I want to give a little heads up that this question is very similar to Product of independent random variables.
Let $X_1,X_2,\ldots,X_d$ be independent r.v.'s with distributions $$P(X_i=1)=P(X_i=-1)=\frac{1}{2}\quad\tag{*}$$. For each $S \in [d]$, define $Y_S = \prod\limits_{i\in S}X_i.$ Show that the variables $\{Y_S\}$ are pairwise independent.
In the other question it has been shown show that $(Z_i)_{i=1}^m$ are mutually(jointly) independent where $Z_i = X_1X_2\ldots X_i$. The key distinction between the questions is that the sets of $[d]$ that are used to take the product are different and pairwise independence is required instead of joint mutual independence.
