Lomax distribution

The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero. The probability density function (pdf) for the Lomax distribution is given by with shape parameter and scale parameter . The density can be rewritten in such a way that more clearly shows the relation to the Pareto Type I distribution. That is: The th non-central moment exists only if the shape parameter strictly exceeds , when the moment has the value The Lomax distribution is a Pareto Type I distribution shifted so that its support begins at zero. Specifically: The Lomax distribution is a Pareto Type II distribution with xm=λ and μ=0: The Lomax distribution is a special case of the generalized Pareto distribution. Specifically: The Lomax distribution with scale parameter λ = 1 is a special case of the beta prime distribution. If X has a Lomax distribution, then . The Lomax distribution with shape parameter α = 1 and scale parameter λ = 1 has density , the same distribution as an F(2,2) distribution. This is the distribution of the ratio of two independent and identically distributed random variables with exponential distributions. The Lomax distribution is a special case of the q-exponential distribution. The q-exponential extends this distribution to support on a bounded interval. The Lomax parameters are given by: The logarithm of a Lomax(shape = 1.0, scale = λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0. This implies that a Lomax(shape = 1.0, scale = λ)-distribution equals a log-logistic distribution with shape β = 1.0 and scale α = log(λ). The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution.

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Loi bêta prime

En théorie des probabilités et en statistique, la loi bêta prime (également connue sous les noms loi bêta II ou loi bêta du second type) est une loi de probabilité continue définie dont le support est et dépendant de deux paramètres de forme. Si une variable aléatoire X suit une loi bêta prime, on notera . Sa densité de probabilité est donnée par : où B est la fonction bêta. Cette loi est une loi de Pearson de type VI. Le mode d'une variable aléatoire de loi bêta prime est .

Compound probability distribution

In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. If the parameter is a scale parameter, the resulting mixture is also called a scale mixture.

Conjugate prior

In Bayesian probability theory, if the posterior distribution is in the same probability distribution family as the prior probability distribution , the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function . A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise, numerical integration may be necessary. Further, conjugate priors may give intuition by more transparently showing how a likelihood function updates a prior distribution.

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