Probability Integrals of the MultivariatetDistribution

Abstract

Results on probability integrals of multivariate t distributions are reviewed. The results discussed include: Dunnett and Sobel's probability integrals, Gupta and Sobel's probability integrals, John's probability integrals, Amos and Bulgren's probability integrals, Steffens' non-central probabilities, Dutt's probability integrals, Amos' probability integral, Fujikoshi's probability integrals, probabilities of cone, probabilities of convex polyhedra, probabilities of linear inequalities, maximum probability content, and Monte Carlo evaluation.

Autonomous Learning Multimodel Systems From Data Streams

“…of x is defined as the weighted sum of continuous multivariate Cauchy type kernel functions [22]- [24] corresponding to the data clouds. An interesting property of τ M M is that it integrates to 1 [21]:…”

A Generalized Methodology for Data Analysis

“…Traditionally, this requires identifying local peaks/modes by clustering, expectation maximization, optimization, etc. [1]- [3], [21]- [23]. Within EDA, the discrete global typicality (τ D ) is derived automatically from the data with no user input and can quantify multimodality.…”

“…A similar property having a maximum, though its value is 1  , is also valid for the traditional probability by definition and according to the central limit theorem [1]- [3]. In reality, data distributions are usually multimodal [21]- [24], therefore the local description should be improved. In order to address this issue, the traditional probability theory often involves mixture of unimodal distributions, which requires estimation of number of modes and it is not easy [24].…”

“…As we said before, we consider the well-known expression of the multivariate Cauchy distribution [21]- [23] , ,..., ,... 1…”