Other One-Parameter Models and Their Conjugate Priors
You encountered the Poisson distribution in problems at the end of the previous chapter. The Poisson distribution is useful when the random variable is a count of the number of rare events occurring per unit time, unit volume, unit distance, etc. For exam
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Other One-Parameter Models and Their Conjugate Priors
6.1 Poisson You encountered the Poisson distribution in problems at the end of the previous chapter. The Poisson distribution is useful when the random variable is a count of the number of rare events occurring per unit time, unit volume, unit distance, etc. For example, the number of new cases of rhabdomyosarcoma (a rare form of cancer) occurring in Johnson County, Iowa, each year might be represented as a Poisson random variable. So might the number of flaws in each 1,000 feet of yarn produced by a spinning machine. A Poisson random variable can take on only nonnegative integer values. In order for the Poisson distribution to be appropriate, there is a constant average rate at which the events occur, and the numbers of events in disjoint intervals (different years, different segments of yarn, etc.) must be independent. In our examples, this implies that if there were an unusually large number of new rhabdomyosarcoma cases in Johnson County in one particular year, that would not affect the probability distribution for the number of new cases in the following year. Thus, the Poisson distribution would not be appropriate for counts of a contagious disease.
6.2 Normal: Unknown Mean, Variance Assumed Known So far, we have been considering discrete data—binary responses to survey questions and integer counts of rare events. Thus, the distributions of the random variables of interest, and the resulting likelihoods, have been probability mass functions. Now we will begin to consider cases in which the data are realizations of continuous random variables, which are described by probability density functions (pdfs).
M.K. Cowles, Applied Bayesian Statistics: With R and OpenBUGS Examples, Springer Texts in Statistics 98, DOI 10.1007/978-1-4614-5696-4 6, © Springer Science+Business Media New York 2013
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6 Other One-Parameter Models and Their Conjugate Priors
As you know, there are many parametric families of continuous pdfs. Both frequentists and Bayesians must use care in choosing the density that is likely to best describe the population of values from which their sample data is going to be drawn. The normal (also called the Gaussian) density is one of the most commonly used pdfs, and I am sure you are familiar with its bell-shaped density curve. The normal density is a good model for data when the random variable is continuous-valued, the distribution of values in the population is likely to be symmetric around a single mode, and the tails of the distribution are not heavy. It is a good choice for many variables that are measurements on living things, like weights, body temperatures, or heart rates of a species of mammals. The normal density is not appropriate for variables for which the population distribution is likely to be skewed, such as household incomes. You should be familiar with the normal probability density function, here shown for a random variable Y from a normal density with mean μ and population variance σ 2 : Y ∼ N(μ , σ 2 )
1 (y − μ )2
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