A generalization of the Geometric distribution. Consider a sequence of independent Bernoulli trials, each with success probability . Assume the trials continue until at least the th success, where .

Let be the r.v. that equals the number of failures before the th success. Then has the Negative Binomial distribution and the textbook writes this as .

Notice that when , then this is just the Geometric distribution. Hence it’s a generalization.


Similar to the intuition for the Geometric distribution’s PMF, we now have multiple successes along with our failures. However, the order in which the successes matters, because the sample space is all the sequences of trials that lead to the th success.

Given , this means we have total trials. The th trial is the one where we get our th success, so we need to place the successes among the other trials that occur before it. We fill in the remaining spots with our failures.

Let . the PMF is given by


Consider . We can define it in terms of 3 new r.v.s:

  • equals the number of failures until the 1st success.
  • equals the number of failures between the 1st and 2nd success.
  • equals the number of failures between the 2nd and 3rd success.

Then it must be true that . In addition, they are all independent of one another, and all have the Geometric distribution.

By LOE, we have , and so the expectation of is given by

for successes and success probability of .