Does anyone understand why the negative binomial distribution is the sum of r geometric distributions? See the example given below. Could someone please explain the logic behind this statement and perhaps give a simple example?
2026-02-24 00:01:43.1771891303
Why is the negative binomial distribution the sum of r geometric distributions?
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a geometric distribution counts the trials to get the first success...
think you have r independent geometric distributions...each rv counts how many trials you need to get the first (one) success.
It is very intuitive to realize that the sum of these r rv's count the number of trials to get r successes....this is a negative binomial
a different parametrization of geometric distribution (counting the failures before the first success) generates a sum that is a Pascal distribution
Here you can find all the details duly explained