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<=theta<=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?