Fundamentals of Probability: A First Course pp 321-341 | Cite as

# Convolutions and Transformations

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## Abstract

Very naturally, in applications we often want to study suitable functions or transformations of an original collection of variables *X* _{1}, *X* _{2},…, *X* _{n}. For example, the original variables *X* _{1}, *X* _{2},…, *X* _{n} could be the inputs into some process or system, and we may be interested in the output, which is some suitable function of these input variables. We dealt with the problem of finding distributions of functions of one continuous variable in Chapter 7. Similar, but technically more involved, techniques for studying distributions of functions of many continuous variables are presented with illustrations in this chapter. Sums, products, and quotients are special functions that arise quite naturally in applications. These will be discussed with special emphasis, although the general theory is also presented. Specifically, we present in this chapter the classic theory of polar transformations, the Helmert transformation in arbitrary dimensions, and the development of the *t*, and *F* distributions.

## Keywords

Joint Density Marginal Density Continuous Random Variable Jacobian Determinant Joint Density Function## Preview

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## References

- Hinkley, D. (1969). On the ratio of two correlated normal random variables, Biometrika, 56(3), 635–639.zbMATHCrossRefMathSciNetGoogle Scholar