Gamma distribution

, eta_m = V_{3m} e^{-xi_m}.
# If $eta\_m\; >\; xi\_m^\{delta\; -\; 1\}\; e^\{-xi\_m\}$, then increment "m" and go to step 2.
# Assume $xi\; =\; xi\_m$ to be the realization of $Gamma\; (delta,\; 1)$Now, to summarize,

:$heta\; left(\; xi\; -\; sum\; \_\{i=1\}\; ^\{\; [k]\; \}\; \{ln\; U\_i\}\; ight)\; sim\; Gamma\; (k,\; heta),$where ["k"] is the integral part of "k", and "ξ" has been generated 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.

The GNU Scientific Library has robust routines for sampling many distributions including the Gamma distribution.

Related distributions

Specializations

* If $X\; sim\; \{Gamma\}(k=1,\; heta=1/lambda),$, then "X" has an exponential distribution with rate parameter λ.
* If $X\; sim\; \{Gamma\}(k=\; u/2,\; heta=2),$, 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 $X^2\; sim\; \{Gamma\}(3/2,\; 2a^2),$, then "X" has a Maxwell-Boltzmann distribution with parameter "a".
*$X\; sim\; mathrm\{SkewLogistic\}(\; heta),$, then $mathrm\{log\}(1\; +\; e^\{-X\})\; sim\; Gamma\; (1,\; heta),$

Others

* If "X" has a Γ("k", θ) distribution, then 1/"X" has an inverse-gamma distribution with parameters "k" and θ^-1.
* If "X" and "Y" are independently distributed Γ(α, θ) and Γ(β, θ) respectively, then "X" / ("X" + "Y") has a beta distribution with parameters α and β.
* If "X_i" are independently distributed Γ(α_"i",θ) respectively, then the vector ("X"₁ / "S", ..., "X_n" / "S"), where "S" = "X"₁ + ... + "X_n", follows a Dirichlet distribution with parameters α₁, ..., α_"n".
* For large "k" the gamma distribution converges to Gaussian distribution with mean $mu\; =\; k\; heta$ and variance $sigma^2\; =\; k\; heta^2$.
* The Gamma distribution is the conjugate prior for the precision of the normal distribution with known mean.

Applications

The gamma distribution is frequently a probability model for waiting times; for instance, in life testing, the waiting time until death is a random variable which is frequently modeled with a gamma distribution. [See Hogg and Craig Remark 3.3.1. for an explicit motivation.test]

See also

* Gamma process
*

Notes

References

* R. V. Hogg and A. T. Craig. "Introduction to Mathematical Statistics", 4th edition. New York: Macmillan, 1978. "(See Section 3.3.)"
* MathWorld|urlname=GammaDistribution|title=Gamma distribution
* [http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm Engineering Statistics Handbook]
* S. C. Choi and R. Wette. (1969) "Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias", Technometrics, 11(4) 683-690