1. Let X,Y and Z be 3 random variables, and X and Z are independent. Prove that Cov(X+Y,Z)=Cov(Y,Z).
For this i expanded into
E(Z(X+Y)) - E(X+Y)E(Z)=E(YZ)-E(Y)E(Z)
now i am not sure what to do here.....
2. Prove that the variance of the uniform[L,R] distribution is given by the expression ((R-L)^2) / 12.
3. Let X and Y be independent, X~Bernoulli(1/2), and Y~N(0,1). Let Z=X+Y, W=X-Y. Compute Var(Z), Var(W), Cov(Z,W), Corr(Z,W).