Products of independent random variables. 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.
Let $T\gt S$, then $P(Y_T=1|Y_S)=\frac{1}{2}=P(Y_T=1)$.  The point being that the product of $X_i$ for $i\in T\ and\ i\notin S$ is independent of $Y_S$.
Let $T\gt S$, then $P(Y_T=1|Y_S)=\frac{1}{2}=P(Y_T=1)$.  The point being that the product of $X_i$ for $i\in T\ and\ i\notin S$ is independent of $Y_S$.