Further Remarks on the Expectation of the Reciprocal of a Positive Random Variable
Stanley L. Sclove, Gordon D. Simons, John Van Ryzin · The American Statistician · 1967
E(XJ ) > (EX)-1 (1) for a non-degenerate positive random variable X. Fleiss [2] gave a simple proof, in pedagogic interest, and an application. Chian, [1] contributed a simpler proof. Finally, Gurland [3] gave a still simpler argument and obtained in the same stroke a wider result. The present note (a) attaches intuitive meaning to these results by exhibiting them as consequences of an obvious property of covariance and (b) proves the property by the same device used in [3]. Thus, while retaining the simplicity achieved in [3], we can gain further generality and, more importantly, augment the intuitive basis implicit there. The property is: If a random variable Y has a nondegener-ate increasing (dec.) r egression E(YIX) on a non-degenerate random variable X, then X and Y have positive (neg.) covatriance. Putting Y X-'= E(YIX), we have 0 > Cov [X,Y] I E(X) E(X-'), which is (1). Again, putting X= J(Z), Y = g(Z) for a nondegenerate random variable Z and monotone functions / and g, one of which is continuous, the main result of [3] is given. A simple argument proves the property. The function q(x) = E(YIX = x) EY is monotone and Eq(X) = 0, so there is a point 4 at which q (x) either becomes zero or changes sign. Then the random variable (X-4)q(X) cannot change sign and moreover E(X-4)q(X) =A 0 since neither X nor q (X) is degenerate. In view of