In this post I will show, using generating functions, that the sum of independent, identically distributed geometric random variables has a negative binomial distribution. More precisely, suppose is a sequence of independent random variables with , i.e.

Then the generating function of is

For each positive integer , let . Then the generating function of is

Now let , i.e.

Then the generating function of is

To compute this infinite series, let . We show that and hence for each nonnegative integer . For , the claim is evident, as . Assume the claim for some , then

It follows that

thus proving the claim. Now, by Taylor’s theorem,

and hence

Since and have the same generating function, we conclude that they have the same distribution.

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