Let $\{X_i | 1 \leq i \leq N\}$ be N Bernoulli random variables with the same success probability $p$ $$ \forall i, X_i\sim Bernoulli(p) $$ The variables are not i.i.d, but the correlation coefficient for each pair of different variables is equal to $\rho$, and $\rho > 0$ $$ \forall i,j, i\neq j, \rho_{i,j}=\rho, \rho>0 $$
I want to find the probability that more than half of the variables succeeded, define: $$ Y=\sum_{i=1}^{N}{X_i} $$ Looking for: $$ P(Y\geq \frac{N}{2})=? $$
If the Bernoulli variables were i.i.d this could be calculated with a binomial distribution, but how can this be calculated when the variables are correlated?