Some applications of Mellin transforms to the theory of bivariate statistical distributions

A method is described for finding the frequency functions of bivariate random variables which are the products or ratios of other bivariate random variables. If (ξ, n ) are a pair of bivariate random variables with joint frequency function f ( x, y ) then the method depends upon the fact that the expectation of │ ξ │ r –1 │η│ s –1 is related to the Mellin transform of f ( x, y ) in two dimensions. Knowing the expectation we can then recover the frequency function by means of the inverse Mellin transform. Some examples are given to illustrate the theory.