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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 & ContextProbability 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:
Closed form for symbolic order:
Closed form for symbolic order:
Closed form for symbolic order:
Different moments of generalized gamma distribution:
Hazard function of a gamma distribution:
Hazard function of a generalized gamma distribution with :
Quantile function of a gamma distribution:
Quantile function of a generalized gamma distribution:
Consistent use of Quantity in parameters yields QuantityDistribution:
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 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). TextWolfram Research (2007), GammaDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GammaDistribution.html (updated 2016).
CMSWolfram Language. 2007. "GammaDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/GammaDistribution.html.
APAWolfram 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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