Functions of Random Variables

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


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?


Random Vector Joint Distribution Quantile Function Borel Function Cauchy Distribution 
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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  1. 1.Department of Mathematical SciencesRice UniversityHoustonUSA

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