Gamma Distribution
The distribution with density on .
Example 4.34 (Gamma Distribution)
For and , the gamma distribution with parameters and is a continuous random variable with density
Proof of density validity. We have
We denote . Then,
If , then .
Moment generating function
For ,
obtained by recognising the integral as a constant multiple of the integral of the density.
Sums of gamma random variables
If and are independent, then for ,
so by uniqueness of the moment generating function. In particular, if are i.i.d. random variables, then .
General shape parameter
Replacing in the density by the gamma function
extends the definition to a general shape parameter: for we say when
Related
Stated in
- Example 4.34 (Gamma Distribution)§4.8.1 Gamma Distribution