Question

概率题,设(X,Y)服从区域D{(x,y):0＜x＜1,0＜y＜x}上的均匀分布,求相关系数ρ

Answer 1

E(x)=∫[0,1]xdx∫[0,x]2dy=∫[0,1]2x^2dx=2/3

E(y)=∫[0,1]dx∫[0,x]2ydy=∫[0,1]x^2dx=1/3

E(x^2)=∫[0,1]x^2dx∫[0,x]2dy=∫[0,1]2x^3 dx=1/2

E(y^2)=∫[0,1]dx∫[0,x]2y^2dy=∫[0,1]2x^3/3dx=1/6

D(x)=E(x^2)-[E(x)]^2)=1/18

D(y)=1/18

E(xy)=∫[0,1]xdx∫[0,x]2ydy=∫[0,1]x^3dx=1/4

cov(x,y)=E(xy)-E(x)E(y)=1/36

相关系数ρ=0.5