Let $Y=(Y_1,Y_2,\dots,Y_{10})$, be a random vector with integer coordinates, uniformly distributed over the set $$\left\{(y_1,\dots,y_n)\;\middle|\,\, \sum_{i=1}^{10}y_i=100,y_i\in \mathbb Z,y_i\ge 0\right\}.$$ What are $\operatorname{var}(Y_i)$ and $\operatorname{cov}(Y_i, Y_j), i \neq j$?
I tried to solve this problem by finding the distribution of each $y_i$s. In fact, $f_{Y_i}(y) = \frac{{108-y} \choose 8}{109 \choose 9} $.
It is clear that $E(Y_i) = 10 \forall i$ as the distributions are identical, so $ \sum_{i=1}^{10} E(Y_i) = 100 \Rightarrow E(Y_i) = 10 \forall i$.
But how do i evaluate $$\sum_{y=0}^{100} y^2 \frac{{108-y} \choose 8}{109 \choose 9}$$ to find variance of any $Y_i$?
I also found that if the variance is known, covariance between $Y_i$ and $Y_j$ $(i\neq j)$ can be computed from the equation $\operatorname{var}(\sum_{i=0}^{10} Y_i) = 0$