## Gamma Distribution

A general type of statistical Distribution which 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.

Given a Poisson Distribution with a rate of change , the Distribution Function giving the waiting times until the th change is

 (1)

for . The probability function is then obtained by differentiating ,
 (2)

Now let and define to be the time between changes. Then the above equation can be written
 (3)

The Characteristic Function describing this distribution is
 (4)

and the Moment-Generating Function is
 (5)

In order to find the Moments of the distribution, let
 (6) (7)

so
 (8)

and the logarithmic Moment-Generating function is
 (9) (10) (11)

The Mean, Variance, Skewness, and Kurtosis are then
 (12) (13) (14) (15)

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

 (16) (17)

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)

Let
 (19)

 (20)

then the Jacobian is
 (21)

so
 (22)

 (23)

The sum therefore has the distribution
 (24)

which is a gamma distribution, and the ratio has the distribution
 (25)

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

 (26)

then the Jacobian is
 (27)

so
 (28)

 (29)

The ratio therefore has the distribution
 (30)

which is a Beta Prime Distribution with parameters .

The standard form'' of the gamma distribution is given by letting , so and

 (31)

so the Moments about 0 are
 (32)

where is the Pochhammer Symbol. The Moments about are then
 (33) (34) (35) (36)

The Moment-Generating Function is
 (37)

and the Cumulant-Generating Function is
 (38)

so the Cumulants are
 (39)

If is a Normal variate with Mean and Standard Deviation , then
 (40)

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.