Question

- 联合机率密度共三问.jiont pdf:联合几率密度independent random variable:相互独立

Answer 1

1.∫∫c*exp(-(x^2+y^2-2x-8y)dxdy=1==>c=e^(-17)/2π

2.f(x,y)=(1/2π)^(1/2)exp(-(x-1)^2/2)*(1/2π)^(1/2)exp(-(y-4)^2/2)

=f(x)*f(y) Xand Y are independent random variable

N(1,1),N(4,1)

3.N(4,1)==>(Y-4)~N(0,1)==>(Y-4)^2~Γ(-1/2,2)==>E[Y-4)^2]=2(-1/2+1)=1

==>E[Y-4)^2]=E(Y^2)-8E(Y)+16=1==>E(Y^2)=17

forthe same reason E(X^2)=1

==>E(Y^2+X^2Y)=E(Y^2)+E(X^2Y)=17+1*4=21