A two-point condition for characterizing the exponential distribution by means of identifically distributed properties related to order statistics

Abstract

Let X₁, X₂, ..., X_n be independent and identically distributed random variables subject to a continuous distribution function F. Let X_1∶n, X_2∶n, ..., X_n∶n denote the corresponding order statistics. Write

$$P(X_{k + s:n} - X_{k:n} \geqslant x) = P(X_{s:n - k} \geqslant x),$$

((*))

where n, k are fixed integers. We apply a result of Marsaglia and Tubilla on the lack of memory of the exponential distribution finction assuming that certain distribution functions involving the above order statistics are equal in two incommensurable points τ₁, τ₂ > 0; this characterizes the exponential distribution. As a special case it turns out that the equality (*) assumed for s=1, 2 and x=τ₁, τ₂ implies that F is exponential.