In solving the following problem:
Let $X$ be the number of 1's and Y the number of 2's that occur in
$n$ rolls of a fair die. Compute $Cov( X, Y)$.
I cam up with this solution:
$ Cov( X, Y) = E[ XY] - E[ X]E[ Y]\\ = -\sum\limits_1^n x p(x) \sum\limits_1^n y p(y)\\ = -\sum\limits_1^n x \frac{ 1}{ 6} \sum\limits_1^n y \frac{ 1}{ 6}\\ = -\frac{ 1}{ 36} \sum\limits_1^n x \sum\limits_1^n y\\ = -\frac{ 1}{ 36} \frac{ 1}{ 4} n^2 (n + 1)^2\\ = -\frac{ 1}{ 144} n^2 (n + 1)^2\\$
However, the answer in the back of the book and a work solution here (Problem 7.36) indicate the solution is $-\frac{ n}{36}$ and I do not understand why?
Specifically, I do not understand why they are treating the problem as the covariance of sequences?