Gamma function
The gamma function and gamma distribution are two different things. This post is about the former and is a preparatory step to study the latter.
Definition. The gamma function is defined by
The integrand is smooth on
so its integrability is determined by its behavior at
and
. Because of the exponent, it is integrable in the neighborhood of
The singularity at
is integrable if
In all calculations involving the gamma function one should remember that its argument should be positive.
Properties
1) Factorial-like property. Integration by parts shows that
if
2) because
3) Combining the first two properties we see that for a natural
Thus the gamma function extends the factorial to non-integer
4)
Indeed, using the density of the standard normal
we see that
(replacing )
(replacing )
Many other properties are not required in this course.
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