Jointly Bernoulli Distribution

ait.abd

Let

[itex]
x_i \sim \text{Bernoulli}(p_i) : p_i \in [0,1], x_i \in \{ +1, -1\}.
[/itex]
What is the expression for the joint distribution of the the vector [itex]X = [x_1, x_2, ... , x_N][/itex].
What is the entropy [itex]h(X)[/itex] of the vector [itex]X[/itex] where

[itex]
h(X) = -\sum_{\forall X \in \{+1,-1\}^N} p(X) \log (p(X)).
[/itex]