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

For any discrete random variable X, with probability mass function P(X = j) = pj, pj 0, j \0, , N\, and j=0N pj = 1, define the polynomial function gX(z) = j=0N pj zj. For a certain discrete random variable Y, there exists a scalar [0, 1] such that gY(z) = (1 - + z)N. The expectation of Y is

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