Question

设二维随机变量（X,Y）的联合概率密度为 f(x,y)={cx^2y 0<x<1 0<y<1} 0 其他 求：(1)常数c; (2) P{X<0.5,Y<

设二维随机变量（X,Y）的联合概率密度为
f(x,y)={cx^2y 0<x<1 0<y<1}
0 其他

求：(1)常数c; (2) P{X<0.5,Y<0.5}; (3) X的边缘概率密度;
(4) EX; (5) 协方差Cov(X,Y)

Answer 1

(1)F(X,Y)=f(0,1)f(01)cx^2ydydx=c/2f(0,1)x^2dx=c/6x^3(0,1)=c/6=1
c=6
(2)P{X<0.5,Y<0.5}=f(0,0.5)F(0,0.5)6x^2ydydx=f(0,1)3/4x^2dx=1/32
(3)fx(X)=f(0,1)6x^2ydy=3x^2
(4)E(X)=f(0,1)xfx(X)dx=f(0,1)3x^3dx=3/4
(5)cov(x,y)=E(XY)-E(X)E(Y)后面的自己算啊，有事要走开