Editing Negative binomial distribution (section)

===Gamma–Poisson mixture===
The negative binomial distribution also arises as a continuous mixture of [[Poisson distribution]]s (i.e. a [[compound probability distribution]]) where the mixing distribution of the Poisson rate is a [[gamma distribution]]. That is, we can view the negative binomial as a {{math|Poisson(''λ'')}} distribution, where {{mvar|λ}} is itself a random variable, distributed as a gamma distribution with shape {{mvar|r}} and scale {{math|1=''θ'' = (1 − ''p'')/''p''}} or correspondingly rate {{math|1=''β'' = ''p''/(1 − ''p'')}}.

To display the intuition behind this statement, consider two independent Poisson processes, "Success" and "Failure", with intensities {{mvar|p}} and {{math|1 − ''p''}}. Together, the Success and Failure processes are equivalent to a single Poisson process of intensity 1, where an occurrence of the process is a success if a corresponding independent coin toss comes up heads with probability {{mvar|p}}; otherwise, it is a failure. If {{mvar|r}} is a counting number, the coin tosses show that the count of successes before the {{mvar|r}}-th failure follows a negative binomial distribution with parameters {{mvar|r}} and {{mvar|p}}. The count is also, however, the count of the Success Poisson process at the random time {{mvar|T}} of the {{mvar|r}}-th occurrence in the Failure Poisson process. The Success count follows a Poisson distribution with mean {{math|''pT''}}, where {{mvar|T}} is the waiting time for {{mvar|r}} occurrences in a Poisson process of intensity {{math|1 − ''p''}}, i.e., {{mvar|T}} is gamma-distributed with shape parameter {{mvar|r}} and intensity {{math|1 − ''p''}}. Thus, the negative binomial distribution is equivalent to a Poisson distribution with mean {{math|''pT''}}, where the random variate {{mvar|T}} is gamma-distributed with shape parameter {{mvar|r}} and intensity {{math|(1 − ''p'')}}. The preceding paragraph follows, because {{math|1=''λ'' = ''pT''}} is gamma-distributed with shape parameter {{mvar|r}} and intensity {{math|(1 − ''p'')/''p''}}.

The following formal derivation (which does not depend on {{mvar|r}} being a counting number) confirms the intuition.

: <math>\begin{align}
& \int_0^\infty f_{\operatorname{Poisson}(\lambda)}(k) \times f_{\operatorname{Gamma}\left(r,\, \frac{p}{1-p}\right)}(\lambda) \, \mathrm{d}\lambda \\[8pt]
= {} & \int_0^\infty \frac{\lambda^k}{k!} e^{-\lambda} \times \frac 1 {\Gamma(r)} \left(\frac{p}{1-p} \lambda \right)^{r-1} e^{- \frac{p}{1-p} \lambda} \, \left( \frac p{1-p} \, \right)\mathrm{d}\lambda \\[8pt]
= {} & \left(\frac{p}{1-p}\right)^r \frac{1}{k!\,\Gamma(r)} \int_0^\infty \lambda^{r+k-1} e^{-\lambda \frac{p+1-p}{1-p}} \;\mathrm{d}\lambda \\[8pt]
= {} & \left(\frac{p}{1-p}\right)^r \frac{1}{k!\,\Gamma(r)} \Gamma(r+k) (1-p)^{k+r} \int_0^\infty f_{\operatorname{Gamma}\left(k+r, \frac{1}{1-p}\right)}(\lambda) \;\mathrm{d}\lambda \\[8pt]
= {} & \frac{\Gamma(r+k)}{k!\;\Gamma(r)} \; (1-p)^k \,p^r  \\[8pt] 
= {} & f(k; r, p).
\end{align}</math>

Because of this, the negative binomial distribution is also known as the '''gamma–Poisson (mixture) distribution'''. The negative binomial distribution was originally derived as a limiting case of the gamma-Poisson distribution.<ref name="Greenwood1920">{{cite journal |last1=Greenwood |first1=M. |last2=Yule |first2=G. U. |year=1920 |title=An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference of multiple attacks of disease or of repeated accidents |journal=[[Journal of the Royal Statistical Society|J R Stat Soc]] |volume=83 |issue=2 |pages=255–279 |doi=10.2307/2341080 |jstor=2341080 |url=https://zenodo.org/record/1449492 }}</ref>