The Gamma Distribution

Java Applet Simulation of the gamma experiment

In this section we will study a family of distributions that has special importance in probability statistics.

The Gamma Function

The gamma function is defined by

1. Show that the integral defining the gamma function converges for any a > 0.

The graph of the gamma function on the interval (0, 5) is shown below:

2. Integrate by parts to show that for any a > 0,

3. Use Exercise 2 to show that for any positive integer k,

4. Use the standard normal density function to show that

The Gamma Density

5. Show that the following function is a probability density function for any r > 0 and a > 0:

A random variable X with the density in Exercise 5 is said to have the gamma distribution with parameters a and r. The gamma distribution is important because of the Poisson process and because of the chi-square distribution that occurs in statistics. The following exercise shows that the family of densities has a rich variety of shapes.

6. Draw a careful sketch of the gamma probability density functions in each of the following cases:

1. 0 < a < 1.
2. a = 1.
3. a > 1. Show that the mode occurs at (a - 1) / r.

Because a controls the shape of the density function, it is sometimes referred to as the shape parameter.

Simulation

The applet for this page simulates a random variable with the gamma distribution. The graph of the density function is shown in blue. The mean and standard deviation are recorded in the first table and shown graphically in terms of the horizontal blue bar, which is centered at the mean and extends one standard deviation on either side. You can vary the parameters with a scroll bars. When you run the experiment, independent values of the gamma distribution are generated and recorded in the second table. The empirical density function is graphed in red and the sample mean and variance recorded in the first table and shown graphically in terms of the horizontal red bar.

7. Start the applet and vary the parameters. Note the shape and location of the gamma density function and the location and size of the mean-standard deviation bar.

8. Run the simulation with an update frequency of 10 for several different values of the parameters. Note the apparent convergence of the empirical density and statistics to the corresponding distribution density and parameters.

Moments

9. Suppose that X has the gamma distribution with parameters a > 0 and r > 0. Show that the mean and variance are given by

1. E(X) = a / r.
2. var(X) = a / r².

10. Suppose that X has the gamma distribution with parameters a > 0 and r > 0. Show that the moment generating function of X is

Transformations

11. Suppose that X has the gamma distribution with parameters a > 0 and r > 0. Show that if c > 0 then cX has the gamma distribution with parameters r/c and a.

Because of the result in Exercise 11, r is sometimes referred to as the scale parameter.

12. Suppose that X has the gamma distribution with parameters a > 0 and r > 0; that Y has the gamma distribution with parameters b > 0 and r; and that X and Y are independent. Use the result of Exercise 10 to show that X + Y has the gamma distribution with parameters a + b and r.

Special Distributions