设(X,Y)服从二维正态分布,且E(X)=E(Y)=0,E(X*X)=E(Y*Y)=1,E(XY)=相关系数（r）.Z=MAX(X,Y),求E(Z)

Question

设(X,Y)服从二维正态分布,且E(X)=E(Y)=0,E(X*X)=E(Y*Y)=1,E(XY)=相关系数（r）.Z=MAX(X,Y),求E(Z)

Answer 1

另W=MIN(X,Y),故 W+Z = X+Y（因为W,Z是X,Y的排序）
X,Y都为标准正态分布
Z-W = | X-Y |,
EZ +EW = E(Z+W) =E(X+Y) = 0
EZ - EW = E(Z-W) = E | X - Y | = sqrt (2/PI) ,PI=3.14159.
从上2式知：EZ = sqrt(1/2PI)