The gamma distribution is a probability distribution that is useful in actuarial modeling. From a mathematical point of view, the gamma function is the starting point of defining the gamma distribution. This post discusses the basic facts that are needed for defining the gamma distribution. Here’s the definition of the gamma function.
For any real number , define:
The following is the graph of the gamma function. It has a U shape. As , the graph goes up to infinity. As , the graph increases without bound. As is seen below, the gamma function coincides with the factorial function at the integers.
The gamma function is defined by the improper integral as described in . The improper integral converges. In fact, the integral converges for all complex numbers with positive real part. For our purposes at hand, we restrict to be positive real numbers. Showing that the integral in converges is important. For example, it will be nice to know that the density function of the gamma distribution sums to 1.0. To show that the integral in converges, we first make the observation: for some positive real number and for all ,
This is saying that the quantity dominates the quantity when is sufficiently large. This is because the exponential function increases at a much faster rate than the polynomial function . Then multiply both sides of by to obtain the following:
Now, break up the interval in into two pieces:
A basic fact. For any continuous function defined over a closed interval of finite length, the integral exists and has a finite value. As a result, the first integral in exists. So we just focus on the second integral.
Note that in , the inequality in is used. Due to , the second integral in is finite. With both integrals in finite, it follows that the improper integral in the definition of is finite.
The gamma function is a well known function in mathematics and has wide and deep implications. Here, we simply state several basic facts that are needed. In the discussion here, One useful fact is that the gamma function is the factorial function shifted down by one when the argument is a positive integer. Thus the gamma function generalizes the factorial function. Another useful fact is that the gamma function satisfies a recursive relation.
- If is a positive integer, .
- For all , .
The recursive relation is derived by using integration by parts. It is clear from definition that . By an induction argument, for all integers greater than 1.
As result, , and and so on. Given that , the recursive relation tells us that and and so on. The above recursive relation can be further extended:
- For any positive integer , .
When the integral in has “incomplete” limits, the resulting functions are called incomplete gamma functions. The following are called the upper incomplete gamma function and lower incomplete gamma function, respectively.
The next post shows how the gamma distribution arises naturally from the gamma function.