Let $X_1$, $X_2, \dots, X_n$ be identically distributed random variables with given mean, and let $N$ be a random variable with $N \ge 0$ that is independent of $X_1$, $X_2, \dots, X_n$. If $Y=X_1+X_2+...X_n$; how do we find $\mathrm{E}(Y|N=n)$? With $D$ being the domain of $y$, I know the conditional expectation is defined as
$\mathrm{E}(Y|N=n) = \sum\limits_{y \in D} y \cdot \mathrm{P}(Y=y|N=n) = \sum\limits_{y \in D} y \cdot \dfrac{P(Y=y,N=n)}{P(N=n)}$.
I am not sure how to count/compute $P(Y=y,N=n)$. How can it be computed?