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.
1) Factorial-like property. Integration by parts shows that
3) Combining the first two properties we see that for a natural
Thus the gamma function extends the factorial to non-integer
Indeed, using the density of the standard normal we see that
Many other properties are not required in this course.