Gamma distribution

Gamma(alpha,beta)

Creates a gamma distribution with shape parameter alpha and scale parameter beta. The scale parameter, beta, is optional and defaults to beta=1. The gamma distribution is bounded below by zero (all sample points are positive) and is unbounded from above. It has a theoretical mean of A*B and a theoretical variance of alpha*beta^2. When alpha>1, the distribution is unimodal with the mode at (alpha-1)*beta. An exponential distribution results when alpha=1. As alpha-->oo, the gamma distribution approaches a normal distribution in shape.

The gamma distribution encodes the time required for alpha events to occur in a Poisson process with mean arrival time of beta.

Note

Some textbooks use Rate=1/beta, instead of beta, as the scale parameter.

When to use

Use the gamma distribution with alpha>1 if you have a sharp lower bound of zero but no sharp upper bound, a single mode, and a positive skew. The LogNormal distribution is also an option in this case. Gamma() is especially appropriate when encoding arrival times for sets of events. A gamma distribution with a large value for alpha is also useful when you wish to use a bell-shaped curve for a positive-only quantity.

Library

Distribution

Parameter Estimation

Suppose X contains sampled historical data indexed by I. To estimate the parameters of the gamma distribution that best fits this sampled data, the following parameter estimation formulae can be used:

alpha := Mean(X,I) / Variance(X,I)

beta := Variance(X,I) / Mean(X,I)

See Also

• Dens_Gamma - probability density at x
• GammaI -- cumulative density at x, incomplete gamma function
• GammaIInv -- inverse cumulative density
• GammaFn -- the gamma function
• Beta, Exponential, LogNormal -- related distributions