Expected Value of RV Sums

[Half Full]

Suppose there are 3 independent random variables s.t. E[x] = E[y] = E[z] = 0 and Var[x] = Var[y] = Var[z] = 1. Calculate E[(x^2)(3Y + 3Z)^2].

Can someone take me through the steps of solving for this. I'm assuming the first step would be looking at E[(x^2)(3X+3x)^2].

 

[Half Full]

I think I've got it.

E[X^2 * (3*Y+3*Z)^2]

Since Var[X] = E[X^2] - E[X]^2 and we have Var[X]=1, E[X]=0 then: 1 = E[X^2] - (0)^2, therefore E[X^2] = 1

E[X^2 * (3*Y+3*Z)^2]
E[X^2] * E[(3*Y+3*Z)^2] -> E[X^2] = 1
E[(3*Y+3*Z)^2]
E[9Y^2 + 6*Y*Z + 9Z^2]
E[9Y^2] + E[6*Y*Z] + E[9Z^2]
9*E[Y^2] + 6*E[YZ] + 9*E[Z^2]
9*E[Y^2] + 6*E[Y]*E[Z] + 9*E[Z^2]

Now we can plug in some actual numbers: E[Y^2] = 1, E[Z^2] = 1, E[Y] = 0, E[Z] = 0

9*[1] + 6*[0]*[0] + 9*[1]
9 + 0 + 9 =18

 

[tkhunny]

No problem here:

9*E[Y^2] + 6*E[YZ] + 9*E[Z^2]
9*E[Y^2] + 6*E[Y]*E[Z] + 9*E[Z^2]

Since we know Y and Z are independent.

Are you SURE...

E[X^2 * (3*Y+3*Z)^2]
E[X^2] * E[(3*Y+3*Z)^2]

...That X^2 and (3*Y+3*Z)^2 are independent?