A Characterization of Exponential Distributions Through Conditional Independence

SUMMARY Let Y  1, Y  2, . . ., Yn be mutually independent non-negative random variables having an absolutely continuous distribution function Fi(y) over its support [0, ∞) and the corresponding density function f  i(y) > 0 for y > 0. Let A denote the event that Yi – Y  i+1 > 0 for all i = 1, 2, . . ., n – 1. Then we show that, conditional on the event A, Yi – Y  i+1 and Y  i+1 are independent for all i = 1, 2,. . ., k if and only if Yi (i = 1, 2, . . ., k) are exponentially distributed random variables, where 1 ≤ k ≤ n – 1. We note that the k exponential distributions can have different scale parameters.