Functions of Random Variables
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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?
KeywordsRandom Vector Joint Distribution Quantile Function Borel Function Cauchy Distribution
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