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Multiple Random Variables and Their Characteristics
Book chapter

Multiple Random Variables and Their Characteristics

P. Mohana Shankar
Probability, Random Variables, and Data Analytics with Engineering Applications, pp 233-336
09 Feb 2021
DOI: https://doi.org/10.1007/978-3-030-56259-5_4

Abstract

Bayes’ rule and correlation Bessel functions and modified Bessel functions Central limit theorem for the products Central limit theorem for the sums Conditional, joint, and marginal densities Convolution Covariance and correlation Density of function of two variables Density of the sum and difference of two variables Density of the sum of the squares of normal variables functions Jacobian Joint cumulative distributions Joint density Joint density of two functions of two random variables Joint moments K-distribution Leibniz theorem Maximum and minimum of two or more variables Meijer Mellin transforms Moment generating functions and Laplace transforms Order statistics Product and ratio of two variables Rician and Hoyt densities Two stage experiments and conditional densities
This chapter is devoted to multiple random variables (mainly two variables). The transformation of two variables is presented initially by expanding on the notions of conditional densities in Chap. 3, before invoking the approaches requiring the use of Leibniz theorem and Jacobian. Modeling of outcomes in an experiment is presented as a two-stage experiment. Characteristic functions and Laplace transforms are offered for the determination of the densities of the sum and difference of variables. Mellin transforms (a topic not covered in textbooks) are presented as an approach to finding the densities of the products and ratios of two or more random variables. Meijer G functions are introduced to express the densities of products of random variables. The chapter offers detailed descriptions of the central limit theorem (sums and products) and order statistics. The examples and exercises are applications oriented and often involve the use of computational approaches.

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