Approximating the tail probabilities of the longest run in a sequence of Bernoulli trials

Abstract

We consider the longest run of either successes or failures in a sequence of (Formula presented.) Bernoulli trials. The exact distribution of this random variable is obtained using probability generating function techniques. We consider approximating the tail areas of this random variable using a partial fraction approximation and a saddlepoint approximation with various continuity corrections. We investigate and compare the performance of the approximations for various values of sequence length and various values of the probability of success.