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Some distributions have been specially named as compounds: beta-binomial distribution, Beta negative binomial 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).
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
The matrix gamma distribution and the Wishart distribution are multivariate generalizations of the gamma distribution (samples are positive-definite matrices rather than positive real numbers). The gamma distribution is a special case of the generalized gamma distribution , the generalized integer gamma distribution , and the generalized ...
The beta function is also important in statistics, e.g. for the beta distribution and beta prime distribution. As briefly alluded to previously, the beta function is closely tied with the gamma function and plays an important role in calculus .
The Gamma distribution, which describes the time until n consecutive rare random events occur in a process with no memory. The Erlang distribution, which is a special case of the gamma distribution with integral shape parameter, developed to predict waiting times in queuing systems; The inverse-gamma distribution; The generalized gamma distribution
Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required. [1]
The compound gamma distribution [3] is the generalization of the beta prime when the scale parameter, q is added, but where p = 1. It is so named because it is formed by compounding two gamma distributions :
In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision .