## Thursday, October 17, 2013

### Does Anyone Understand the Gamma Function or a Gamma Distribution?

I have to take another test in my current master's course and I feel clueless.  The worst part is not getting any feedback, or answers to questions, from my instructor.

I thought I was paying a lot of money to get an education in a difficult subject, I didn't realize I was going to pay a lot of money just to watch videos and have to figure everything else out on my own (or not figure it out, as the case may be).

Yes, I'm frustrated.

Anonymous said...

You can use the Gamma function to express fractional factorials.

PeggyU said...

No, but you've made me curious.

Darren said...

Why does the gamma function have the form it has? How was it created/discovered in the first place, such that it has that form?

Why does a gamma distribution have the form it has?

I get the idea about "fractional factorials" but still can't calculate them. And that doesn't answer the questions above.

I don't want just to memorize a formula. I want to understand what the heck I'm doing.

Joshua Sasmor said...

The gamma function is a result of applying the analytic continuation to a polynomial multiple of the power series of e^z, which gives the n! when evaluated at (n+1). So it was _rigged_ to work out so that Gamma(n)= (n-1)! The analytic continuation is discussed in complex analysis (usually). So I'm not sure the ideas are easily explainable without that.

Unfortunately, I never mastered this part of probability, and had to just memorize formulas here.

Anonymous said...

Truthfully, I don't know the whole history. I can find a couple things that let me make some guesses, though. Start with Wolfram's history information:
http://functions.wolfram.com/GammaBetaErf/Gamma/35/

If you start with the factorial function and try to imagine a continuous version of it, you'd be led to looking at functions which have some special properties. The first is a simple recurrence relationship:
f(x+1)=xf(x)
which the factorial function satisfies, and should be true for any generalization of it. There is only one function which meets all of the desired criteria: this is the gamma function. If you went looking for the work by Bohr and Mollerup you could probably learn what all the criteria are, and how every other function gets eliminated (Wikipedia has a brief summary and some further links to explore- try Bohr Mollerup).

The gamma distribution is, in a sense, a generalization of an exponential distribution. If you have a set of independent variables which have exponential distributions, their sum will have a gamma distribution. This provides some connection to other distributions as well (e.g. Poisson, which is a commonly found experimental distribution in the physical sciences). A cursory glance finds this page: