ABSTRACT
A question that naturally arises is, given a random variable X and associated distribution function F(x), what is the distribution function of the random variable g(X) given a Borel measurable function g(x)? When X has a density function, what is the density function of g(X), and must this exist? More generally, what are the distribution functions and densities of sums and ratios of random variables, where now g(x) is a multivariate function? The first section addresses the distribution function question for strictly monotonic g(x), and the density question when such g(x) is differentiable. For more general transformations, the change of variable results from the integration theory of Book V are needed, and these details are deferred to Book VI. A number of results are then derived for the distribution and density functions of sums of independent random variables using the integration theories of Book III. The various forms for such distribution functions then reflect the assumptions made on the underlying distribution functions and/or density functions. Examples and exercises connect the theory with the Chapter 1 catalog of distribution functions. The chapter ends with an investigation into ratios of independent random variables, as well as a special example with dependent random variables. Again, examples connect the theory with Chapter 1 distribution examples as well as introduce new distributions.
