On Probabilistic Proofs of Certain Binomial Identities

This short note gives a simple statistical proof of a binomial identity, by evaluating the Laplace transform of the maximum of n independent exponential random variables in two different ways. As a by-product, we obtain a rigorous proof of an interesting result concerning the exponential distribution. The connections between a probabilistic approach and our approach are discussed. In the process, several new binomial identities are also obtained.