4. Special Distributions

Summary

In this chapter, we study a number of special distributions that are important in probability and statistics. We also explore two general classes of distributions: location-scale families and exponential families.

An incredible variety of special distributions have been studied over the years, and new ones are constantly being added to the literature. To truly deserve the adjective special, a distribution should have a certain level of mathematical elegance and economy, and should arise in interesting and diverse applications.

Topics

1. Introduction
2. The Normal Distribution
3. The Gamma Distribution
4. The Chi-Square Distribution
5. The Student t Distribution
6. The F Distribution
7. The Multivariate Normal Distribution
8. The Beta Distribution
9. The Weibull Distribution
10. The Zeta Distribution
11. The Pareto Distribution
12. The Logistic Distribution
13. The Lognormal Distribution
14. The Extreme Value Distribution
15. Benford's Law
16. The Irwin-Hall Distribution
17. The Folded Normal Distribution
18. The Semicircle Distribution
19. The Hyperbolic Secant Distribution
20. The Cauchy Distribution
21. The Triangle Distribution

Applets

• Special Distribution Simulator
• Special Distribution Calculator
• Random Quantile Experiment
• Rejection Method Experiment
• Bivariate Normal Experiment

External Resources

The best online site for information on special distributions is the Distributome project. The definitive printed works on special distributions are the books by Johnson and Kotz and their co-authors.

• Distributome: An Interactive Web-Based Resource for Probability Distributions
• Univariate Discrete Distributions, by Norman L. Johnson, Samuel Kotz, and Andrienne W. Kemp.
• Continuous Univariate Distributions by Norman L. Johnson, Samuel Kotz, and N. Balakrishnan
• Continuous Univariate Distributions by Norman L. Johnson, Samuel Kotz, and N. Balakrishnan
• Discrete Multivariate Distributions, by Norman L. Johnson, Samuel Kotz, and N. Balakrishnan,
• Continuous Multivariate Distributions: Models and Applications, by Samuel Kotz, N. Balakrishnan, and Normal L. Johnson

Quote

• Beauty is the first test: there is no permanent place in the world for ugly mathematics.--GH Hardy, A Mathematician's Apology.