ANALYSIS OF THE LURIA-DELBRUCK DISTRIBUTION USING DISCRETE CONVOLUTION POWERS

The Luria-Delbruck 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); p(n) = (m/n) SIGMA(i = 0)n-1 p(i)/(n - i + 1) where m is the expected number of mutations. A new relation for the asymptotic behavior of p(n) (almost-equal-to c/n2) is also derived. This corresponds to the probability of finding a very large number of mutants. A formula for the z-transform of the distribution is also reported.