Distributional relationships arising from simple trigonometric formulas

Jones, M. C. (1999). Distributional relationships arising from simple trigonometric formulas. The American Statistician, 53(2) pp. 99–102.

DOI: https://doi.org/10.1080/00031305.1999.10474439

URL: http://www.jstor.org/stable/2685725

Abstract

The polar representation of a pair (X, Y) of spherically symmetrically distributed random variables provides an attractive route to the known fact that their ratio has a Cauchy distribution. In this note, a variety of other distributional relationships involving X and Y is observed to arise very straightforwardly from the simplest of trigonometric formulas, namely multiple-of-angle formulas and sum-of-angles formulas. Cos and sin formulas yield functions of X and Y-which may be independent standard normals-that have the same distribution as X, and tan formulas yield functions of Cauchy variables that remain Cauchy distributed.

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