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Title: Exponential order statistics and some combinatorial identities Authors:  Palaniappan Vellaisamy - Indian Institute of Technology Bombay (India)
Aklilu Zeleke - Michigan State University (United States) [presenting]
Abstract: It is known that order statistics from exponential distribution have several interesting properties. We consider, without loss of generality, the exponential distribution with mean unity. For example, the $k$-th order statistic, $ 1 \leq k \leq n$, has the distribution of sum of independent exponential random variables (rvs) with different parameters. The usual proof of this result uses the transformation to the set of spacings from the set of order statistics and by applying Jacobian density theorem. We prove the above-mentioned result using the Laplace transform methods. The main purpose is to bring out the connections between exponential order statistics and several combinatorial identities. In fact, we give simpler proofs of several combinatorial/binomial identities by evaluating the Laplace transformation of the $k$-th exponential order statistic by two different ways and equating them. A probabilistic interpretation and some extensions of these combinatorial identities are also discussed.