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Suppose I have a huge (effectively infinite) population of widgets. The number of widgets that are broken is given by a random variable $X$, whose probability generating function is $p(z) = E(z^X)$.

Now suppose I look at a proportion $\theta$ ($0 \le \theta \le 1$) of the population. Let $Y$ be the random variable of the number of broken widgets in this proportion. Then it is easily shown that the probability generating function for $Y$ is $p(1-\theta(1-z))$.

What is a book that contains this fact so that I can reference it?

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up vote 2 down vote accepted

I just realized that my desired result follows from the second bullet point of this section from wikipedia. Presumably this can be found in any book on probabilty generating functions, so I guess I'll do a literature search.

http://en.wikipedia.org/wiki/Probability-generating_function#Functions_of_independent_random_variables

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