Gamma(L,b,a) where b = Scale and a = Shape (L is a shift from zero and by default 0)
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:
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
4. Prior distribution for Normal Bayesian inference
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).
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.