Distributional Identities of Beta and Chi-Squared Variates: A Geometrical Interpretation

A bivariate change of variable is proposed that elucidates the major distributional identities involving the two beta distributions and the chi-squared distribution. These identities are then seen to be the consequence of simple trigonometric relationships. The extension to n dimensions clarifies further identities. There are, however, identities that seem not to fit into this scheme; three of these are reported.