GammaDistribution Class 
Namespace: Accord.Statistics.Distributions.Univariate
[SerializableAttribute] public class GammaDistribution : UnivariateContinuousDistribution, IFittableDistribution<double, GammaOptions>, IFittable<double, GammaOptions>, IFittable<double>, IFittableDistribution<double>, IDistribution<double>, IDistribution, ICloneable, ISampleableDistribution<double>, IRandomNumberGenerator<double>
The GammaDistribution type exposes the following members.
Name  Description  

GammaDistribution 
Constructs a Gamma distribution.
 
GammaDistribution(Double, Double) 
Constructs a Gamma distribution.

Name  Description  

Entropy 
Gets the entropy for this distribution.
(Overrides UnivariateContinuousDistributionEntropy.)  
Mean 
Gets the mean for this distribution.
(Overrides UnivariateContinuousDistributionMean.)  
Median 
Gets the median for this distribution.
(Inherited from UnivariateContinuousDistribution.)  
Mode 
Gets the mode for this distribution.
(Overrides UnivariateContinuousDistributionMode.)  
Quartiles 
Gets the Quartiles for this distribution.
(Inherited from UnivariateContinuousDistribution.)  
Rate 
Gets the inverse scale parameter β = 1/θ.
 
Scale 
Gets the distribution's scale
parameter θ (theta).
 
Shape 
Gets the distribution's
shape parameter k.
 
Standard 
Gets the standard Gamma distribution,
with scale θ = 1 and location k = 1.
 
StandardDeviation 
Gets the Standard Deviation (the square root of
the variance) for the current distribution.
(Inherited from UnivariateContinuousDistribution.)  
Support 
Gets the support interval for this distribution.
(Overrides UnivariateContinuousDistributionSupport.)  
Variance 
Gets the variance for this distribution.
(Overrides UnivariateContinuousDistributionVariance.) 
Name  Description  

Clone 
Creates a new object that is a copy of the current instance.
(Overrides DistributionBaseClone.)  
ComplementaryDistributionFunction 
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
(Inherited from UnivariateContinuousDistribution.)  
CumulativeHazardFunction 
Gets the cumulative hazard function for this
distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)  
DistributionFunction(Double) 
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionDistributionFunction(Double).)  
DistributionFunction(Double, Double) 
Gets the cumulative distribution function (cdf) for this
distribution in the semiclosed interval (a; b] given as
P(a < X ≤ b).
(Inherited from UnivariateContinuousDistribution.)  
Equals  Determines whether the specified object is equal to the current object. (Inherited from Object.)  
Estimate 
Estimates a new Gamma distribution from a given set of observations.
 
Finalize  Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)  
Fit(Double) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)  
Fit(Double, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)  
Fit(Double, Double) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)  
Fit(Double, Int32) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)  
Fit(Double, Double, GammaOptions) 
Fits the underlying distribution to a given set of observations.
 
Fit(Double, Double, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Overrides UnivariateContinuousDistributionFit(Double, Double, IFittingOptions).)  
Fit(Double, Int32, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)  
FromBayesian 
Constructs a Gamma distribution using α and β parameterization.
 
FromMean 
Constructs a Gamma distribution using k and μ parameterization.
 
Generate 
Generates a random observation from the current distribution.
(Overrides UnivariateContinuousDistributionGenerate.)  
Generate(Int32) 
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.)  
Generate(Int32, Double) 
Generates a random vector of observations from the current distribution.
(Overrides UnivariateContinuousDistributionGenerate(Int32, Double).)  
GetHashCode  Serves as the default hash function. (Inherited from Object.)  
GetRange 
Gets the distribution range within a given percentile.
(Inherited from UnivariateContinuousDistribution.)  
GetType  Gets the Type of the current instance. (Inherited from Object.)  
HazardFunction 
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)  
InverseDistributionFunction 
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
(Overrides UnivariateContinuousDistributionInverseDistributionFunction(Double).)  
LogCumulativeHazardFunction 
Gets the log of the cumulative hazard function for this
distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)  
LogProbabilityDensityFunction 
Gets the logprobability density function (pdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionLogProbabilityDensityFunction(Double).)  
Marsaglia 
Random Gammadistribution number generation
based on Marsaglia's Simple Method (2000).
 
MemberwiseClone  Creates a shallow copy of the current Object. (Inherited from Object.)  
ProbabilityDensityFunction 
Gets the probability density function (pdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionProbabilityDensityFunction(Double).)  
QuantileDensityFunction 
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
(Inherited from UnivariateContinuousDistribution.)  
Random(Double, Double) 
Generates a random observation from the
Gamma distribution with the given parameters.
 
Random(Double, Double, Int32) 
Generates a random vector of observations from the
Gamma distribution with the given parameters.
 
Random(Double, Double, Int32, Double) 
Generates a random vector of observations from the
Gamma distribution with the given parameters.
 
ToString 
Returns a String that represents this instance.
(Inherited from DistributionBase.)  
ToString(IFormatProvider) 
Returns a String that represents this instance.
(Inherited from DistributionBase.)  
ToString(String) 
Returns a String that represents this instance.
(Inherited from DistributionBase.)  
ToString(String, IFormatProvider) 
Returns a String that represents this instance.
(Overrides DistributionBaseToString(String, IFormatProvider).) 
Name  Description  

HasMethod 
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.)  
ToT  Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.)  
ToT  Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by Matrix.) 
The gamma distribution is a twoparameter family of continuous probability distributions. There are three different parameterizations in common use:
In each of these three forms, both parameters are positive real numbers. The parameterization with k and θ appears to be more common in econometrics and certain other applied fields, where e.g. the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution. This is the default construction method for this class.
The parameterization with α and β is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (aka rate) parameters, such as the λ of an exponential distribution or a Poisson distribution – or for that matter, the β of the gamma distribution itself. (The closely related inverse gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution.) In order to create a Gamma distribution using the Bayesian parameterization, you can use FromBayesian(Double, Double).
If k is an integer, then the distribution represents an Erlang distribution; i.e., the sum of k independent exponentially distributed random variables, each of which has a mean of θ (which is equivalent to a rate parameter of 1/θ).
The gamma distribution is the maximum entropy probability distribution for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).
References:
The following example shows how to create, test and compute the main functions of a Gamma distribution given parameters θ = 4 and k = 2:
// Create a Γdistribution with k = 2 and θ = 4 var gamma = new GammaDistribution(theta: 4, k: 2); // Common measures double mean = gamma.Mean; // 8.0 double median = gamma.Median; // 6.7133878418421506 double var = gamma.Variance; // 32.0 double mode = gamma.Mode; // 4.0 // Cumulative distribution functions double cdf = gamma.DistributionFunction(x: 0.27); // 0.002178158242390601 double ccdf = gamma.ComplementaryDistributionFunction(x: 0.27); // 0.99782184175760935 double icdf = gamma.InverseDistributionFunction(p: cdf); // 0.26999998689819171 // Probability density functions double pdf = gamma.ProbabilityDensityFunction(x: 0.27); // 0.015773530285395465 double lpdf = gamma.LogProbabilityDensityFunction(x: 0.27); // 4.1494220422235433 // Hazard (failure rate) functions double hf = gamma.HazardFunction(x: 0.27); // 0.015807962529274005 double chf = gamma.CumulativeHazardFunction(x: 0.27); // 0.0021805338793574793 // String representation string str = gamma.ToString(CultureInfo.InvariantCulture); // "Γ(x; k = 2, θ = 4)"