设二维随机变量(X，Y)服从二维正态分布，其概率密度为
f(x,y)=1/(50π)e-(x2+y2)/50
证明X与Y相互独立．
证明：fX(x) =∫+∞-∞f(x,y)dy =∫+∞-∞1/(50π)e-(x2+y2)/50dy =1/(50π)∫+∞-∞e-y2/50dy =1/[5√(2π)]e-x2/50 fY(y) =∫+∞-∞f(x,y)dx =∫+∞-∞1/(50π)e-(x2+y2)/50dx =1/(50π)e-(y2)/50∫+∞-∞e-(x2/50)dx =1/[5√(2π)]e-y2/50 ∴f(x,y)=fX(x)•fY(y) ∴X与Y相互独立．
