Question about Independent Normal Variables

[erica1451]

1. The problem statement, all variables and given/known data
Let X,Y be two independent normal N(0,1) and N(0,2) variables, and Z=XY. Suppose that E(X^4)=3
a) Compute Var(X), Var(Y), Var(Z), Cov(X,Y), Cov(Y,Z), Var(X+Y), Var(X+Z)
b) By computing E[(X^2)(Z^2)], prove that X,Z are not independent.


2. Relevant equations
Var(X)=E(X^2)-E(X)^2
Cov(X,Y)=E[(X-E(X))(Y-E(Y))]


3. The attempt at a solution
I know that Var(X)=1, Var(Y)=2, Cov(X,Y)=0. The others I'm not sure about. I found that Var(Z)=E[(X^2)(Y^2)] but I'm not sure if E[(X^2)(Y^2)]=E[X^2]E[Y^2]? I'm having similar problems with the other parts.

[EnumaElish]

X+Y should be simple. Do you know the variance formula for addition?

I'm not sure if E[(X^2)(Y^2)]=E[X^2]E[Y^2]Does (X, Y) independent imply that their squares are independent? Can you go from "(X, Y) independent" to "(X^2, Y^2) independent," e.g. by applying FX,Y(x,y) = FX(x)FY(y)?