## Distributions and Transformations

This chapter reviews the most important univariate distributions and shows how to derive their expectations, variances, moment-generating functions (if they exist), and characteristic functions. Many distributions arise as transformations of random variables or vectors. Therefore, the problem of how the distribution of Y = g(X) is related to the distribution of X for a Borel-measure function or mapping g(x) is also addressed.

In Chapter 1 I introduced three “natural” discrete distributions, namely, the hypergeometric, binomial, and Poisson distributions. The first two are natural in the sense that they arise from the way the random sample involved is drawn, and the last is natural because it is a limit of the binomial distribution...

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