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 alpha*beta 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 [math]\displaystyle{ \alpha \to \infty }[/math] , 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.

The probability density of the Gamma distribution is

[math]\displaystyle{ p(x) = {{\beta^{-\alpha} x^{\alpha-1} \exp(-x/\beta)}\over{\Gamma(\alpha)}} }[/math]

If you need to compute the density, use the Dens_Gamma(x, alpha, beta) function from the "Distribution Densities.ana" library. For the cumulative probability, use the GammaI(x, a, b) function.

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 -- and above, related distributions
• SDeviation()