Computing $\int_{0}^1 \prod_{x \in \mathcal{X}} (p_xy + (1-p_x)) dy$

Question

How would I compute this integral: $$\int_{0}^1 \prod_{x \in \mathcal{X}} (p_xy + (1-p_x)) dy$$ I have tried using the chain rule and not made any headway on that front. Any advice is welcome.

Answer 1

I assume $|\chi|<\infty.$

If we are able to rewrite the product as follows $$\prod_{x\in\chi}(p_xy+(1-p_x))=\sum_{n=0}^{|\chi|}a_ny^n$$ then the integral is very easy to compute and it results in $$\int_0^1\prod_{x\in\chi}(p_xy+(1-p_x))dy=\int_0^1\sum_{n=0}^{|\chi|}a_ny^ndy=\sum_{n=0}^{|\chi|}\frac{a_n}{n+1}.$$ Now the terms $a_n$ are those where you multiply exactly $n$ $p_xy$ together and $|\chi|-n$ terms of the form $(1-p_x)$ (look at the degreee). Thus $$a_n=\sum_{A\subset \chi\\ |A|=n}\prod_{p_x\in A} p_x\cdot\prod_{p_x\notin A}(1-p_x)$$