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GammaDistribution—Wolfram Language Documentation

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BUILT-IN SYMBOL

GammaDistribution[α,β]

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

GammaDistribution[α,β,γ,μ]

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

Details Background & Context Examplesopen allclose all Basic Examples  (8)

Probability density function of a gamma distribution:

Cumulative distribution function of a gamma distribution:

Mean and variance of a gamma distribution:

Median of a gamma distribution:

Probability density function of a generalized gamma distribution:

Cumulative distribution function of a generalized gamma distribution:

Mean and variance of a generalized gamma distribution:

Median of a generalized gamma distribution:

Scope  (12)

Generate a sample of pseudorandom numbers from a gamma distribution:

Compare its histogram to the PDF:

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

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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

Skewness depends only on the shape parameters α and γ:

Skewness of gamma distribution:

In the limit, gamma distribution becomes symmetric:

Skewness of generalized gamma distribution:

Kurtosis depends only on the shape parameters α and γ:

Kurtosis of gamma distribution:

In the limit kurtosis nears the kurtosis of NormalDistribution:

Kurtosis of generalized gamma distribution:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Closed form for symbolic order:

Different moments of generalized gamma distribution:

Moment:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function of a gamma distribution:

Hazard function of a generalized gamma distribution with :

With :

Quantile function of a gamma distribution:

Quantile function of a generalized gamma distribution:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find median time:

Applications  (6)

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

The hazard function increasing in time for :

Find the reliability of two such devices in series:

Find the reliability of two such devices in parallel:

Compare the reliability of both systems for and :

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:

Find the mean time to failure:

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

Simulate time to failure for 30 independent devices:

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:

Find the average waiting time until the 40 customer arrives:

Find the probability that the time until the 40 customer arrives is at least 1 hour:

Simulate the waiting time until the 40 customer arrives during rush hour over 30 days:

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

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

Fit a MixtureDistribution to the data:

Compare the histogram to the PDF of estimated distribution:

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

Find average waiting time:

Find most common waiting times:

Simulate waiting times for the next 60 eruptions:

LogNormalDistribution data can be modeled by a gamma distribution:

Compare the histogram to the PDF of estimated distribution:

Comparing log-likelihoods with estimation by lognormal distribution:

Stacy distribution is a special case of generalized GammaDistribution:

Properties & Relations  (32) Possible Issues  (2)

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

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

Neat Examples  (1)

PDFs for different β values with CDF contours:

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).

CMS

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

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: 12-July-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: 12-July-2025 ]}


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