I wonder if there is any way that can allow me to approximate the distribution of the sum of products of Binomial random variables with a closed form?
For a binomial random variable $X_i^{(k)} \sim Bin(2, \pi_i^{(k)})$, and I hope to find a closed form for: $$ Y_{i,j} = \sum_{k=0}^{p}X_i^{(k)}X_j^{(k)} $$ and each of $X$ is an independent drawn. Also, we have: $$ \pi_i^{(k)} \sim Beta(\alpha, \beta) $$ I wonder if there is any way to approximate the distribution of $Y$ with a simpler closed form other than directly write it out.
Some extra information:
I need a simpler form mainly because ultimately, I will need to the expectation: $$ \mathbf{E}[\dfrac{(\sum_{i,j\in S_1}Y_{i,j})^2}{\sum_{i,j\in S_2}Y_{i,j}^2}] $$ where $S_1$ and $S_2$ are two different sets. If I directly write the $Y$ out as how it is defined, the expectation will be way too complicated for me to derive anything, so can I approximate $Y$ in a simpler way?