Poisson as a Limit of the Negative Binomial Distribution
Introduction
John D. Cook has a great sequence of posts1 about the negative binomial distribution. The standard form of the negative binomial distribution has two parameters, \(r\) and \(p\), and has a probability mass function (pmf) given by: \[ f(y; r, p) = \binom{r + y - 1}{y} \times (1 - p)^{y} \times p^{r} \]
This distribution has mean \(\mu = \frac{r(1-p)}{p}\) and variance \(\sigma^2 = \frac{r(1-p)}{p^{2}}\). Although \(r\) is sometimes interpreted as the “number of successes” that must be reached, the distribution can be generalized to noninteger positive values of \(r\) by expressing the binomial coefficient in terms of the \(\Gamma\) function. To see this, note that for positive-integer values of \(r\): \[ \binom{r + y - 1}{y} = \frac{(r + y - 1)!}{(r - 1)! \cdot y!} . \] Since the \(\Gamma\) function is the generalization of the factorial function, we can extend the binomial distribution to noninteger positive \(r\) by writing: \[ f(y;\ r, p) = \frac{\Gamma(r + y)}{\Gamma(r) \cdot y!} \times (1 - p)^{y} \times p^{r} . \]
In his notes, John D. Cook mentions the following result:
Proposition: If \(r \rightarrow \infty\) and \(p \rightarrow 1\) as \(\mu\) stays constant, \(f(y; r, p)\) converges to a Poisson probability mass function with mean \(\mu\).
Let’s prove this result. But first, we need to be more careful when writing the proposition; more specifically we need to be clear about what it means for \(r \rightarrow \infty\) and \(p \rightarrow 1\) while \(\mu\) is constant. To do so, note that the parameter \(p = \frac{r}{\mu + r}\). Now if we fix \(\mu\), as \(r \rightarrow \infty\) we have \(p \rightarrow 1\). We can now rewrite the pmf in terms of the parameters \(r\) and \(\mu\) by slightly overloading the notation for the pmf as: \[ f(y; r, \mu) := f\left(y;\ r, \frac{r}{r + \mu} \right) = \frac{\Gamma(r + y)}{\Gamma(r) \cdot y!} \times \frac{r^r \cdot \mu^y}{(r + \mu)^{y + r}} \]
We can now formally write the result we want to prove as follows:
Proof: Starting with the reparameterized pmf, multiply and divide by \(r^{y}\) and rearrange: \[ f(y; r, \mu) = \frac{\Gamma(r + y)}{\Gamma(r)\cdot r^{y}} \times \left(\frac{r}{r + \mu}\right)^{y + r} \frac{\mu^y}{y!} . \] Next, since \(y\) is a non-negative integer we can use the definition of the \(\Gamma\)-function to write: \(\Gamma(r + y) = \Gamma(r) \times \prod_{j = 1}^{y} (r + j)\). Plugging this and again re-arranging terms we get: \[ f(y; r, \mu) = \prod_{j=1}^{y} \left(1 + \frac{j}{r}\right) \times \left(\frac{1}{1+ \mu / r}\right)^{y} \times \left(\frac{1}{1+ \mu / r}\right)^{r} \times \frac{\mu^y}{y!} . \] Now note that \(\frac{j}{r} \rightarrow 0\) and \(\left(\frac{1}{1+ \mu / r}\right)^{y} \rightarrow 1\) as \(r \rightarrow \infty\), so we have: \[ \lim_{r \rightarrow \infty} f(y; r, \mu)\ =\ \frac{\mu^y}{y!} \times \lim_{r \rightarrow \infty} \left(\frac{1}{1+ \mu / r}\right)^{r} . \] The last bit is easy once we note that: \[ \begin{aligned} \lim_{r \rightarrow \infty} \left(\frac{1}{1+ \mu / r}\right)^{r}\ &=\ \lim_{r \rightarrow \infty} \left(1 + \frac{\mu}{r}\right)^{-r}\\ &=\ \left[\lim_{r \rightarrow \infty} \left(1 + \frac{\mu}{r}\right)^{r/\mu}\right]^{-\mu}\\ &=\ e^{-\mu} . \end{aligned} \]
Putting all of this together, we have: \[ \lim_{r \rightarrow \infty} f(y; r, \mu)\ =\ \frac{e^{-\mu} \cdot \mu^y}{y!}. \]
Figure 1 shows the negative binomial pmf (in black) with increasing \(r\), and the Poisson pmf in blue. We can clearly see the negative binomial pmf resembles the poisson more and more as \(r\) increases. The animation was created using the excellent Julia packages Makie.jl and AlgebraOfGraphics.jl, see Danisch and Krumbiegel (2021).
Acknowledgements
Thanks to Steven Moen, Margaret Turner, and Sam Ozminkowski for suggestions and edits.
References
Footnotes
Citation
@online{kokandakar2025,
author = {Kokandakar, Ajinkya},
title = {Poisson as a {Limit} of the {Negative} {Binomial}
{Distribution}},
date = {2025-04-15},
url = {https://ajinkya-k.github.io/posts/nb-pois},
langid = {en}
}