已知随机变量X服从标准正态分布N(0，1)，令Y=X3，求X与
Y的相关系数pXY
【正确答案】：E(XY)=E(X4) =(1/√2π)∫+∞-∞x4e-(x2/2)dx=(1/√2π)∫+∞-∞x3de-(x2/2) =-(1/√2π)x3e-(x2/2)∣+∞-∞+(3/2π)∫+∞-∞x2e -(x2/2)dx=(3/√2π)∫+∞-∞x2e-(x2/2)dx=3 D(X)=1， D(Y)=E(Y2)-(E(Y))2 =E(X6)-(E(X3))2． 而E(X6)=(1/√2π)∫+∞-∞x6•e-(x2/2)dx=15， E(X3)=(3/√2π)∫+∞-∞x2e-(x2/2)dx=0． 则D(Y)=1 5， pXY=Cov(X,Y)/√D(X)√D(Y)=[E(XY)-E(X)E(Y)]/√D(X)√D(Y)=(3-0)/(1•√15)=√15/5 注意：当X—N(μ，σ)，则有 (1)μ2m-1=0(m=1，2，…)， (2)μ2m=(2m-1)(2m-3)…3•σ2m(m=1，2…)．
