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Gamma

Probability density function
Cumulative distribution function
Parameters	shape (real) scale (real)
Support
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean
Median	no simple closed form
Mode	for
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)	for
Characteristic function

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of k exponentially distributed random variables, each of which has mean θ.

Contents 1 Characterization 1.1 Probability density function 1.2 Cumulative distribution function 2 Properties 2.1 Summation 2.2 Scaling 2.3 Exponential family 2.4 Information entropy 2.5 Kullback-Leibler divergence 3 Maximum likelihood estimation 4 Generating Gamma random variables 5 Related distributions 5.1 Specializations 5.2 Others 6 References 7 See also

The probability density function of the gamma distribution can be expressed in terms of the gamma function:

where k > 0 is the shape parameter and θ > 0 is the scale parameter of the gamma distribution. (This parameterization is used in the infobox and the plots.)

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:

Both parameterizations are common because either can be more convenient depending on the situation.

The cumulative distribution function can be expressed in terms of the incomplete gamma function,

If X_i has a Gamma(α_i, β) distribution for i = 1, 2, ..., N, then

provided all X_i are independent.

The gamma distribution exhibits infinite divisibility.

For any t > 0 it holds that tX is distributed Gamma(k, tθ). That demonstrates that θ is a scale parameter.

The Gamma distribution is a two-parameter exponential family with natural parameters k − 1 and 1 / θ, and natural statistics X and ln(X).

The information entropy is given by:

where ψ(k) is the digamma function.

The directed Kullback-Leibler divergence between Gamma(α₀, β₀) and Gamma(α, β) is given by

The likelihood function for N iid observations is

from which we calculate the log-likelihood function

Finding the maximum with respect to θ by taking the derivative and setting it equal to zero yields the maximum likelihood estimate of the θ parameter:

Substituting this into the log-likelihood function gives:

Finding the maximum with respect to k by taking the derivative and setting it equal to zero yields:

where

is the digamma function.

There is no closed-form 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.^{[citation needed]} An explicit form for the Newton-Raphson update of this initial guess is given by Choi and Wette (1969) as the following expression

where denotes the trigamma function (the derivative of the digamma function).

The digamma and trigamma functions can be difficult to calculate with high precision. However, approximations known to be good to several significant figures can be computed using the following approximation formulae:

and

For details, see Choi and Wette (1969).

Given the scaling property above, it is enough to generate Gamma variables with β = 1 as we can later convert to any value of β with simple division.

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 "α-addition" 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 "α-addition" property once more. This is the most difficult part.

We provide an algorithm without proof. It is an instance of the acceptance-rejection method:

1. Let m be 1.
2. Generate V_{2m − 1} and V_2m — independent uniformly distributed on (0, 1] variables.
3. If , where , then go to step 4, else go to step 5.
4. Let . Go to step 6.
5. Let .
6. If , then increment m and go to step 2.
7. Assume ξ = ξ_m to be the realization of Gamma(δ,1).

Now, to summarize,

where [k] is the integral part of k, and ξ has been generating using the algorithm above with δ = {k} (the fractional part of k), U_k and V_l are distributed as explained above and are all independent.

• If , then X has an exponential distribution with rate parameter λ.
• If , then X is identical to χ²(ν), the chi-square distribution with ν degrees of freedom.
• If k is an integer, the gamma distribution is an Erlang distribution and is the probability distribution of the waiting time until the k-th "arrival" in a one-dimensional Poisson process with intensity 1/θ.
• If , then X has a Maxwell-Boltzmann distribution with parameter a.

• If X has a Gamma(k, θ) distribution, then 1/X has an inverse-gamma distribution with parameters k and θ^-1.
• If X and Y are independently distributed Gamma(α, θ) and Gamma(β, θ) respectively, then X / (X + Y) has a beta distribution with parameters α and β.
• If X_i are independently distributed Gamma(α_i,θ) respectively, then the vector (X₁ / S, ..., X_n / S), where S = X₁ + ... + X_n, follows a Dirichlet distribution with parameters α₁, ..., α_n. This holds true for any θ.

• inverse-gamma distribution
• chi-square distribution

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