Show that the following method of generating discrete random variables works (D. R. Fredkin). Suppose, for concreteness, that X takes on values 0, 1, 2, . . . with probabilities p0, p1, p2, . . . . Let U be a uniform random variable. If U < p0, return X = 0. If not, replace U by U − p0, and if the new U is less than p1, return X = 1. If not, decrement U by p1, compare U to p2, etc.
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