A Probabilistic Proof of a Binomial Identity

We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity. The goal of this note is to give a simple (and interesting) probabilistic proof of the binomial identity n ∑ k=0 ( n k ) (−1) θ θ + k = n ∏ k=1 k θ + k , for all θ > 0 and all n ∈ N. (1) If one is only concerned with giving a proof of this equality, other proofs than the probabilistic one given below may be more natural. For instance, a proof may be given by identifying the left side of (1) as the evaluation of a hypergeometric function 2F1 (−n, θ; θ + 1|1) and then applying the Chu-Vandermonde formula [2, equation (1.2.9)] to obtain the right side of (1). Another approach would be to use the Rice integral formulas [1, 3] to equate the left side of (1) with a complex contour integral that can be seen to equal the right side of (1). These approaches give short proofs of (1), but they both use a good deal of advanced mathematics. With a bit of work, one can also obtain an elementary proof of (1) using only basic properties of the binomial coefficients and mathematical induction. The proof of (1) given below arose not in a search for a new proof of this identity, but as a result of some independent probability research . . . and a cluttered desk. Being unable to find a probability calculation I had done the day before, I sought to repeat the calculation but obtained a different expression for the same quantity. After some initial confusion, I realized that my computations gave a simple proof of the identity (1). 1 Probability theory background. Before giving the probabilistic proof of (1), I will recall some basic facts from probability theory. All of the probability needed for this paper can be found in a basic undergraduate probability book such as [4]. Recall that a random variable Y has an exponential distribution with parameter λ if P (Y ≤ y) = { 1− e−λy y ≥ 0 0 y < 0.