# Functions of Random Variables

Chapter
Part of the Springer Texts in Statistics book series (STS)

## Abstract

A random variable X is a function defined on the basic space Ω. If the outcome of the experiment is ω, the corresponding number is t = X(ω). Frequently, if value t is observed, we are interested in a corresponding value υ = g(t) obtained by applying the rule for the function g to the observed value. If we make the assignment υ = g(X(ω)) for all ω, we have a new function Z = g(X) defined on the basic space. Similarly, we may be interested in Z = h(X, Y). If X(ω) — t and Y(ω) = u, then Z(ω) = h(t, u). Elementary mapping properties show that if g is a Borel function, then Z = g(X) is a random variable; similarly, if h is a Borel function of two variables, then Z = h(X, Y) is a random variable. In this chapter, we address the basic problem: Given the distribution for X, or the joint distribution for (X, Y), how can we assign probabilities to events determined by the new random variable Z?

## Keywords

Random Vector Joint Distribution Quantile Function Borel Function Cauchy Distribution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.