| 释义 |
Gamma DistributionA general type of statistical Distribution which is related to the Beta Distribution and arises naturally inprocesses for which the waiting times between Poisson Distributed events are relevant. Gammadistributions have two free parameters, labeled  and  , a few of which are illustrated above.
Given a Poisson Distribution with a rate of change  , the Distribution Function  giving thewaiting times until the  th change is
for  . The probability function  is then obtained by differentiating ,
Now let  and define  to be the time between changes. Then the above equationcan be written
 | (3) |
 | (4) |
In order to find the Moments of the distribution, let
so
and the logarithmic Moment-Generating function is
The Mean, Variance, Skewness, and Kurtosis 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 withparameters
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.
 | (18) |
 | (19) |
 | (20) |
 | (21) |
 | (22) |
The sum  therefore has the distribution
 | (24) |
where  is the Beta Function, which is a Beta Distribution.
If  and  are gamma variates with parameters  and  , the  is a variate with a BetaPrime Distribution with parameters and . Let
 | (26) |
 | (27) |
 | (28) |
The ratio therefore has the distribution
 | (30) |
The ``standard form'' of the gamma distribution is given by letting  , so  and
so the Moments about 0 are
 | (32) |
 is the Pochhammer Symbol. The Moments about  are then
The Moment-Generating Function is
 | (37) |
 | (38) |
 | (39) |
 is a Normal variate with Mean  and Standard Deviation  ,then
 | (40) |
 .See also Beta Distribution, Chi-Squared Distribution
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 534, 1987. |
