Characterizations of the Exponential Distribution

In this note, two characterizations of the exponential distribution are established. Theorem 1: Let a > -1, a ~ O. A positive random variable X has an exponential distribution if and only if E[CX-y)aI X > y] is a finite constant for all y ~ O. Theorem 2: Let Xl ,X2, ..• ,Xn be i.i.d. random variables having a continuous distribution FCx), and XCl)'XC2)' ... 'XCn) their order statistics. Let a > 0 and m = 1,2, ..• ,n-l be given. Then FCx) is an exponential distribution up to a location parameter if and only if E[CX Cm l ) XCm))aIXCm) = y] is a finite constant for almost all y with respect to FCy). *This research was supported by the Air Force Office of Scientific Research under contract AFOSR-75-2796 and the Office of Naval Research under contract NOOOI4-75-C-0809. CHARACTERIZATIONS OF THE EXPONENTIAL DISTRIBUTION BY CONDITIONAL MOMENTS