Exponential Order Statistics, the Basel problem and Combinatorial Identities

TL;DR

This paper explores the properties of exponential order statistics, provides a new proof of the Basel problem, and derives combinatorial identities through probabilistic methods involving exponential distributions.

Contribution

It introduces a novel probabilistic approach to analyze exponential order statistics and derives new combinatorial identities via Laplace transform techniques.

Findings

Representation of k-th order statistic as sum of independent exponentials

Probabilistic proof of Basel problem

Derivation of combinatorial identities from Laplace transforms

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.