# Sum of dependent Random Variables

Let $X_i \sim \operatorname{Ber(p_i)}$ be dependent random variables. Consider the modulo two sum $S_n=\sum \limits_{i=1}^nX_i\operatorname{mod 2}$. For a fixed $n$, I am interested in finding $\mathbb{P}(S_n=0)$.

Furthermore to restrict the space of all joint distributions we induce the following condition,

For an $n$ length vector $(X_1,X_2,...,X_n), \delta<n$ coordinates are uniformly distributed over the set $\{0,1 \}$ and the remaining coordinates distributions are $\{0,1\}$ valued i.e, $p_i=0$ or $1$(deterministic coordinates).

Any help would be appreciated.

Thank you

• It can be anything in $[0,1]$. You need to somehow limit dependencies between $X_i$. – A.S. Mar 5 '16 at 7:03
• No. I mean you need to somehow restrict the space of all possible joint distributions of $X_i$. – A.S. Mar 5 '16 at 7:06
• I didnt understand as what you mean by 'limit dependencies'. – lebesgue Mar 5 '16 at 7:07
• The question has been edited based on the comments given. – lebesgue Mar 5 '16 at 8:12