BIGpedia - Gamma distribution - Encyclopedia and Dictionary Online

In probability theory and statistics, the gamma distribution is a continuous probability distribution.

Contents

1 Specification of the gamma distribution

1.1 Probability density function
1.2 Cumulative distribution function

2 Properties

3 Related distributions

4 References

5 See also

Specification of the gamma distribution

Probability density function

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

$f(x) = x^{k-1} \frac{e^{-x/\theta}}{\theta^k \, \Gamma(k)} \ \mathrm{for}\ x > 0 \,\!$

where $k > 0$ is the shape parameter and $θ > 0$ is the scale parameter of the gamma distribution. (NOTE: this parameterization is what 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:

$g(x) = x^{\alpha-1} \frac{\beta^{\alpha} \, e^{-\beta\,x} }{\Gamma(\alpha)} \ \mathrm{for}\ x > 0 \,\!$

Both are common because they are more convenient to use in certain fields with different parameterizations.

Cumulative distribution function

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

$F(x) = \int_0^x f(u)\,du = \frac{\gamma(k, x/\theta)}{\Gamma(k)} \,\!$

Properties

If $X 1$ has a gamma distribution with parameters $k 1$ and $θ$ , and $X 2$ has a gamma distribution with parameters $k 2$ and $θ$ , then $X 1 + X 2$ has a gamma distribution with parameters $k 1 + k 2$ and $θ$ . Or alternatively:

X 1

Gamma(k 1,θ)

and

X 2

Gamma(k 2,θ)

then

X 1 + X 2

Gamma(k 1 + k 2,θ)

The gamma distributions are infinitely divisible probability distributions.

Related distributions

$X$ ~ $Exponential(θ)$ is an exponential distribution if $X$ ~ $Gamma(1,θ)$ .
$Y$ ~ $Gamma(N,θ)$ is a gamma distribution if $Y = X_1 + \cdots + X_N$ and if the $X i$ ~ $Exponential(θ)$ are all independent and share the same parameter $θ$ .
$X$ ~ $χ 2 (ν)$ is a chi-square distribution if $X$ ~ $Gamma(ν / 2,1 / 2)$ .
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 $k th$ "arrival" in a one-dimensional Poisson process with intensity $1 / θ$ .
$X$ ~ $Gamma(k,θ)$ then $Y$ ~ $InvGamma(k,θ)$ if $Y = 1 / X$ , where $InvGamma$ is the family of inverse-gamma distributions.

References

R. V. Hogg and A. T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)