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．
【正确答案】：f x(x)=∫+∞-∞f(x,y)dy= ∫+∞-∞(1/50π)ex2/50•e-(y2/50)dy =(1/50π)e-(x2/50)∫+∞-∞e-(y2/50)dy =(√2/10√π)e-(x2/50=(1/5√2π)e-(x2/50 fY(y)=∫+∞-∞f(x,y)dx=∫+∞-∞ (1/50π)e-(x2/50)∫+∞-∞e-(y2/50)dx =(1/50π)e-(y2/50)∫+∞-∞e-(x2/50)dx= (1/5√2π)e-(y2/50 ∴f(x,y)=fX(x)•fY(y) 因此X与Y相互独立．
