Question

二维随机变量（X,Y）的概率密度为f(x,y)=12y² 当

Answer 1

Cov(X,Y) = E{ [X-E(X)] [Y-E(Y)] }

= E{ XY - E(X)Y - E(Y)X +E(X)E(Y) }

= E(XY) - E(X)E(Y)

ρxy = Cov(X,Y)/[√D(X)√D(Y)]

= [E(XY) - E(X)E(Y)]/[√D(X)√D(Y)]

E(X) = ∫∫xf(x)dydx = 4/5

E(Y) = ∫∫yf(x,y)dydx = 3/5

E(X²) = ∫∫x²f(x,y)dydx = 2/3

E(y²) = ∫∫y²f(x,y)dydx = 2/5

E(XY) = ∫∫xyf(x,y)dydx = 1/2

D(X) = E(X²) - E²(X) = 2/75

D(Y) = E(Y²) - E²(Y) = 1/25

所以：

ρxy = [E(XY) - E(X)E(Y)]/[√D(X)√D(Y)]

= (1/2 - 12/25)/√(2/(75*25))

= (1/2)*√(3/2）