Random vectors and joint distributions

Introduction

A single, real-valued random variable is a function (mapping) from the basic space Ω to the real line. That is, to each possible outcome ω of an experiment there corresponds a real value $t=X\left(\omega \right)$ . The mapping induces a probability mass distribution on the real line, which provides a means of making probabilitycalculations. The distribution is described by a distribution function F _X . In the absolutely continuous case, with no point mass concentrations, the distribution may also bedescribed by a probability density function f _X . The probability density is the linear density of the probability mass along the real line (i.e., mass per unit length).The density is thus the derivative of the distribution function. For a simple random variable, the probability distribution consists of a point mass p _i at each possible value t _i of the random variable. Various m-procedures and m-functions aid calculations for simple distributions. In the absolutely continuous case, a simple approximationmay be set up, so that calculations for the random variable are approximated by calculations on this simple distribution.

Often we have more than one random variable. Each can be considered separately, but usually they have some probabilistic ties which must be taken into account when theyare considered jointly. We treat the joint case by considering the individual random variables as coordinates of a random vector . We extend the techniques for a single random variable to the multidimensional case. To simplify exposition and to keep calculationsmanageable, we consider a pair of random variables as coordinates of a two-dimensional random vector. The concepts and results extend directly to any finite number of randomvariables considered jointly.

Random variables considered jointly; random vectors

As a starting point, consider a simple example in which the probabilistic interaction between two random quantities is evident.