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Probability density function 

Cumulative distribution function 

Parameters  

Support  
Probability density function (pdf)  
Cumulative distribution function (CDF)  
Mean  (see digamma function) 
(see digamma function) 
Median  No simple closed form  No simple closed form 
Mode  
Variance  (see trigamma function) 
(see trigamma function) 
Skewness  
Excess kurtosis  
Entropy  
Momentgenerating function (mgf)  
Characteristic function 
In probability theory and statistics, the gamma distribution is a twoparameter family of continuous probability distributions. The common exponential distribution and chisquared distribution are special cases of the gamma distribution. There are three different parametrizations 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.^{[1]}
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^{[2]} – 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.)
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).^{[3]}
A random variable X that is gammadistributed with shape k and scale θ is denoted by
The probability density function using the shapescale parametrization is
Here Γ(k) is the gamma function evaluated at k.
The cumulative distribution function is the regularized gamma function:
where γ(k, x/θ) is the lower incomplete gamma function.
It can also be expressed as follows, if k is a positive integer (i.e., the distribution is an Erlang distribution):^{[4]}
Alternatively, the gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter. A random variable X that is gammadistributed with shape α and rate β is denoted
The corresponding density function in the shaperate parametrization is
Both parametrizations are common because either can be more convenient depending on the situation.
The cumulative distribution function is the regularized gamma function:
where γ(α, βx) is the lower incomplete gamma function.
If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion:^{[4]}
The skewness is equal to , it depends only on the shape parameter (k) and approaches a normal distribution when k is large (approximately when k > 10).
Unlike the mode and the mean which have readily calculable formulas based on the parameters, the median does not have an easy closed form equation. The median for this distribution is defined as the value ν such that
A formula for approximating the median for any gamma distribution, when the mean is known, has been derived based on the fact that the ratio μ/(μ − ν) is approximately a linear function of k when k ≥ 1.^{[5]} The approximation formula is
where is the mean.
If X_{i} has a Gamma(k_{i}, θ) distribution for i = 1, 2, ..., N (i.e., all distributions have the same scale parameter θ), then
provided all X_{i} are independent.
For the cases where the X_{i} are independent but have different scale parameters see Mathai (1982) and Moschopoulos (1984).
The gamma distribution exhibits infinite divisibility.
If
then, for any c > 0,
Indeed, we know that if X is an exponential r.v. with rate λ then c X is an exponential r.v. with rate λ / c; the same thing is valid with Gamma variates (and this can be checked using the momentgenerating function, see, e.g., these notes, 10.4(ii)): multiplication by a positive constant c divides the rate (or, equivalently, multiplies the scale).
The gamma distribution is a twoparameter exponential family with natural parameters k − 1 and −1/θ (equivalently, α − 1 and −β), and natural statistics X and ln(X).
If the shape parameter k is held fixed, the resulting oneparameter family of distributions is a natural exponential family.
One can show that
or equivalently,
where ψ is the digamma function.
This can be derived using the exponential family formula for the moment generating function of the sufficient statistic, because one of the sufficient statistics of the gamma distribution is ln(x).
The information entropy is
In the k, θ parameterization, the information entropy is given by
The Kullback–Leibler divergence (KLdivergence), of Gamma(α_{p}, β_{p}) ("true" distribution) from Gamma(α_{q}, β_{q}) ("approximating" distribution) is given by^{[6]}
Written using the k, θ parameterization, the KLdivergence of Gamma(k_{p}, θ_{p}) from Gamma(k_{q}, θ_{q}) is given by
The Laplace transform of the gamma PDF is
The likelihood function for N iid observations (x_{1}, ..., x_{N}) is
from which we calculate the loglikelihood function
Finding the maximum with respect to θ by taking the derivative and setting it equal to zero yields the maximum likelihood estimator of the θ parameter:
Substituting this into the loglikelihood function gives
Finding the maximum with respect to k by taking the derivative and setting it equal to zero yields
There is no closedform solution for k. The function is numerically very well behaved, so if a numerical solution is desired, it can be found using, for example, Newton's method. An initial value of k can be found either using the method of moments, or using the approximation
If we let
then k is approximately
which is within 1.5% of the correct value.^{[7]} An explicit form for the Newton–Raphson update of this initial guess is:^{[8]}
With known k and unknown θ, the posterior density function for theta (using the standard scaleinvariant prior for θ) is
Denoting
Integration over θ can be carried out using a change of variables, revealing that 1/θ is gammadistributed with parameters α = Nk, β = y.
The moments can be computed by taking the ratio (m by m = 0)
which shows that the mean ± standard deviation estimate of the posterior distribution for θ is
Given the scaling property above, it is enough to generate gamma variables with θ = 1 as we can later convert to any value of β with simple division.
Suppose we wish to generate random variables from Gamma(n+δ,1), where n is a nonnegative integer and 0 < δ < 1. Using the fact that a Gamma(1, 1) distribution is the same as an Exp(1) distribution, and noting the method of generating exponential variables, we conclude that if U is uniformly distributed on (0, 1], then −ln(U) is distributed Gamma(1, 1). Now, using the "αaddition" property of gamma distribution, we expand this result:
where U_{k} are all uniformly distributed on (0, 1] and independent. All that is left now is to generate a variable distributed as Gamma(δ, 1) for 0 < δ < 1 and apply the "αaddition" property once more. This is the most difficult part.
Random generation of gamma variates is discussed in detail by Devroye,^{[9]}^{:401–428} noting that none are uniformly fast for all shape parameters. For small values of the shape parameter, the algorithms are often not valid.^{[9]}^{:406} For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter^{[10]} modified acceptance–rejection method Algorithm GD (shape k ≥ 1), or transformation method^{[11]} when 0 < k < 1. Also see Cheng and Feast Algorithm GKM 3^{[12]} or Marsaglia's squeeze method.^{[13]}
The following is a version of the AhrensDieter acceptance–rejection method:^{[10]}
A summary of this is
where
While the above approach is technically correct, Devroye notes that it is linear in the value of k and in general is not a good choice. Instead he recommends using either rejectionbased or tablebased methods, depending on context.^{[9]}^{:401–428}
In Bayesian inference, the gamma distribution is the conjugate prior to many likelihood distributions: the Poisson, exponential, normal (with known mean), Pareto, gamma with known shape σ, inverse gamma with known shape parameter, and Gompertz with known scale parameter.
The gamma distribution's conjugate prior is:^{[14]}
where Z is the normalizing constant, which has no closedform solution. The posterior distribution can be found by updating the parameters as follows:
where n is the number of observations, and x_{i} is the ith observation.
If the shape parameter of the gamma distribution is known, but the inversescale parameter is unknown, then a gamma distribution for the inversescale forms a conjugate prior. The compound distribution, which results from integrating out the inversescale, has a closed form solution, known as the compound gamma distribution.^{[15]}
This section requires expansion. (March 2009) 
The gamma distribution has been used to model the size of insurance claims^{[16]} and rainfalls.^{[17]} This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process. The gamma distribution is also used to model errors in multilevel Poisson regression models, because the combination of the Poisson distribution and a gamma distribution is a negative binomial distribution.
In neuroscience, the gamma distribution is often used to describe the distribution of interspike intervals.^{[18]} Although in practice the gamma distribution often provides a good fit, there is no underlying biophysical motivation for using it.
In bacterial gene expression, the copy number of a constitutively expressed protein often follows the gamma distribution, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the mean number of protein molecules produced by a single mRNA during its lifetime.^{[19]}
In genomics, the gamma distribution was applied in peak calling step (i.e. in recognition of signal) in ChIPchip^{[20]} and ChIPseq^{[21]} data analysis.
The gamma distribution is widely used as a conjugate prior in Bayesian statistics. It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution. It is also the conjugate prior for the exponential distribution.
The Wikibook Statistics has a page on the topic of: Gamma distribution 
