Suppose $X_i$ are not-independent random variables, and $N$ an integer random variable. I now define the "composite" random variable $$ Y(N) = \sum_{i=1}^N X_i \;, $$ and I want to calculate $$ \text{Var}[Y(N)] = ... $$
I'm interested in the specific situation where $\langle X_i^k \rangle$ is independent on $i$, and interested in obtaining an expression involving moments of the distributions for $X$ and $N$.
Is there any trick to derive a compact form for this variance?
EDIT: at the end I might want to consider N from a Binomial distribution.
Assuming that for $i\neq j$, $\mathrm{Cov}(X_i,X_j)=a$ is independent on $i$ and $j$ and $\mathrm{Var}(X_i)=b$ is independent of $i$. Then \begin{align*} \mathbb E[(Y-\mathbb E[Y])^2] &= \mathbb E\left[\mathbb E\left[(Y-\mathbb E[Y])^2|N\right]\right]\\ &=\mathbb E\left[ \sum_{i=1}^N\sum_{j=1}^N \mathrm{Cov}(X_i,X_j) \right]\\ &= \mathbb E\left[N a + N(N-1) b\right]\\ &= \mathbb E[N] (a-b) + \mathbb E[N^2] b \end{align*} You can then compute the first and second moment of $N$ to get your answer.