Gamma Distribution
المؤلف:
Beyer, W. H.
المصدر:
CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press
الجزء والصفحة:
...
6-4-2021
1622
Gamma Distribution
A gamma distribution is a general type of statistical distribution that is related to the beta distribution and arises naturally in processes for which the waiting times between Poisson distributed events are relevant. Gamma distributions have two free parameters, labeled 
and 
, a few of which are illustrated above.
Consider the distribution function 
of waiting times until the 
th Poisson event given a Poisson distribution with a rate of change 
,
for 
, where 
is a complete gamma function, and 
an incomplete gamma function. With 
an integer, this distribution is a special case known as the Erlang distribution.
The corresponding probability function 
of waiting times until the 
th Poisson event is then obtained by differentiating 
,
Now let 
(not necessarily an integer) and define 
to be the time between changes. Then the above equation can be written
 |
(13)
|
for 
. This is the probability function for the gamma distribution, and the corresponding distribution function is
 |
(14)
|
where 
is a regularized gamma function.
It is implemented in the Wolfram Language as the function GammaDistribution[alpha, theta].
The characteristic function describing this distribution is
where ](https://mathworld.wolfram.com/images/equations/GammaDistribution/Inline59.gif)
is the Fourier transform with parameters 
, and the moment-generating function is
giving moments about 0 of
 |
(19)
|
(Papoulis 1984, p. 147).
In order to explicitly find the moments of the distribution using the moment-generating function, let
so
giving the logarithmic moment-generating function as
The mean, variance, skewness, and kurtosis excess are then
The gamma distribution is closely related to other statistical distributions. If 
, 
, ..., 
are independent random variates with a gamma distribution having parameters 
, 
, ..., 
, then 
is distributed as gamma with parameters
Also, if 
and 
are independent random variates with a gamma distribution having parameters 
and 
, then 
is a beta distribution variate with parameters 
. Both can be derived as follows.
 |
(34)
|
Let
 |
(35)
|
 |
(36)
|
then the Jacobian is
 |
(37)
|
so
 |
(38)
|
The sum 
therefore has the distribution
 |
(41)
|
which is a gamma distribution, and the ratio 
has the distribution
where 
is the beta function, which is a beta distribution.
If 
and 
are gamma variates with parameters 
and 
, the 
is a variate with a beta prime distribution with parameters 
and 
. Let
 |
(45)
|
then the Jacobian is
 |
(46)
|
so
 |
(47)
|
The ratio 
therefore has the distribution
 |
(50)
|
which is a beta prime distribution with parameters 
.
The "standard form" of the gamma distribution is given by letting 
, so 
and
so the moments about 0 are
where 
is the Pochhammer symbol. The moments about 
are then
The moment-generating function is
 |
(61)
|
and the cumulant-generating function is
 |
(62)
|
so the cumulants are
 |
(63)
|
If 
is a normal variate with mean 
and standard deviation 
, then
 |
(64)
|
is a standard gamma variate with parameter 
.
REFERENCES:
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 534, 1987.
Jambunathan, M. V. "Some Properties of Beta and Gamma Distributions." Ann. Math. Stat. 25, 401-405, 1954.
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 103-104, 1984.
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