A Characteristic Property of the Exponential Distribution

Let X be a nonnegative random variable with probability distribution function F. Suppose X₈, ₍ (i = 1, \, n) is the ith smallest order statistics in a random sample of size n from F. A necessary and sufficient condition for F to be exponential is given which involves the identical distribution of the random variables X and (n - i) (X₈+₁, ₍ - X₈, ₍) for some i and n, (1 \ i < n).