Gamma distribution | Vose Software

# Gamma distribution

Format: Gamma(a, b)

The Gamma distribution is right-skewed and bounded at zero. 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(a,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 b. For example, if we know that major flooding occurs in a town on average every six years, Gamma(4,6) models how many years it will take before the next four floods have occurred.

Example: An insurance company observes that large commercial fire claims occur randomly in time with a mean of 0.7 years between claims. For its financial planning it would like to estimate how long it will be before it pays out the 5th such claim, The time is given by Gamma(5,0.7).

### 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 identities:

Poisson(Gamma(a,b)) = NegBin(a, 1/(b+1))         where a is an integer

Poisson(Gamma(a,b)) = Polya(a, b)

Poisson(l+Gamma(a,b)) = Delaporte(a, b, l)

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 NegBin distribution becomes a neat combination of the two.

### 3. Conjugate prior distribution in Bayesian inference

In Bayesian inference, the Gamma(a,b) distribution is the conjugate to the Poisson likelihood function, which makes it a useful distribution to describe the uncertainty about the Poisson mean.

### 4. Prior distribution for Normal Bayesian inference

If X is Gamma(a, b) distributed, then Y=X^(-1/2) is an Inverted Gamma distribution (InvGamma(a, b)) which is sometimes used as a Bayesian prior for s for a Normal distribution

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, b) = Gamma(a, b) where m is an integer.

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

The definition of a Gamma(a,b) distribution as the time to wait until a observations leads naturally to the useful identity: Gamma(x,b) + Gamma(y,b) = Gamma(x+y,b).

A Gamma(a,b) distribution is the sum of a Expon(b) distributions. Thus, from Central Limit Theorem, when a is large, the Gamma distribution is approximately Normal.

The MaxEnt uncertainty distribution for a parameter with known mean and geometric mean is a Gamma.

## ModelRisk functions added to Microsoft Excel for the Gamma distribution

VoseGamma generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseGammaObject constructs a distribution object for this distribution.

VoseGammaProb returns the probability density or cumulative distribution function for this distribution.

VoseGammaProb10 returns the log10 of the probability density or cumulative distribution function.

VoseGammaFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseGammaFitObject constructs a distribution object of this distribution fitted to data.

VoseGammaFitP returns the parameters of this distribution fitted to data.