# Thread: help with expected value

1. ## help with expected value

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).

2. q1.

$Cov(X+Y,Z) = E[(X+Y)Z]-E[(X+Y)]E(Z)$

$= E(XZ)+E(YZ)-[E(X)+E(Y)]E(Z)$

$= E(XZ)+E(YZ)-E(X)E(Z)-E(Y)E(Z)$

$=E(YZ)-E(Y)E(Z) + (E(XZ)-E(X)(EZ))$

$=Cov(Y,Z)+ (\mbox{use the property of covariance of 2 random vars that are independent})$

3. 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).

4. for 3
for var(Z)
i am getting
Var(Z) = Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y)
=Var(X)+Var(Y)+2(E(XY)-E(X)E(Y))
=Var(X)+Var(Y)+2(E(X)E(Y)-E(X)E(Y))
=Var(X)+Var(Y)
=E((x-0.5)^2) + E(y^2)

Is this the last step or can i go further?

5. X and Y are indep so their covariance is ZERO.

So V(X+Y)=V(X)+V(Y)=(1/2)(1/2)+1.

Likewise V(X-Y)=V(X)+V(Y)=(1/2)(1/2)+1.

6. i am not understanding how your calculating the variances...