Negative Binomial Distribution

We use the Negative Binomial Distribution to answer the following question:

  • Given a probability of success per trial, what is the probability that we got the -th success after trials, i.e. ?
title:Negative Binomial Distribution (Definition)
We say $X ∼ NB(r, p)$, where: 
1. $X$ is the number of trials required to observe the $r$-th success
2. Each trial is part of a sequence of independent Bernoulli experiments each with a probability of success $p$.
  • Support of

  • p.m.f. of : where

  • is the number of trials

  • is the number of successes

  • is the probability of success for 1 trial


Consider the situation where you have = 3, , and you want to find .

Intuitively, this asks what are the odds that you get the 3rd success on the 7th try?

So for the first six tries, you have options of successes, multiplied by odds of getting successes, and probability of failure, then a probability of getting the 7th try. Which yields the p.m.f. of the equation you see above.