STATS 426 Lecture Notes - Lecture 8: Gamma Distribution, Independent And Identically Distributed Random Variables, Orthogonal Matrix
Document Summary
The gamma distribution: the gamma function is a real-valued non-negative function de ned on (0, ) in the following manner. ( ) =z x 1 e x dx , > 0 . Two of these are listed below: (a) ( + 1) = ( ) , (b) (n) = (n 1)! (n integer) . Property (b) is an easy consequence of property (a). Start o with (n) and use property (a) recursively along with the fact that (1) = 1 (why?). To prove property (a), use integration by parts. Check that this is a bona de density function. 0 f (x, , ) dx through the substitution y = x and using the de nition of ( ) (verify!). The rst parameter is called the shape parameter and the second parameter is called the scale parameter. Xed the shape parameter regulates the shape of the gamma density. 1 exercise that justi es the term scale parameter for .
