This is a description of the gamma distribution in the context of the kth interarrival time of a Poisson process. It is NOT a general description of the gamma distribution, but a link on the page takes you to another page that is a general description.
The student/reader derives the probability density function of the gamma, clicks on an applet that generates gamma random variables for various values of the two parameters, and is led through exploring properties of the distribution.
There are problems posed to the student, some of which allow the student to click on a link to find answers.

Type of Material:

Web page with links to an applet.

Recommended Uses:

To be used in an upper-level probability class that is familiar with Virtual Laboratories. Could be used in conjunction with a homework assignment.

Technical Requirements:

The material requires MathML, JAVA, and access to web browser.

Identify Major Learning Goals:

The student should learn to think of a gamma random variable with integer parameters as the time of the kth arrival. The student should be able to describe when a gamma can be approximated with a normal random variable. The student should be able to calculate basic probabilities, the mean, and the variance of the gamma. The student should be able to use the gamma to answer questions about the arrival time of the kth item and the rate of the process.

Target Student Population:

This would be appropriate for a mathematical statistics class: upper level undergraduate or introductory graduate level.

Prerequisite Knowledge or Skills:

Calculus and an understanding of convolutions, the Poisson distribution, mean, and variance. At least one or two classes in probability and statistics.

Content Quality

Rating:

Strengths:

Full description of how the gamma distribution fits into the application of a Poisson process. Contains a development of the probability distribution function, discussion of the moments, approximation with the normal distribution, and discussion of how this gives an unbiased, consistent estimator of (one over) the rate of the process. Many of the properties from the gamma distribution are discussed in this material. Additionally, in-depth exercises to get students acquainted with the gamma distribution are present. The graph illustrating the gamma distribution is also very nice.

Concerns:

The title being the Gamma Distribution sounds like it is a description of the general gamma distribution. Instead, it is specific to this particular application. A few phrases throughout the document should be revised. For example, in exercise 3, the student is asked to sketch the graph of the gamma density (I assume for various k and r values) and observe how the density increases first then decreases. This is true, except in the case when k = 1. When k=1, the density does not increase first. Also, under the moments section, the wording of "corresponding results for the exponential distribution" should be changed. In addition, it should be stated in the material that as an individual changes the rate parameter, the graph of the gamma changes the axes to maintain the overall shape.

Potential Effectiveness as a Teaching Tool

Rating:

Strengths:

The material can be very effective, if used in conjunction with other material from Virtual Laboratories. As mentioned previously, the properties of the gamma distribution and exercises are nicely displayed, along with an easy to use graphical display of the distribution. It's nice that the student/reader is asked to do some of the derivations. Also, questions with numeric answers have the answers available.

Concerns:

This material should not be used as a stand alone document. To maneuver through the material, an individual should be familiar with the set up and needs to have a number of required software packages. This is a long assignment to give for one homework grade. It contains ten show that... statements, which my students are frightened by. However, just because that scares students doesn't mean that it should be taken out.

Ease of Use for Both Students and Faculty

Rating:

Strengths:

The material provides much information about the gamma distribution (as it relates to Poisson processes). Assuming that you've already heard of a Poisson process, this is a completely self-contained lesson. Each technical term is linked to a page that describes that term.

Concerns:

This material should NOT be used as a stand alone document. It is also not easy to maneuver through the document when clicking on the various links to get definitions. It would be helpful if the numbering on the answers matched the questions (e.g., in the key, we might see something like 3.15, but it might not be clear that this refers to problem 15 on page 3) so that a teacher could assign certain questions but not others.

Creative Commons:

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