# ⓘ Systems of probability distributions ..

## (a,b,0) class of distributions

In probability theory, the distribution of a discrete random variable N whose values are nonnegative integers is said to be a member of the class of distributions if its probability mass function obeys p k p k − 1 = a + b k, k = 1, 2, 3, … {\displaystyle {\frac {p_{k}}{p_{k-1}}}=a+{\frac {b}{k}},\qquad k=1.2.3,\dots } where p k = P N = k {\displaystyle p_{k}=PN=k} provided a {\displaystyle a} and b {\displaystyle b} exist and are real. There are only three discrete distributions that satisfy the full form of this relationship: the Poisson, binomial and negative binomial distributions. Thes ...

## Copula (probability theory)

In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform. Copulas are used to describe the dependence between random variables. Their name comes from the Latin for "link" or "tie", similar but unrelated to grammatical copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications. Sklars theorem states that any multivariate joint distribution can be written in terms of univariate marginal ...

## Mixture distribution

In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors, in which case the mixture distribution is a multivariate distribution. In cases where each of the underlying random variables is continuous, the out ...

## Pearson distribution

The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.

## Tweedie distribution

In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models. The Tweedie distributions were named by Bent Jorgensen after Maurice Tweedie, a statistician and medica ...

## Vine copula

A vine is a graphical tool for labeling constraints in high-dimensional probability distributions. A regular vine is a special case for which all constraints are two-dimensional or conditional two-dimensional. Regular vines generalize trees, and are themselves specializations of Cantor trees. Combined with bivariate copulas, regular vines have proven to be a flexible tool in high-dimensional dependence modeling. Copulas are multivariate distributions with uniform univariate margins. Representing a joint distribution as univariate margins plus copulas allows the separation of the problems o ...

## ⓘ Systems of probability distributions

• In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different
• In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random
• distributions Important continuous distributions include the continuous uniform, normal, exponential, gamma and beta distributions In probability theory
• probabilists List of probability distributions List of probability topics List of scientific journals in probability Timeline of probability and statistics
• The Pearson distribution is a family of continuous probability distributions It was first published by Karl Pearson in 1895 and subsequently extended
• entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions According
• In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and
• inference, a prior probability distribution often simply called the prior, of an uncertain quantity is the probability distribution that would express
• Many probability distributions that are important in theory or applications have been given specific names. The Bernoulli distribution which takes value
• quasiprobability distributions also counterintuitively have regions of negative probability density, contradicting the first axiom. Quasiprobability distributions arise
• Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if