Compound (or Bayesian) Relationships
When one or more parameter(s) of a distribution are random variables, the compound distribution is the marginal distribution of the variable.
Examples:
- If X|N is a binomial (N,p) random variable, where parameter N is a random variable with negative-binomial (m, r) distribution, then X is distributed as a negative-binomial (m, r/(p+qr)).
- If X|N is a binomial (N,p) random variable, where parameter N is a random variable with Poisson (μ) distribution, then X is distributed as a Poisson (μp).
- If X|μ is a Poisson (μ) random variable and parameter μ is random variable with gamma (m, β) distribution, then X is distributed as a negative-binomial (m, μβ/(μ+β)), sometimes called Gamma-Poisson distribution if m is not integer.
Some distributions have been specially named as compounds: Beta-Binomial distribution, Beta-Pascal distribution, Gamma-Normal distribution.
Examples:
- If X is a Binomial (n,p) random variable, and parameter p is a random variable with beta (α, β) distribution, then X is distributed as a Beta-Binomial(α, β,n).
- If X is a negative-binomial (m,p) random variable, and parameter p is a random variable with beta (α, β) distribution, then X is distributed as a Beta-Pascal(α, β,m).
Read more about this topic: Relationships Among Probability Distributions
Famous quotes containing the word compound:
“We are all aware that speech, like chemistry, has a structure. There is a limited set of elements—vowels and consonants—and these are combined to produce words which, in turn, compound into sentences.”
—Roger Brown (b. 1925)