Negative Binomial Distribution

Compact study note.

Summary

The negative binomial distribution models the number of failures before a fixed number of successes in independent Bernoulli trials.^[1]

Prerequisites

• Geometric Distribution

Definition

$X\sim \mathrm{NegBin}(r,p)$ here means $X$ counts failures before the $r$ th success.

Notation and Assumptions

Trials are independent, success probability is constant, and $r$ is fixed in advance.

Parameters

$r\in \{1,2,\dots \}$ and $0<p\le 1$ .

Support

$\{0,1,2,\dots \}$ .

PMF or PDF

$P(X=k)=(\genfrac{}{}{0ex}{}{k+r-1}{k})p^r(1-p{)}^k$ for $k=0,1,2,\dots$ .

CDF

Usually evaluated by summing the PMF; closed forms use special functions.

Moments

$E[X]=r(1-p)/p$ and $\mathrm{Var}(X)=r(1-p)/p^2$ .

Essential Result

For $r=1$ , this is the failure-count version of the geometric distribution.

Small Example

For $r=3$ , $p=0.5$ , $P(X=2)=(\genfrac{}{}{0ex}{}{4}{2})(0.5{)}^3(0.5{)}^2=6/32=0.1875$ .

Common Mistakes

• Not stating whether $X$ counts failures or total trials.
• Replacing 'fixed successes' with 'fixed trials', which is binomial.

Connections

• Geometric Distribution
• Binomial Distribution
• Bernoulli Distribution

References