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

The polynomial function $g_X(z) = \sum_{j=0}^N p_j z^j = E[z^X]$ is the probability generating function (PGF) of the random variable $X$.

The expectation $E[Y]$ can be computed by taking the first derivative of $g_Y(z)$ with respect to $z$ and evaluating at $z = 1$:

Given:

Differentiating with respect to $z$ using the chain rule:

Evaluating at $z = 1$:

Alternatively, recognizing $g_Y(z) = ((1-\beta) + \beta z)^N$ as the PGF of a Binomial distribution $Y \sim \text{Binomial}(N, \beta)$, the mean is known to be $E[Y] = N\beta$.

Hence, the correct option is (B).