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We consider the k-th order statistic from unit exponential distribution andshow that it can be represented as a sum of independent exponential randomvariables. Our proof is simple and different. It readily proves that thestandardized exponential spacings also follow unit exponential distribution. Aninteresting probabilistic proof of the Basel problem is also given. Anotheradvantage of our approach is that by computing the Laplace transform of thek-th order statistic in two different ways, we derive several interestingcombinatorial identities. A probabilistic interpretation of these identitiesand their generalizations are also given.

Exponential Order Statistics, the Basel problem and Combinatorial Identities