设二维随见变量（X,Y）的概率密度为f(x)=3/16xy,0

Question

设二维随见变量（X,Y）的概率密度为f(x)=3/16xy,0

Answer 1

f(x)=∫(0,x^2)(3/16)xydy=(3/16)x(y^2/2)(0,x^2)=(3/32)x^5 0≤x≤2
f(y)=∫(√y,2)(3/16)xydx=(3/16)y(x^2/2)(√y,2)=(3/32)(4y-y^2) 0≤y≤4
E(X)=∫(0,2)(3/32)x^6dx=(3/32)(1/7)2^7=12/7
E(Y)=∫(0,4)(3/32)y(4y-y^2)dy=(3/32)[(4/3)y^3-(1/4)y^4](0,4)=(3/32)[(4/3)×4^3-(1/4)4^4]=2
D(X)=E(X^2)-[E(X)]^2=∫(0,2)(3/32)x^7dx-(12/7)^2=(3/32)(1/8)2^8-144/49=3/49
D(Y)=E(Y^2)-[E(Y)]^2=∫(0,4))(3/32)y^2(4y-y^2)dy-4=(3/32)[4^4-(1/5)4^5]-4=4/5
cov(X,Y)=E(XY)-E(X)E(Y)=∫(0,2)dx∫(0,x^2)(3/16)x^2y^2dy-(12/7)×2=(32/9)-(24/7)=8/63
R(X,Y)=cov(X,Y)/[√D(X)√D(Y)]=(8/63)/[√(3/49)√(4/5)]=(4√15)/27