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In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

In applied probability, a probability distribution can be specified in a number of different ways, often chosen for mathematical convenience:

by supplying a valid probability mass function or probability density function

by supplying a valid cumulative distribution function or survival function

by supplying a valid hazard function

by supplying a valid characteristic function by supplying a rule for constructing a new random variable from other random variables whose joint probability distribution is known.

A probability distribution can either be univariate or multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

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