Probability | Part IA, 2006

Let KK be a fixed positive integer and XX a discrete random variable with values in {1,2,,K}\{1,2, \ldots, K\}. Define the probability generating function of XX. Express the mean of XX in terms of its probability generating function. The Dirichlet probability generating function of XX is defined as

q(z)=n=1K1nzP(X=n)q(z)=\sum_{n=1}^{K} \frac{1}{n^{z}} P(X=n)

Express the mean of XX and the mean of logX\log X in terms of q(z)q(z).

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