a die is rolled twice, let X be the sum of the outcomes and let Y be the first outcome minus the second outcome. Compute Cov(X,Y)
If $\displaystyle A$ is the outcome of your first roll and $\displaystyle B$ is the outcome of the second roll, then according to the question$\displaystyle X=A+B \; and\; Y=A-B$
Now use the properties of covariancesto find:
$\displaystyle Cov(X,Y) = Cov(A+B, A-B) $
$\displaystyle = Cov(A,A)-Cov(A,B)+Cov(A,B)-Cov(B,B)=.... $
Or you can say $\displaystyle \displaystyle cov(X,Y) = E(X,Y)-E(X)\times E(Y)$
Find the expected values from these tables
Code:First X 1 2 3 4 5 6 Second 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 First Y 1 2 3 4 5 6 Second 1 0 1 2 3 4 5 2 -1 0 1 2 3 4 3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 5 -4 -3 -2 -1 0 1 6 -5 -4 -3 -2 -1 0 XY First Second 1 2 3 4 5 6 1 0 3 8 15 24 35 2 -3 0 5 12 21 32 3 -8 -5 0 7 16 27 4 -15 -12 -7 0 9 20 5 -24 -21 -16 -9 0 11 6 -35 -32 -27 -20 -11 0
Seems like you're messed up.
look at the second post and go over the link
$\displaystyle Cov(X,Y) = Cov(A+B, A-B)$
$\displaystyle =Cov(A,A)-Cov(A,B)+Cov(A,B)-Cov(B,B)$
$\displaystyle =Cov(A,A)-Cov(B,B)$
$\displaystyle =Var(A)-Var(B)$
$\displaystyle =0 \;\;\; WHY??$