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Concept# Multivariate t-distribution

Summary

In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t-distribution is distinct and makes particular use of the matrix structure.
Definition
One common method of construction of a multivariate t-distribution, for the case of p dimensions, is based on the observation that if \mathbf y and u are independent and distributed as N({\mathbf 0},{\boldsymbol\Sigma}) and \chi^2_\nu (i.e. multivariate normal and chi-squared distributions) respectively, the matrix \mathbf{\Sigma}, is a p × p matrix, and {\boldsymbol\mu} is a constant vector then the random variable

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