Gamma distribution
In probability theory and statistics, the gamma distribution is a continuous probability distribution. Its probability density function can be expressed in terms of the gamma function:
- <math> f(x) = x^{k-1} \frac{e^{-x/\theta}}{\Gamma(k)\,\theta^k}
\ \ \ \ \mathrm{for\ } x > 0<math>
where k > 0 is the shape parameter and θ > 0 is the scale parameter of the gamma distribution.
The cumulative distribution function can be expressed in terms of the incomplete gamma function,
- <math> F(x) = \int_0^x f(u)\,du
= \frac{\gamma(k, x/\theta)}{\Gamma(k)} <math>
The expected value and variance of a gamma random variable X are:
- <math>
\begin{matrix}
E(X) = k \theta \\
\\
\mathrm{var}(X) = k \theta^2
\end{matrix}
<math>
If <math>X_1<math> has a gamma distribution with parameters <math>k_1<math> and θ,
and <math>X_2<math> has a gamma distribution with parameters <math>k_2<math> and θ,
then <math>X_1 + X_2<math> has a gamma distribution with parameters <math>k_1 + k_2<math> and θ.
If k is equal to 1, the gamma distribution is an exponential distribution with parameter θ.
The sum of n exponential variables, all with the same parameter θ, is a gamma variable with parameters n and θ.
If k is an integer, the gamma distribution is an Erlang distribution (so named in honor of A.K. Erlang) and is the probability distribution of the waiting time of the kth "arrival" in a one-dimensional Poisson process with intensity 1/θ.
If k is a half-integer and θ = 2, then the gamma distribution is a chi-square distribution with 2 k degrees of freedom.
The gamma distributions are infinitely divisible probability distributions.
References
- R.V. Hogg and A.T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)
