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title: "Gamma"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575231/Gamma"
format: markdown
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Gamma(L,*b,a*) *where* *b* *= Scale and* *a =* *Shape (L is a shift from zero and by default 0)*

[Gamma equations](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575671/)

[Crystal Ball parameter restrictions](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575280/)

 

 

The Gamma (L,*b,a*) distribution is right-skewed and bounded at L. It is a parametric distribution based on Poisson mathematics. Examples of the Gamma distribution are given below:

 

![image](media://d88092a5-1ef9-4473-85a5-eed4aa56f522)

 

 

#### Uses

The Gamma distribution is extremely important in risk analysis modeling, with a number of different uses:

##### 1. Poisson waiting time

The Gamma(0,*b*,*a*) distribution models the [time required for ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981/)*[a](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981/)*[ events to occur](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981/), given that the events occur randomly in a [Poisson process](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575359) with a mean time between events of *b*. For example, if we know that major flooding occurs in a town on average every six years, Gamma(0,6,4) models how many years it will take before the next four floods have occurred.

##### 2. Random variation of a Poisson intensity *l*

The Gamma distribution is used for its convenience as a description of random variability of *l* in a Poisson process. It is convenient because of the identity:

 

Poisson(Gamma(0,*b*,*a*)) = NegBinomial(1/(*b*+1),*a*)-*a*

 

The Gamma distribution can take a variety of shapes, from an [Exponential](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575227/) to a [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246/), so random variations in *l* for a Poisson can often be well approximated by some Gamma, in which case the [Negative Binomial](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575213/) distribution becomes a neat combination of the two.

##### 3. Conjugate prior distribution in Bayesian inference

In [Bayesian inference](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575368), the Gamma(0,*b*,*a*) distribution is the<span style="color: #0b0820"> </span><span style="color: #3366ff">[conjugate](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575373/)</span> to the [Poisson](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575214/) likelihood function, which makes it a useful distribution to describe [the uncertainty about the Poisson mean ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574982/)*[l](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574982/)*.

##### 4. Prior distribution for Normal Bayesian inference

If X is Gamma(0,*b*,*a*) distributed, then Y=X^(-1/2) is an Inverted Gamma distribution (InvGamma(0,*b*,*a*)) which is sometimes used as a [Bayesian prior for ](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575399/Bayesian+analysis+example+The+Monty+Hall+problem)*[s](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575399/Bayesian+analysis+example+The+Monty+Hall+problem)* for a [Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575246/) distribution

 

#### Comments

The Gamma distribution has also found use in meteorology, inventory theory, insurance risk, economics and queuing theory.

 

The *[Erlang](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575225)* distribution is the Gamma distribution for integer values of *a,* i.e. Erlang(*m, b*) = Gamma(0*, b*,*a*) where *m* is an integer.

 

The Exponential distribution is a special case of the Gamma and Erlang: Gamma*(0*,*b,1)* =Erlang*(*1,*b)* = Exponential*(1/b)*.

 

The [definition](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574981/) of a Gamma(0,*b*,*a*) distribution as the time to wait until *a* observations leads naturally to the useful identity: Gamma(0,*b,x*) + Gamma(0,*b,y*) = Gamma(0,*b,x+y*).

 

A Gamma(0,*b*,*a*) distribution is the sum of *a* Exponential(1/*b*) distributions. Thus, from [Central Limit Theorem](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574953/), when *a* is large, the Gamma distribution is [approximately Normal](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575266/).

 

The Excel function GAMMADIST(x,*a,b*,**0**) returns the probability density function for the Gamma(0,*b,a*) distribution, while GAMMADIST(x,*a,b*,1) returns its cumulative distribution function.

 

 

 

 

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