On the probability of the longest run length in an independent series

Abstract

Various problems concerning the analysis of droughts lead to the problem of finding the probability distribution function (PDF), of the longest run lengths in a series of finite length. Unfortunately, the exact PDF of such runs is not available in the literature. This paper is concerned with the derivation of the exact PDF of the longest run length to be expected in an independent series. In the derivation of the PDF, different definitions of runs encountered in the hydrology literature have been considered. The analytical derivations are solved numerically on digital computers and the results are presented in forms of cumulative PDF and the expected maximum run length changes with the sample lengths. The application of the methodology developed herein is presented for three annual flow series.