q1.
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).
1) It is true that Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z), but they may want you to prove that.
2) That's easy, just compute the first and second moments.
Clearly the mean is the midpoint, (L+R)/2.
3) Most of this is straight forward.
Use Cov(X+Y,X-Y)=Cov(X,X)+Cov(X,-Y)+Cov(Y,X)+Cov(Y,-Y)= V(X)-Cov(X,Y)+Cov(X,Y)-V(Y)=V(X)-V(Y).