Negative Binomial Distribution

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

• Given a probability of success $p$ per trial, what is the probability that we got the $r$-th success after $k$ trials, i.e. $P(X=k)$?

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 $X=\{r,r+1,r+2,\dots \}$

• p.m.f. of $X$: $P(X=k)=\left(\genfrac{}{}{0ex}{}{k-1}{r-1}\right)p^r(1-p{)}^{k-r}$ where

• $k$ is the number of trials

• $r$ is the number of successes

• $p$ is the probability of success for 1 trial

• $E(X)=\frac{r}{p}$

• $Var(X)=\frac{r(1-p)}{p^2}$

Intuition

Consider the situation where you have $r$ = 3, $p=0.5$, and you want to find $k=7$.

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 $\left(\genfrac{}{}{0ex}{}{6}{2}\right)$ options of successes, multiplied by $p^2$ odds of getting successes, and $(1-p{)}^{6-2}$ probability of failure, then a probability $p$ of getting the 7th try. Which yields the p.m.f. of the equation you see above.

Related

• A Negative Binomial Distribution is a sequence of Negative Binomial Distribution is a sequence of Geometric Distributions.