设随机变量（X,Y)具有概率密度f(x)=1/8(x+y),0 设随机变量（X,Y)具有概率密度f(x,y)=1/8(x+y),0

设随机变量（X,Y)具有概率密度f(x,y)=1/8(x+y),0<x,y<2,求E(X),cov(X,Y),ρXY
f(x)=1/4*(x+1),0<x<2
f(y)=1/4*(y+1),0<y<2

EX=∫xf(x)dx=7/6
EY=∫yf(y)dy=7/6

EX^2=∫x^2f(x)dx=5/3
EY^2=∫y^2f(y)dy=5/3

DX=EX^2-(EX)^2=11/36
DY=EY^2-(EY)^2=11/36

EXY=∫∫xyf(x,y)dxdy=4/3

cov(X,Y)=EXY-EXEY=4/3-7/6*7/6=-1/36

ρXY=cov(X,Y)/√DXDY=-1/11

解毕