Gamma(L,*b**,** a*)

*where*

*b**= Scale and*

*a =**Shape (L is a shift from zero and by default 0)*

Crystal Ball parameter restrictions

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:

#### 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*,

*) distribution models the time required for*

*a**events to occur, given that the events occur randomly in a Poisson process with a mean time between events of*

*a**. 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.*

*b*##### 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 to a Normal, so random variations in *l* for a Poisson can often be well approximated by some Gamma, in which case the Negative Binomial distribution becomes a neat combination of the two.

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

In Bayesian inference, the Gamma(0,* b*,

*) distribution is the conjugate to the Poisson likelihood function, which makes it a useful distribution to describe the uncertainty about the Poisson mean*

*a**l*.

##### 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*,

*)) which is sometimes used as a Bayesian prior for*

*a**s*for a Normal distribution

#### Comments

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

The *Erlang* distribution is the Gamma distribution for integer values of * a,* i.e. Erlang(

*m,*) = Gamma(0

*b**,*

*, b**) where*

*a**m*is an integer.

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

*(*1,

*= Exponential*

*b*)*(1/*

*.*

*b*)

The definition of a Gamma(0,* b*,

*) distribution as the time to wait until*

*a**observations leads naturally to the useful identity: Gamma(0,*

*a**) + Gamma(0,*

*b,**x*

*b,**y*) = Gamma(0,

*b,**x+y*).

A Gamma(0,* b*,

*) distribution is the sum of*

*a**a*Exponential(1/

*) distributions. Thus, from Central Limit Theorem, when*

*b**a*is large, the Gamma distribution is approximately Normal.

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.