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The probability-generating function of a random variable $X$ is $$ G(z) = E[z^X] $$

Assume that $X$ is a discrete random variable on $\{0,1,2,\ldots\}$ then $$ G(z) = \sum_{j = 0}^n z^j P(X = j) $$

I have always used the PGF when dealing with real $z$. I was curious if there are any interesting problems involving complex $z$ when $X$ is real.

Stoof
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    Well, often we just treat the pgf as a formal power series and don't care what the value of $z$ is (real or complex). – Prasiortle May 15 '20 at 19:47
  • I was thinking of cases like computing the probability of seeing an even as (G(1) + G(-1))/2. Thought there might be similar computations where we could use G(i). – Stoof May 16 '20 at 04:52
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    Ah, I see. You may note that the fourth roots of unity are $1$, $-1$, $i$, and $-i$, and so the probability of $X$ taking a value that is divisible by $4$ is $\frac{G(1)+G(-1)+G(i)+G(-i)}{4}$. Similarly you can do divisible by $n$ by considering the $n$th roots of unity. This is an example of the discrete Fourier transform (also known in some circles as the 'roots of unity filter'). – Prasiortle May 16 '20 at 06:12

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