For any discrete random variable X with probability mass function P(X = j) = p_j, j in {0, 1, ..., N} and sum_{j=0}^{N} ...

Question

For any discrete random variable X with probability mass function P(X = j) = p_j, j in {0, 1, ..., N} and sum_{j=0}^{N} p_j = 1, define the polynomial function g(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:

A. N(1 - beta)
B. N*beta
C. N*beta*(1 - beta)
D. Not expressible in terms of N and beta alone

Accepted Answer

B. N*beta The function g_Y(z) is the probability generating function (PGF) of Y. The given PGF (1 - beta + beta*z)^N is the PGF of a Binomial(N, beta) distribution, since each of the N independent Bernoulli(beta) trials contributes a factor (1 - beta + beta*z). For a Binomial(N, beta) random variable, the expected value is N*beta. The PGF evaluated at z=1 gives g(1) = 1 (confirming total probability), and the derivative at z=1 gives E[Y].