Integral Identities for Random Variables

Using a general method for deriving identities for random variables, we find a number of new results involving characteristic functions and generating functions.The method is simply to promote a parameter in an integral relation to the status of a random variable and then take expected values of both sides of the equation.Results include formulas for calculating the characteristic functions for x 2, .x, 1/x, x 2 + x, R 2 = x 2 + y 2, and so forth in terms of integral transforms of the characteristic functions for x and (x, y), and so forth.Generalizations to higher dimensions can be obtained using the same method.Expressions for inverse/fractional moments, E{n!}, and so forth are also presented, demonstrating the method.