Q: You have two i.i.d Rv's X~N(0,1) Y~(0,1). Let Z=(X+Y)^2.
a) Find the mean on Z i.e E[Z}.
b) Find Corr(X,Z) & Corr(Y,Z).
c) Determine if Z & Y are uncorrelated.
Ans: Finding E[Z] was easy. I just did the E[(X+Y)^2] = E[X^2] + 2E[X]E[Y] + E[Y^2]. Where E[Y]=E[X]=0
and Var(X) = Var(Y) = 1 = E[X^2] = E[Y^2]
solving I got E[Z] = 2.
I am stuck on part B and C however. I know the Corr(X,Z) = E[XZ] = E[(X-E[X])(Z-E[Z])] also E[XZ] = integral of XZ.fxz(x,z)dzdx for all values of z and x.
But I get stuck I am not given fxz(x,z). Specifically the variance of Z. If I know the variance of Z I can find E[XZ}. Any ideas??