# Gamma

Gamma(L,b,a) where b = Scale and a = Shape (L is a shift from zero and by default 0)

Gamma equations

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

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