Analysis of the Luria–Delbrück distribution using discrete convolution powers

Abstract

The Luria–Delbrück distribution arises in birth-and-mutation processes in population genetics that have been systematically studied for the last fifty years. The central result reported in this paper is a new recursion relation for computing this distribution which supersedes all past results in simplicity and computational efficiency:p0=e–m;wheremis the expected number of mutations. A new relation for the asymptotic behavior ofpn(≈c/n2) is also derived. This corresponds to the probability of finding a very large number of mutants. A formula for thez-transform of the distribution is also reported.