Gamma distribution

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"),"Uk" and "Vl" 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 "Xi" are independently distributed Γ(α"i",θ) respectively, then the vector ("X"1 / "S", ..., "Xn" / "S"), where "S" = "X"1 + ... + "Xn", 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


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