Are there other similar discrete probability distributions where the probability of random variable Y taking on value n is given by e.g. the n-th term in the series expansion of a function divided by the closed form of the function?Conversely, does this mean that every function with a series expansion has a corresponding discrete probability distribution? Which are the more commonly known functions and their corresponding probability distributions?

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Kyle George · Accepted Answer

Step 1For any probability distribution p supported on the nonnegative integers, the series   f  (  z  )  =      ∑          n      =      0        ∞    p  (  n  )      z    n   defines the probability generating function of p. It is analytic on (at least)       |    z      |    &amp;lt;  1. If X is a random variable with distribution p,   f  (  z  )  =      E    [      z    X    ].Step 2If f(z) is analytic in       |    z      |    &amp;lt;  1 with   f  (  1  )  =  1 and all its Maclaurin series coefficients are nonnegative, then f(z) is the probability generating function of a probability distribution supported on the nonnegative integers.

Are there other similar discrete probability distributions where the probability of random variable Y taking on value n is given by e.g. the n-th term in the series expansion of a function divided by the closed form of the function?

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