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In Bayesian probability theory, if, given a likelihood function (), the posterior distribution is in the same probability distribution family as the prior probability distribution (), the prior and posterior are then called conjugate distributions with respect to that likelihood function and the prior is called a conjugate prior for the likelihood function ().
If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse scale forms a conjugate prior. The compound distribution , which results from integrating out the inverse scale, has a closed-form solution known as the compound gamma distribution .
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
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
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).
The shift geometric distribution is discrete compound Poisson distribution since it is a trivial case of negative binomial distribution. This distribution can model batch arrivals (such as in a bulk queue [5] [9]). The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total ...
A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that the conditional distribution of a random variable, given the value of the rate parameter, is a Poisson distribution , and that the rate parameter itself is considered as a random variable.
Consider a data set (,), …, (,), where the are Euclidean vectors and the are scalars.The multiple regression model is formulated as = +. where the are random errors. Zellner's g-prior for is a multivariate normal distribution with covariance matrix proportional to the inverse Fisher information matrix for , similar to a Jeffreys prior.