Exponential Order Statistics, the Basel problem and Combinatorial Identities

Abstract

We consider the k-th order statistic from unit exponential distribution and show that it can be represented as a sum of independent exponential random variables. Our proof is simple and different. It readily proves that the standardized exponential spacings also follow unit exponential distribution. An interesting probabilistic proof of the Basel problem is also given. Another advantage of our approach is that by computing the Laplace transform of the k-th order statistic in two different ways, we derive several interesting combinatorial identities. A probabilistic interpretation of these identities and their generalizations are also given.