Introducing the gamma function

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 \alpha>0, define:

    \displaystyle \Gamma(\alpha)=\int_0^\infty t^{\alpha-1} \ e^{-t} \ dt \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)

The following is the graph of the gamma function. It has a U shape. As x \rightarrow 0, the graph goes up to infinity. As x \rightarrow \infty, the graph increases without bound. As is seen below, the gamma function coincides with the factorial function at the integers.

Gamma Function Graph

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The Convergence

The gamma function is defined by the improper integral as described in (0). The improper integral converges. In fact, the integral converges for all complex numbers \alpha with positive real part. For our purposes at hand, we restrict \alpha to be positive real numbers. Showing that the integral in (0) 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 (0) converges, we first make the observation: for some positive real number M and for all t>M,

    \displaystyle t^{\alpha-1} <e^{\frac{t}{2}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

This is saying that the quantity \displaystyle e^{\frac{t}{2}} dominates the quantity \displaystyle t^{\alpha-1} when t is sufficiently large. This is because the exponential function \displaystyle e^{\frac{t}{2}} increases at a much faster rate than the polynomial function \displaystyle t^{\alpha-1}. Then multiply both sides of (1) by \displaystyle e^{-t} to obtain the following:

    \displaystyle t^{\alpha-1} e^{-t}<e^{-\frac{t}{2}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)

Now, break up the interval in (0) into two pieces:

    \displaystyle \Gamma(\alpha)=\int_0^M t^{\alpha-1} \ e^{-t} \ dt +\int_M^\infty t^{\alpha-1} \ e^{-t} \ dt \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)

A basic fact. For any continuous function h(t) defined over a closed interval [a,b] of finite length, the integral \displaystyle \int_a^b h(t) \ dt exists and has a finite value. As a result, the first integral in (3) exists. So we just focus on the second integral.

    \displaystyle \int_M^\infty t^{\alpha-1} \ e^{-t} \ dt < \int_M^\infty e^{-\frac{t}{2}} \ dt=2 e^{-\frac{M}{2}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)

Note that in (4), the inequality in (2) is used. Due to (4), the second integral in (3) is finite. With both integrals in (3) finite, it follows that the improper integral in the definition of \Gamma(\alpha) is finite.

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Some Properties

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 \alpha 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 n is a positive integer, \Gamma(n)=(n-1)!.
  • For all \alpha>0, \Gamma(\alpha+1)=\alpha \Gamma(\alpha).

The recursive relation \Gamma(\alpha+1)=\alpha \Gamma(\alpha) is derived by using integration by parts. It is clear from definition that \Gamma(1)=1. By an induction argument, \Gamma(n)=(n-1)! for all integers greater than 1.

As result, \Gamma(1)=0!=1, \Gamma(2)=1!=1 and \Gamma(3)=2!=2 and so on. Given that \Gamma(\frac{1}{2})=\sqrt{\pi}, the recursive relation tells us that \Gamma(\frac{3}{2})=\frac{1}{2} \sqrt{\pi} and \Gamma(\frac{5}{2})=\frac{3}{4} \sqrt{\pi} and so on. The above recursive relation can be further extended:

  • For any positive integer k, \Gamma(\alpha+k)=\alpha (\alpha+1) \cdots (\alpha + k-1) \ \Gamma(\alpha).

When the integral in (0) 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.

    \displaystyle \Gamma(\alpha, x)=\int_x^\infty t^{\alpha-1} \ e^{-t} \ dt \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

    \displaystyle \gamma(\alpha, x)=\int_0^x t^{\alpha-1} \ e^{-t} \ dt \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)

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The next post shows how the gamma distribution arises naturally from the gamma function.

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\copyright \ 2016 - \text{Dan Ma}

8 thoughts on “Introducing the gamma function

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