}^{1}\int_{0}^{x}(x^{2}+y^{2})\cdot12y^{2}dydx$
E(X) = $\int_{0}^{1}\int_{0}^{x}x\cdot12y^{2}dydx$
步骤 2：计算E(Y)
E(Y) = $\int_{0}^{1}\int_{0}^{x}y\cdot12y^{2}dydx$
步骤 3：计算E(XY)
E(XY) = $\int_{0}^{1}\int_{0}^{x}xy\cdot12y^{2}dydx$
步骤 4：计算 $E({X}^{2}+{Y}^{2})$
$E({X}^{2}+{Y}^{2})$ = $\int_{0}^{1}\int_{0}^{x}(x^{2}+y^{2})\cdot12y^{2}dydx$