Gamma-distribution
A continuous probability distribution concentrated on the positive semi-axis with density
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where is a parameter assuming positive values, and
is Euler's gamma-function:
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The corresponding distribution function for is zero, and for
it is expressed by the formula
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The integral on the right-hand side is called the incomplete gamma-function. The density is unimodal and for
it attains the maximum
at the point
. If
the density
decreases monotonically with increasing
, and if
,
increases without limit. The characteristic function of the gamma-distribution has the form
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The moments of the gamma-distribution are given by the formula
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In particular, the mathematical expectation and variance are equal to . The set of gamma-distributions is closed with respect to the operation of convolution:
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Gamma-distributions play a significant, though not always an explicit, role in applications. In the particular case of one obtains the exponential density. In queueing theory, the gamma-distribution for an
which assumes integer values is known as the Erlang distribution. In mathematical statistics gamma-distributions frequently occur owing to the close connection with the normal distribution, since the sum of the squares
of independent
normally-distributed random variables has density
and is known as the "chi-squared" distribution with
degrees of freedom. For this reason the gamma-distribution is involved in many important distributions in problems of mathematical statistics dealing with quadratic forms of normally-distributed random variables (e.g. the Student distribution, the Fisher
-distribution and the Fisher
-distribution). If
and
are independent and are distributed with densities
and
, then the random variable
has density
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which is known as the density of the beta-distribution. The densities of linear functions of random variables
obeying the gamma-distribution constitute a special class of distributions — the so-called "type III" family of Pearson distributions. The density of the gamma-distribution is the weight function of the system of orthogonal Laguerre polynomials. The values of the gamma-distribution may be calculated from tables of the incomplete gamma-function [1], [2].
References
[1] | V.I. Pagurova, "Tables of the incomplete gamma-function" , Moscow (1963) (In Russian) MR0159040 |
[2] | K. Pearson (ed.) , Tables of the incomplete gamma function , Cambridge Univ. Press (1957) |
Comments
References
[a1] | N.L. Johnson, S. Kotz, "Distributions in statistics" , 1. Continuous univariate distributions , Wiley (1970) MR0270476 MR0270475 Zbl 0213.21101 |
[a2] | L.J. Comrie, "Chambers's six-figure mathematical tables" , II , Chambers (1949) |
Gamma-distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gamma-distribution&oldid=33879