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GammaDistribution
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GammaDistribution

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represents a gamma distribution with shape parameter α and scale parameter β.

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GammaDistribution[α,β,γ,μ]

represents a generalized gamma distribution with shape parameters α and γ, scale parameter β, and location parameter μ.

Details

  • The probability density for value in a gamma distribution is proportional to for , and is zero for . »
  • The probability density for value in a generalized gamma distribution is proportional to for , and is zero elsewhere.
  • GammaDistribution allows α, β, and γ to be any positive real numbers and μ to be any real number.
  • GammaDistribution allows β and μ to be any quantities of the same unit dimensions, and α, γ to be dimensionless quantities. »
  • GammaDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

Background & Context

  • GammaDistribution[α,β,γ,μ] represents a continuous statistical distribution defined over the interval and parametrized by a real number μ (called a "location parameter"), two positive real numbers α and γ (called "shape parameters") and a positive real number β (called a "scale parameter"). The parameter μ determines the horizontal location of the probability density function (PDF) of the gamma distribution. The shape of the PDF is entirely dependent upon the combination of values taken by α, β, and γ and may be either unimodal or monotonically decreasing, with a potential singularity approaching the lower boundary of its domain. In addition, the tails of the PDF are "thin", in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The four-parameter version is sometimes referred to as the generalized gamma distribution, while the two-parameter form GammaDistribution[α,β] (which is equivalent to GammaDistribution[α,β,1,0]) is often referred to as "the" gamma distribution.
  • The (two-parameter) gamma distribution dates back to the 1830s work of Laplace, who obtained it as a posterior conjugate prior to distribution for the precision of normal variates, though the generalizations to three- and four-parameter forms can be traced back to Liouville's work on the Dirichlet integral formula. The name of the gamma distribution derives from the presence of the gamma function in its PDF. The gamma distribution is used to model a number of quantities across various fields. In statistics, the gamma distribution is the distribution associated with the sum of squares of independent unit normal variables and has been used to approximate the distribution of positive definite quadratic forms (i.e. those having the form ) in multinormally distributed variables. The gamma distribution has also been used in many other fields, including meteorology, mathematical finance, statistical ecology, population dynamics, genomics, neuroscience, and actuarial science.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a gamma distribution. Distributed[x,GammaDistribution[α,β,γ,μ]], written more concisely as xGammaDistribution[α,β,γ,μ], can be used to assert that a random variable x is distributed according to a gamma distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions may be given using PDF[GammaDistribution[α,β,γ,μ],x] and CDF[GammaDistribution[α,β,γ,μ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a gamma distribution, EstimatedDistribution to estimate a gamma parametric distribution from given data, and FindDistributionParameters to fit data to a gamma distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic gamma distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic gamma distribution.
  • TransformedDistribution can be used to represent a transformed gamma distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a gamma distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving gamma distributions.
  • The gamma distribution is related to several other distributions. As previously noted, GammaDistribution is firmly rooted in its relationship to NormalDistribution and MultinormalDistribution and is the conjugate prior (in Bayesian inference) to a handful of distributions, including PoissonDistribution, NormalDistribution, ExponentialDistribution, and GompertzMakehamDistribution. GammaDistribution generalizes ChiSquareDistribution (the PDF of GammaDistribution[ν/2,,2,0] is the same as that of ChiSquareDistribution[ν]), ExponentialDistribution (the PDF of ExponentialDistribution[1/λ] is the same as that of GammaDistribution[1,λ]), and MaxwellDistribution (the PDF of MaxwellDistribution[σ] is precisely the same as GammaDistribution[3/2,σ,2,0]). It can be transformed to obtain distributions such as InverseGammaDistribution, MoyalDistribution, and LogGammaDistribution. GammaDistribution is also related to PearsonDistribution, ErlangDistribution, BetaDistribution, ExpGammaDistribution, RayleighDistribution, ChiDistribution, WeibullDistribution, and StudentTDistribution.

Examples

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Basic Examples  (8)Summary of the most common use cases

Probability density function of a gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-jgw0ur
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-46x0tj
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Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0g7i85n6c86-5k5ir5
Direct link to example
Out[3]=3

Cumulative distribution function of a gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-0gdx3h
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-wn1d14
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Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0g7i85n6c86-pxmuhh
Direct link to example
Out[3]=3

Mean and variance of a gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-wdk
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-r2d
Direct link to example
Out[2]=2

Median of a gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-jpc9ef
Direct link to example
Out[1]=1

Probability density function of a generalized gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-xm25ss
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-x7nrjq
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0g7i85n6c86-1ydlgx
Direct link to example
Out[3]=3

Cumulative distribution function of a generalized gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-3rcwk7
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-zfgyku
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0g7i85n6c86-odto6g
Direct link to example
Out[3]=3

Mean and variance of a generalized gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-hdr8qb
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-y3azkg
Direct link to example
Out[2]=2

Median of a generalized gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-f28src
Direct link to example
Out[1]=1

Scope  (12)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-qhtk5j
Direct link to example

Compare its histogram to the PDF:

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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-03mwaz
Direct link to example
Out[2]=2

Generate a set of pseudorandom numbers that have generalized gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-wrjnbe
Direct link to example

Compare its histogram to the PDF:

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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-jxitl8
Direct link to example
Out[2]=2

Distribution parameters estimation:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-45b7g2
Direct link to example

Estimate the distribution parameters from sample data:

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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-epi747
Direct link to example
Out[2]=2

Compare the density histogram of the sample with the PDF of the estimated distribution:

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In[3]:=3
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https://wolfram.com/xid/0g7i85n6c86-f8ui5o
Direct link to example
Out[3]=3

Skewness depends only on the shape parameters α and γ:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-dkwm33
Direct link to example
Out[1]=1

Skewness of gamma distribution:

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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-smp
Direct link to example
Out[2]=2

In the limit, gamma distribution becomes symmetric:

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In[3]:=3
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https://wolfram.com/xid/0g7i85n6c86-k422in
Direct link to example
Out[3]=3

Skewness of generalized gamma distribution:

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In[4]:=4
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https://wolfram.com/xid/0g7i85n6c86-ek34pv
Direct link to example
Out[4]=4

Kurtosis depends only on the shape parameters α and γ:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-9igkqq
Direct link to example
Out[1]=1

Kurtosis of gamma distribution:

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In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-qke
Direct link to example
Out[2]=2

In the limit kurtosis nears the kurtosis of NormalDistribution:

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In[3]:=3
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https://wolfram.com/xid/0g7i85n6c86-bnpxuo
Direct link to example
Out[3]=3

Kurtosis of generalized gamma distribution:

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-yz2fre
Direct link to example
Out[4]=4

Different moments with closed forms as functions of parameters:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-js043h
Direct link to example

Moment:

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In[2]:=2
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https://wolfram.com/xid/0g7i85n6c86-rx074o
Direct link to example
Out[2]=2

Closed form for symbolic order:

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In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-xo1fnb
Direct link to example
Out[3]=3

CentralMoment:

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-pknsqa
Direct link to example
Out[4]=4

Closed form for symbolic order:

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In[5]:=5
✖
https://wolfram.com/xid/0g7i85n6c86-o5ehui
Direct link to example
Out[5]=5

FactorialMoment:

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In[6]:=6
✖
https://wolfram.com/xid/0g7i85n6c86-zg9ct4
Direct link to example
Out[6]=6

Cumulant:

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In[7]:=7
✖
https://wolfram.com/xid/0g7i85n6c86-9gzmth
Direct link to example
Out[7]=7

Closed form for symbolic order:

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In[8]:=8
✖
https://wolfram.com/xid/0g7i85n6c86-t1zkc3
Direct link to example
Out[8]=8

Different moments of generalized gamma distribution:

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In[1]:=1
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https://wolfram.com/xid/0g7i85n6c86-f4yfjm
Direct link to example

Moment:

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In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-xqhud7
Direct link to example
Out[2]=2

CentralMoment:

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In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-5udluv
Direct link to example
Out[3]=3

FactorialMoment:

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-z5zkug
Direct link to example
Out[4]=4

Cumulant:

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In[5]:=5
✖
https://wolfram.com/xid/0g7i85n6c86-kshtug
Direct link to example
Out[5]=5

Hazard function of a gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-w9468z
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-ggals4
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-l4cfan
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-nre5li
Direct link to example
Out[4]=4

Hazard function of a generalized gamma distribution with :

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-q3hboz
Direct link to example
Out[1]=1

With :

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In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-clzwrp
Direct link to example
Out[2]=2

Quantile function of a gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-tp9xac
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-jm6ggb
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-s0begi
Direct link to example
Out[3]=3

Quantile function of a generalized gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-t8khg4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-mcaxii
Direct link to example
Out[2]=2

Consistent use of Quantity in parameters yields QuantityDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-grnpzf
Direct link to example
Out[1]=1

Find median time:

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In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-8tfiv
Direct link to example
Out[2]=2

Applications  (6)Sample problems that can be solved with this function

The lifetime of a device has gamma distribution. Find the reliability of the device:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-4m8ue6
Direct link to example
Out[1]=1

The hazard function increasing in time for :

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In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-xz8iwk
Direct link to example
Out[2]=2

Find the reliability of two such devices in series:

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In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-xfhpfa
Direct link to example
Out[3]=3

Find the reliability of two such devices in parallel:

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-7trpvs
Direct link to example
Out[4]=4

Compare the reliability of both systems for and :

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In[5]:=5
✖
https://wolfram.com/xid/0g7i85n6c86-qqswjr
Direct link to example
Out[5]=5

A device has three lifetime stages: A, B, and C. The time spent in each phase follows an exponential distribution with a mean time of 10 hours; after phase C, a failure occurs. Find the distribution of the time to failure of this device:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-5i8qj1
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-5ya45a
Direct link to example
Out[2]=2

Find the mean time to failure:

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In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-q2gtyu
Direct link to example
Out[3]=3

Find the probability that such a device would be operational for at least 40 hours:

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-unvo2d
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0g7i85n6c86-2qc2wn
Direct link to example
Out[5]=5

Simulate time to failure for 30 independent devices:

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In[6]:=6
✖
https://wolfram.com/xid/0g7i85n6c86-rmigql
Direct link to example
Out[6]=6

In the morning rush hour, customers enter a coffee shop at a rate of 8 customers every 10 minutes. The time between customer arrivals follows an exponential distribution and the time between arrivals follows a GammaDistribution[k,1/λ] distribution. Find the probability of at least 40 customers arriving in 45 minutes:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-feq10c
Direct link to example
Out[1]=1

Find the average waiting time until the 40^(th) customer arrives:

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In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-kq8n70
Direct link to example
Out[2]=2

Find the probability that the time until the 40^(th) customer arrives is at least 1 hour:

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In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-ic4mj
Direct link to example
Out[3]=3

Simulate the waiting time until the 40^(th) customer arrives during rush hour over 30 days:

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-c1wlwk
Direct link to example
Out[4]=4

Mixtures of gamma distributions can be used to model multimodal data:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-z240sm
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-oc3rab
Direct link to example
Out[2]=2

Histogram of waiting times for eruptions of the Old Faithful geyser exhibits two modes:

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In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-uza3m3
Direct link to example
Out[3]=3

Fit a MixtureDistribution to the data:

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-ywlicz
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0g7i85n6c86-4l1127
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0g7i85n6c86-xnf1zt
Direct link to example
Out[6]=6

Compare the histogram to the PDF of estimated distribution:

Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0g7i85n6c86-jj3do1
Direct link to example
Out[7]=7

Find the probability that the waiting time is over 80 minutes:

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In[8]:=8
✖
https://wolfram.com/xid/0g7i85n6c86-34sioc
Direct link to example
Out[8]=8

Find average waiting time:

Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0g7i85n6c86-clhrkj
Direct link to example
Out[9]=9

Find most common waiting times:

Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0g7i85n6c86-pzqe67
Direct link to example
Out[10]=10

Simulate waiting times for the next 60 eruptions:

Copy to clipboard.
In[11]:=11
✖
https://wolfram.com/xid/0g7i85n6c86-rym6x3
Direct link to example
Copy to clipboard.
In[12]:=12
✖
https://wolfram.com/xid/0g7i85n6c86-ekk8we
Direct link to example
Out[12]=12

LogNormalDistribution data can be modeled by a gamma distribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-fg63nf
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-jsaju
Direct link to example
Out[2]=2

Compare the histogram to the PDF of estimated distribution:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-ryv8f6
Direct link to example
Out[3]=3

Comparing log-likelihoods with estimation by lognormal distribution:

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-156q9b
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0g7i85n6c86-2fsujd
Direct link to example
Out[5]=5

Stacy distribution is a special case of generalized GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-xfuj29
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-pn3d0k
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-2h7lw
Direct link to example
Out[3]=3

Properties & Relations  (32)Properties of the function, and connections to other functions

Gamma distribution is closed under scaling by a positive factor:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-mbvwxc
Direct link to example
Out[1]=1

Generalized gamma distribution is closed under translation and scaling by a positive factor:

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-zb8rju
Direct link to example
Out[2]=2

Gamma distribution is closed under addition:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-9aifwc
Direct link to example
Out[1]=1

For identically distributed variables:

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-724t15
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-2bavp5
Direct link to example
Out[3]=3
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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-niknri
Direct link to example
Out[4]=4

GammaDistribution[α,β] converges to a normal distribution as α->∞:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-gf5dne
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-dxb20o
Direct link to example
Out[2]=2

Relationships to other distributions:

ChiSquareDistribution is a special case of gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-n5r
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-d3eh0g
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-36fjyi
Direct link to example
Out[3]=3

Scaled ChiSquareDistribution follows gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-mxcd7c
Direct link to example
Out[1]=1

ChiDistribution is a special case of GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-9lmra4
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-zsx0sz
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-gp649d
Direct link to example
Out[3]=3

ExponentialDistribution is a special case of gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-my7xgl
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-0my6az
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-ejuasl
Direct link to example
Out[3]=3

Sum of variates from ExponentialDistribution has gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-jxz6or
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-pklxrb
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-rszauw
Direct link to example
Out[3]=3

The case :

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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-1ay9ln
Direct link to example
Out[4]=4

Gamma distribution and InverseGammaDistribution have an inverse relationship:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-dnmum3
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-fgdsyu
Direct link to example
Out[2]=2
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In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-bfhwph
Direct link to example
Out[3]=3
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In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-b8s9et
Direct link to example
Out[4]=4

The generalized gamma distribution simplifies to a gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-n50t29
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-bj6kpc
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-l49kr
Direct link to example
Out[3]=3

MaxwellDistribution is a special case of GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-uv8mbt
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-izjymu
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-1vzh66
Direct link to example
Out[3]=3

MoyalDistribution is a transformation of a GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-visqwi
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-6cbn85
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-03w6mi
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-dv9mgp
Direct link to example
Out[4]=4

RayleighDistribution is a special case of GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-i22zjc
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-d75cft
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-ik1ogt
Direct link to example
Out[3]=3

NakagamiDistribution is a special case of GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-qa07k7
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-11xhvn
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-yhulqh
Direct link to example
Out[3]=3

WeibullDistribution is a special case of generalized gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-1ym5gs
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-nn5fw
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-ksry65
Direct link to example
Out[3]=3

HalfNormalDistribution is a special case of generalized gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-u0fr2w
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-bvjfbm
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-dx1m1z
Direct link to example
Out[3]=3

Generalized gamma distribution can be obtained as a transformation from gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-emg0q9
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-kvgv5g
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-nzanrl
Direct link to example
Out[3]=3

ErlangDistribution is a special case of gamma distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-1itjbf
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-brty7o
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-daqy8z
Direct link to example
Out[3]=3

Gamma distribution is related to LogGammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-hsuq8g
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-1et4jg
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-lgm52b
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-u24x8k
Direct link to example
Out[4]=4

GammaDistribution is related to ExpGammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-16lit8
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-82qxqd
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-ntchhx
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-jyf6rv
Direct link to example
Out[4]=4

BetaPrimeDistribution can be obtained as a quotient of generalized GammaDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-i5cosi
Direct link to example
Out[1]=1

ParetoDistribution can be obtained as a quotient of GammaDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-8i6sh8
Direct link to example
Out[1]=1

GammaDistribution is a special case of type 3 PearsonDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-yh1j16
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-bcxz8w
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-sk48d6
Direct link to example
Out[3]=3

BetaDistribution can be obtained as a transformation of two independent gamma variables:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-cxy6u
Direct link to example
Out[1]=1

KDistribution can be obtained from ExponentialDistribution and GammaDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-4n5qhk
Direct link to example
Out[1]=1

Difference of gamma distributions follows VarianceGammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-n7odkt
Direct link to example
Out[1]=1

KDistribution can be represented as a parameter mixture of RayleighDistribution and GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-rp7do3
Direct link to example
Out[1]=1

NegativeBinomialDistribution is a mixture of PoissonDistribution and GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-qkgjoa
Direct link to example
Out[1]=1

GeometricDistribution is a mixture of PoissonDistribution and GammaDistribution:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-rbh64k
Direct link to example
Out[1]=1

StudentTDistribution is a parameter mixture of a NormalDistribution with GammaDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-r3i4sw
Direct link to example
Out[1]=1

ParetoDistribution can be obtained as a quotient of ExponentialDistribution and GammaDistribution:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-gruq01
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0g7i85n6c86-lvqs29
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0g7i85n6c86-eh4z3p
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-7r8hq7
Direct link to example
Out[4]=4

Possible Issues  (2)Common pitfalls and unexpected behavior

GammaDistribution is not defined when either α or β is not a positive real number:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-ebk
Direct link to example
Out[1]=1

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-t70
Direct link to example
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

PDFs for different β values with CDF contours:

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In[1]:=1
✖
https://wolfram.com/xid/0g7i85n6c86-io0104
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0g7i85n6c86-ceikus
Direct link to example
Out[4]=4
Wolfram Research (2007), GammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GammaDistribution.html (updated 2016).
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✖
Wolfram Research (2007), GammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GammaDistribution.html (updated 2016).

Text

Wolfram Research (2007), GammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GammaDistribution.html (updated 2016).

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✖
Wolfram Research (2007), GammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GammaDistribution.html (updated 2016).

CMS

Wolfram Language. 2007. "GammaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/GammaDistribution.html.

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✖
Wolfram Language. 2007. "GammaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/GammaDistribution.html.

APA

Wolfram Language. (2007). GammaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GammaDistribution.html

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✖
Wolfram Language. (2007). GammaDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GammaDistribution.html

BibTeX

@misc{reference.wolfram_2025_gammadistribution, author="Wolfram Research", title="{GammaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/GammaDistribution.html}", note=[Accessed: 29-March-2025 ]}

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✖
@misc{reference.wolfram_2025_gammadistribution, author="Wolfram Research", title="{GammaDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/GammaDistribution.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_gammadistribution, organization={Wolfram Research}, title={GammaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/GammaDistribution.html}, note=[Accessed: 29-March-2025 ]}

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✖
@online{reference.wolfram_2025_gammadistribution, organization={Wolfram Research}, title={GammaDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/GammaDistribution.html}, note=[Accessed: 29-March-2025 ]}