Question

随机变量的数字特征.期望,方差,协方差

Answer 1

1）

E(X)= ∫(0~1)∫(0~x)2x dydx

=2x^3/3|(0~1)

=2/3

E(Y)=∫(0~1)∫(0~x)2ydydx

=x^3/3|(0~1)

=1/3

E(XY)=∫(0~1)∫(0~x)2xydydx

=x^4/4 |(0~1)

=1/4

2)

E(X^2)=∫(0~1)∫(0~x)2x^2 dydx

=x^4/2|(0~1)

=1/2

E(Y^2)=∫(0~1)∫(0~x)2y^2 dydx

=x^4/6|(0~1)

=1/6

D(X)=1/2-(2/3)^2=1/18

D(Y)=1/6-1/9=1/18

3)

Cov(X,Y)=E(XY)-E(X)E(Y)=1/4-2/9=1/36

4)pxy= Cov(X,Y)/根号(D(X)D(Y))=(1/36)/(1/18)=1/2

5)协方矩阵

Cov(X,X) Cov(X,Y)

Cov(Y,X) Cov(Y,Y)

Cov(X,X)=D(X),Cov(Y,Y)=D(Y),Cov(X,Y)=Cov(Y,X)

协方矩阵

=1/18 1/36

1/36 1/18