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GammaDistribution

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GammaDistribution
represents a gamma distribution with shape parameter and scale parameter .
GammaDistribution
represents a generalized gamma distribution with shape parameters and , scale parameter , and location parameter .
  • 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 , , and to be any positive real numbers and to be any real number.
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:
Probability density function of a gamma distribution:
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Cumulative distribution function of a gamma distribution:
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Mean and variance of a gamma distribution:
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Median of a gamma distribution:
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Probability density function of a generalized gamma distribution:
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Cumulative distribution function of a generalized gamma distribution:
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Mean and variance of a generalized gamma distribution:
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Median of a generalized gamma distribution:
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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:
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 :
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^(th) customer arrives:
Find the probability that the time until the 40^(th) customer arrives is at least 1 hour:
Simulate the waiting time until the 40^(th) 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:
MoyalDistribution is a transformation of a 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:
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 represented as a parameter mixture of RayleighDistribution 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:
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