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In probability theory and statistics, the multivariate normal distribution, ... then the distribution of x 1 conditional on x 2 = a is multivariate normal ...
The multinomial distribution, a generalization of the binomial distribution. The multivariate normal distribution, a generalization of the normal distribution. The multivariate t-distribution, a generalization of the Student's t-distribution. The negative multinomial distribution, a generalization of the negative binomial distribution.
The connection of maximum likelihood estimation to OLS arises when this distribution is modeled as a multivariate normal. Specifically, assume that the errors ε have multivariate normal distribution with mean 0 and variance matrix σ 2 I. Then the distribution of y conditionally on X is
If the conditional distribution of given is a continuous distribution, then its probability density function is known as the conditional density function. [1] The properties of a conditional distribution, such as the moments , are often referred to by corresponding names such as the conditional mean and conditional variance .
One common method of construction of a multivariate t-distribution, for the case of dimensions, is based on the observation that if and are independent and distributed as (,) and (i.e. multivariate normal and chi-squared distributions) respectively, the matrix is a p × p matrix, and is a constant vector then the random variable = / / + has the density [1]
There is a set of probability distributions used in multivariate analyses that play a similar role to the corresponding set of distributions that are used in univariate analysis when the normal distribution is appropriate to a dataset. These multivariate distributions are: Multivariate normal distribution; Wishart distribution
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Copulas are used to describe/model the dependence (inter-correlation) between random variables. [1]
In this case the distribution cannot be interpreted as an untruncated normal conditional on < <, of course, but can still be interpreted as a maximum-entropy distribution with first and second moments as constraints, and has an additional peculiar feature: it presents two local maxima instead of one, located at = and =.