  We give 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 simple proof of an
interesting result concerning the exponential distribution. The connections
between a probabilistic approach and the statistical approach are discussed,
which explains why certain binomial identities admit probabilistic
interpretations. In the process, several new binomial identities are also
obtained and discussed.
