# multivariate distribution

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## multivariate distribution

[¦məl·tē′ver·ē·ət ′dis·trə′byü·shən]
(mathematics)
For two or more random variables, X 1, X 2, …, and Xn, the distribution which gives the probability that X 1 = x 1, X 2 = x 2, …, and Xn = xn for all values, x 1, x 2, …, and xn, of X 1, X 2, …, and Xn respectively.
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The copula methodology is based on the results of Sklar who proved that a multivariate distribution can be characterized by the set of its marginal distributions and a "copula" function that describes the dependence between the marginal components (uniquely in the case of continuous distributions).
Formally, a copula is a multivariate distribution function with uniform marginal densities that correspond to the probability integral transformation (PIT) of the individual random variables.
The third section will introduce the copula method for constructing multivariate distributions and discuss some of the more prominent univariate stochastic loss reserving methods.
Balakrishnan, Discrete Multivariate Distributions, John Wiley & Sons, New York, NY (1997).
Johnson, Continuous Multivariate Distributions, Vol.
Topics include special matrices, non-negative matrices, special products and operators, Jacobians, partitioned and patterned matrices, matrix approximation, matrix optimization, multiple integrals and multivariate distributions, linear and quadratic forms, etc.
For sampling from multivariate distributions, functions such as randnormal, randmvt and randmultinomial can be used to generate samples from multivariate normal, multivariate Student's t and multinomial distributions, respectively.
Joint probability, multivariate distributions, and criteria for independence and randomness are then covered.

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