Question

Let X,Y,Z be independent discrete random variables with E[X]=2, E[Y]=0, E[Z]=0, E[X^2]=20, E[Y^2]=E[Y^2]=16, and Var(X)=Var(Y)=Var(Z)= 16. Let A=X(Y+Z) and B=XY.
1.Find E[B].
2.Find Var(B).
3.Find E[AB].
4. are A and B independent?
5.Are A and B are conditionally independent, given X=0.
6.Are A and B are conditionally independent, given X=1.

Answer 1

1. 0
2. 320
3. 320
correct me if am wrong.

Answer 2

1. 0
2. 320
3. 0
E[A*B] = E[A] * E[B]
= E[X(Y+Z)] * E[X*Y]
= (E[X*Y + Y*Z]) * E[X]*E[Y]
= (E[X]*E[Y] + E[Y]*E[Z]) * E[X]*E[Y] since E[X]=2,E[Y]=0
= ((2 * 0) + (0 * 0) ) * 2*0 since E[Y]=0,E[Z]=0
E[AB] = 0

Answer 3

but if you solve it from this method:
E[AB]=E[{x(y+z)}*{xy}]=E[x^2.y^2+x^2.y.z] = E[x^2].E[y^2] + E[x^2].E[y].E[z]
= 20.16+0=320.

is this wrong?

Answer 4

Also, do you know the answer for 4,5, and 6?