Theory of Statistics - George Mason University

Appendix A. Important Probability Distributions 829
We recognize a basic property of any mixture distribution: It is a joint
distribution factored as a marginal (prior) for a random variable, which is often
not observable, and a conditional distribution for another random variable,
which is usually the observable variable of interest.
In Bayesian analyses, the first two assumptions (a prior distribution for
the parameters and a conditional distribution for the observable) lead immediately
to a mixture distribution. The beta-binomial above arises in a canonical
example of Bayesian analysis.
Some families of distributions are recognized because of their relationship
to sampling distributions. These include the t, the chi-squared, and the
Wishart. Other families are recognized because of their use as conjugate priors.
These include the inverted chi-squared and the inverted Wishart.
General References
Evans et al. (2000) give general descriptions of 40 probability distributions.
Balakrishnan and Nevzorov (2003) provide an overview of the important characteristics
that distinguish different distributions and then describe the important
characteristics of many common distributions. Leemis and McQueston
(2008) present an interesting compact graph of the relationships among a
large number of probability distributions. Likewise, Morris and Lock (2009)
give a graph that illustrates various interrelationships among natural exponential
families.
The six books by Johnson, Kotz et al. (Johnson et al. (1995a), Kotz et al.
(2000),Johnson et al. (1997), Johnson et al. (1994),Johnson et al. (1995b),
and Johnson et al. (2005)) contain a wealth of information above many families
of distributions.
Currently, the most readily accessible summary of common probability
distributions is Wikipedia: http://wikipedia.org/ Search under the name
of the distribution.
Theory of Statistics c○2000–2013 James E. Gentle