Editing Negative binomial distribution (section)

==Related distributions==
* The [[geometric distribution]] on {{math|{{mset|0, 1, 2, 3, ... }}}} is a special case of the negative binomial distribution, with
::<math>\operatorname{Geom}(p) = \operatorname{NB}(1,\, p).\,</math>

* The negative binomial distribution is a special case of the [[discrete phase-type distribution]].
* The negative binomial distribution is a special case of discrete [[compound Poisson distribution]].

===Poisson distribution===
Consider a sequence of negative binomial random variables where the stopping parameter {{mvar|r}} goes to infinity, while the probability {{mvar|p}} of success in each trial goes to one, in such a way as to keep the mean of the distribution (i.e. the expected number of failures) constant. Denoting this mean as {{mvar|λ}}, the parameter {{mvar|p}} will be {{math|1=''p'' = ''r''/(''r'' + ''λ'')}}
: <math>
\begin{align}
    \text{Mean:} \quad & \lambda = \frac{(1-p)r}{p} \quad \Rightarrow \quad p = \frac{r}{r+\lambda}, \\
    \text{Variance:} \quad & \lambda \left( 1 + \frac{\lambda}{r} \right) > \lambda, \quad \text{thus always overdispersed}.
\end{align}
  </math>

Under this parametrization the probability mass function will be
: <math>
    f(k; r, p) = \frac{\Gamma(k+r)}{k!\cdot\Gamma(r)}(1-p)^k p^r = \frac{\lambda^k}{k!} \cdot \frac{\Gamma(r+k)}{\Gamma(r)\;(r+\lambda)^k} \cdot \frac{1}{\left(1+\frac{\lambda}{r}\right)^r}
  </math>

Now if we consider the limit as {{math|''r'' → ∞}}, the second factor will converge to one, and the third to the exponent function:
: <math>
    \lim_{r\to\infty} f(k; r, p) = \frac{\lambda^k}{k!} \cdot 1 \cdot \frac{1}{e^\lambda},
  </math>
which is the mass function of a [[Poisson distribution|Poisson-distributed]] random variable with expected value&nbsp;{{mvar|λ}}.

In other words, the alternatively parameterized negative binomial distribution [[convergence in distribution|converges]] to the Poisson distribution and {{mvar|r}} controls the deviation from the Poisson.  This makes the negative binomial distribution suitable as a robust alternative to the Poisson, which approaches the Poisson for large {{mvar|r}}, but which has larger variance than the Poisson for small {{mvar|r}}.
: <math>
    \operatorname{Poisson}(\lambda) = \lim_{r \to \infty} \operatorname{NB} \left(r, \frac{r}{r + \lambda}\right).
  </math>

===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>

===Distribution of a sum of geometrically distributed random variables===

If {{math|''Y''{{sub|''r''}}}} is a random variable following the negative binomial distribution with parameters {{mvar|r}} and {{mvar|p}}, and support {{math|{{mset|0, 1, 2, ...}}}}, then {{math|''Y''{{sub|''r''}}}} is a sum of {{mvar|r}} [[statistical independence|independent]] variables following the [[geometric distribution]] (on {{math|{{mset|0, 1, 2, ...}}}}) with parameter {{mvar|p}}. As a result of the [[central limit theorem]], {{math|''Y''{{sub|''r''}}}} (properly scaled and shifted) is therefore approximately [[normal distribution|normal]] for sufficiently large&nbsp;{{mvar|r}}.

Furthermore, if {{math|''B''{{sub|''s''+''r''}}}} is a random variable following the [[binomial distribution]] with parameters {{math|''s'' + ''r''}} and {{mvar|p}}, then

: <math>
\begin{align}
\Pr(Y_r \leq s) & {} = 1 - I_p(s+1, r) \\[5pt]
& {} = 1 - I_{p}((s+r)-(r-1), (r-1)+1) \\[5pt]
& {} = 1 - \Pr(B_{s+r} \leq r-1) \\[5pt]
& {} = \Pr(B_{s+r} \geq r) \\[5pt]
& {} = \Pr(\text{after } s+r \text{ trials, there are at least } r \text{ successes}).
\end{align}
</math>

In this sense, the negative binomial distribution is the "inverse" of the binomial distribution.

The sum of independent negative-binomially distributed random variables {{math|''r''{{sub|1}}}} and {{math|''r''{{sub|2}}}} with the same value for parameter {{mvar|p}} is negative-binomially distributed with the same {{mvar|p}} but with {{mvar|r}}-value&nbsp;{{math|''r''{{sub|1}} + ''r''{{sub|2}}}}.

The negative binomial distribution is [[Infinite divisibility (probability)|infinitely divisible]], i.e., if {{mvar|Y}} has a negative binomial distribution, then for any positive integer {{mvar|n}}, there exist independent identically distributed random variables {{math|''Y''{{sub|1}}, ..., ''Y''{{sub|''n''}}}} whose sum has the same distribution that {{mvar|Y}} has.

===Representation as compound Poisson distribution===
The negative binomial distribution {{math|NB(''r'', ''p'')}} can be represented as a [[compound Poisson distribution]]: Let <math display=inline> (Y_n)_{n\,\in\,\mathbb N} </math> denote a sequence of [[independent and identically distributed random variables]], each one having the [[logarithmic distribution|logarithmic series distribution]] {{math|Log(''p'')}}, with probability mass function

: <math> f(k; r, p) =  \frac{-p^k}{k\ln(1-p)},\qquad k\in{\mathbb N}.</math>

Let {{mvar|N}} be a random variable, [[independence (probability theory)|independent]] of the sequence, and suppose that {{mvar|N}} has a [[Poisson distribution]] with mean {{math|λ {{=}} −''r'' ln(1 − ''p'')}}. Then the random sum

: <math>X=\sum_{n=1}^N Y_n</math>

is {{math|NB(''r'', ''p'')}}-distributed. To prove this, we calculate the [[probability generating function]] {{math|''G''{{sub|''X''}}}} of {{mvar|X}}, which is the composition of the probability generating functions {{math|''G''{{sub|''N''}}}} and {{math|''G''{{sub|''Y''{{sub|1}}}}}}. Using

:<math>G_N(z)=\exp(\lambda(z-1)),\qquad z\in\mathbb{R},</math>

and

: <math>G_{Y_1}(z)=\frac{\ln(1-pz)}{\ln(1-p)},\qquad |z|<\frac1p,</math>

we obtain

: <math>
\begin{align}G_X(z) & =G_N(G_{Y_1}(z))\\[4pt]
&=\exp\biggl(\lambda\biggl(\frac{\ln(1-pz)}{\ln(1-p)}-1\biggr)\biggr)\\[4pt]
&=\exp\bigl(-r(\ln(1-pz)-\ln(1-p))\bigr)\\[4pt]
&=\biggl(\frac{1-p}{1-pz}\biggr)^r,\qquad |z|<\frac1p,
\end{align}
</math>

which is the probability generating function of the {{math|NB(''r'', ''p'')}} distribution.

The following table describes four distributions related to the number of successes in a sequence of draws:
{| class="wikitable"
|-
!  !! With replacements !! No replacements
|-
| Given number of draws || [[binomial distribution]] || [[hypergeometric distribution]]
|-
| Given number of failures || negative binomial distribution || [[negative hypergeometric distribution]]
|}

===(''a'',''b'',0) class of distributions===

The negative binomial, along with the Poisson and binomial distributions, is a member of the [[(a,b,0) class of distributions|{{math|(''a'', ''b'', 0)}} class of distributions]]. All three of these distributions are special cases of the [[Panjer distribution]]. They are also members of a [[natural exponential family]].