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)^2 / Variance(X,I)

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

The above is not the maximum likelihood parameter estimation, which turns out to be rather complex (see Wikipedia). However, in practice the above estimation formula perform excellently and are so convenient that more complicated methods are hardly justified.

The Gamma distribution with an offset has the form:

Gamma(alpha,beta) - offset

To estimate all three parameters, the following heuristic estimation can be used:

alpha := 4 / Skewness(X,I)^2

offset := Mean(X,I) - SDeviation(X,I) * Sqrt(alpha)

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

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