Distributional Relationships Arising from Simple Trigonometric Formulas
M. C. Jones · The American Statistician · 1999
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.