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title: "Approximations to the Negative Binomial distribution"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575261/Approximations%20to%20the%20Negative%20Binomial%20distribution"
format: markdown
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The [Negative Binomial distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575980/) NegBinomial(*p*,*s*) models the total number of trials (n trials = s *successes plus* n-*s*failures *)* it takes to achieve s successes, where each trial has the same probability of success *p*.
*Normal approximation to the Negative Binomial*
When the number of successes *s* required is large, and *p* is neither very small nor very large, the following approximation works pretty well:
| | |
| --- | --- |
| NegBinomial(*p, s*) *»* [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246/) | > Macro (mathinline)
 |
The approximation can be justified via [Central Limit Theorem](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574953/), because the NegBinomial(*p*,s) distribution can be thought of as the sum of *s* independent NegBinomial(*p*, 1) distributions, each with mean > Macro (mathinline)
and standard deviation > Macro (mathinline)
.
The difficulty lies in knowing whether, for a specific problem, the values for *s* and *p* fall within the bounds for which the Normal distribution is a good approximation. The smaller the value of *p*, the longer the tail of a NegBinomial(*p*,1) distribution:
As *p* gets very small, the NegBinomail(*p*,1) becomes an Exponential distribution (see below), and so we can use a Gamma approximation to the NegBinomial instead of a Normal. On the other hand, as *p* is large, so the NegBinomial(*p*,1) distribution gets more skewed, so *s* would need to be much larger for a Normal approximation (which has to overcome this skewness) to be appropriate:
NegBinomial(0.5,s) distributions and their corresponding Normal distribution approximations
NegBinomial(0.9,*s*) distributions and their corresponding Normal distribution approximations, showing that when *p* is large, *s* needs to be higher for the Normal approximation to work well.
*Gamma approximation to the Negative Binomial*
The [Poisson process](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214/) can be [derived](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574980/) from the Binomial process by making *n* extremely large while *p* becomes very small, but within the constraint that *np* remains finite. In a Poisson process, the [Gamma](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575231/)(0,*b*,*a*) distribution [models](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981/) the 'time' until observing *a* events where *b* is the mean time between events. The NegBinomial distribution is the binomial equivalent, modeling the total number of trials to achieve *s* successes where [(1/*p*)-1] is the mean number of failures per success. The NegBinomial in Crystal Ball includes the s successes which in terms of a Poisson process are not included in the waiting time because each event is assumed to be instantaneous. To make the two approaches more comparable, we subtract the (non-random) number of successes from the NegBinomial(p,s) distribution to obtain the number of failures only (i.e. shift the distribution s to the left). The remaining distribution models the number of failures, with mean (1/p-1) failures for each success. Then, we can make the following approximation:
NegBinomial(p,*s*) - s *»* Gamma(0,1/p-1,*s*) when *p* ® *0*
Or equivalently, using the shift parameter for the Gamma distribution:
NegBinomial(p,*s*) *»* Gamma(s,1/p,*s*) when *p* ® *0*
For *s* = 1, we also have the special case:
[Geometric](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575207/)(*p*) -1 *»* [Exponential](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575227/)(*p/(1-p)*) when *p* ® *0*
When the Exponential distribution is a good approximation to the "Geometric(p) - 1" (p<0.05 is usually good, see below), the Gamma is a good approximation to the NegBinomial.
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