GammaDistribution

Generate a set of pseudorandom numbers that are gamma distributed:

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 and :

Skewness of gamma distribution:

In the limit, gamma distribution becomes symmetric:

Skewness of generalized gamma distribution:

Kurtosis depends only on the shape parameters and :

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:

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 12 customers every 10 minutes. The time between customer arrivals follows an exponential distribution and the time between arrivals follows a GammaDistribution 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:

Parameter influence on the CDF of a gamma distribution for each :

Parameter influence on the CDF of a generalized gamma distribution for each :

Gamma distribution is closed under scaling by a positive factor:

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

GammaDistribution converges to a normal distribution as ->∞:

Sum of gamma-distributed variables follows a gamma distribution:

For identically distributed variables:

Relationships to other distributions:

ChiSquareDistribution is a special case of gamma distribution:

ChiDistribution is a special case of GammaDistribution:

ExponentialDistribution is a special case of gamma distribution:

Sum of variates from ExponentialDistribution has gamma distribution:

The case :

Gamma distribution and InverseGammaDistribution have an inverse relationship:

The generalized gamma distribution simplifies to a gamma distribution:

MaxwellDistribution is a special case of GammaDistribution:

MoyalDistribution is a transformation of a GammaDistribution:

RayleighDistribution is a special case of GammaDistribution:

NakagamiDistribution is a special case of GammaDistribution:

WeibullDistribution is a special case of generalized gamma distribution:

HalfNormalDistribution is a special case of generalized gamma distribution:

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

ErlangDistribution is a special case of gamma distribution:

Gamma distribution is related to LogGammaDistribution:

GammaDistribution is related to ExpGammaDistribution:

Quotient of two independent gamma-distributed random variables has BetaPrimeDistribution:

GammaDistribution is a special case of type 3 PearsonDistribution:

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

KDistribution can be obtained from ExponentialDistribution and GammaDistribution:

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

NegativeBinomialDistribution is a mixture of PoissonDistribution and GammaDistribution:

GeometricDistribution is a mixture of PoissonDistribution and GammaDistribution:

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

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

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