For any discrete random variable $$X,$$ with probability mass function $$P\left( {X = j} \right) = {p_j},$$
$${p_j}\,\, \ge 0,\,j \in \left\{ {0,..........,\,\,\,N} \right\},$$ and $$\,\,\sum\limits_{j = 0}^N {{p_j} = 1,\,\,} $$ define the polynomial function $${g_x}\left( z \right) = \sum\limits_{j = 0}^N {{p_j}{z^j}} .$$ For a certain discrete random variable $$Y$$, there exists a scalar $$\beta $$ $$ \in \left[ {0,1} \right]$$ such that $${g_y}\left( z \right) = {\left\{ {1 - \beta + \left. {\beta z} \right)} \right.^N}.$$ The expectation of $$Y$$ is