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Comments on the Luria–Delbrück distribution

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Adrienne W. Kemp
Adrienne W. Kemp*
Affiliation:
University of St Andrews
*
∗ Postal address: Mathematical Institute, North Haugh, St Andrews KY16 9SS UK.
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Abstract

The long-tailed Luria–Delbrück distribution arises in connection with the ‘random mutation’ hypothesis (whereas the ‘directed adaptation' hypothesis is thought to give a Poisson distribution). At time t the distribution depends on the parameter m = gNt/(a + g) where Nt is the current population size and g/(a + g) is the relative mutation rate (assumed constant). The paper identifies three models for the distribution in the existing literature and gives a fourth model. Ma et al. (1992) recently proved that there is a remarkably simple recursion relation for the Luria–Delbrück probabilities pn and found that asymptotically pn ≈ c/n2; their numerical studies suggested that c = 1 when the parameter m is unity. Cairns et al. (1988) had previously argued and shown numerically that Pn = Σj ≧ n Pj ≈ m/n. Here we prove that n(n + 1)pn < m(1 + 11m/30) for n = 1, 2, ···, and hence prove that as n becomes large n(n + 1)pn, ≈ m; the result mPn ≈ m follows immediately.

Keywords

RANDOM MUTATIONINFINITE DIVISIBILITYPOISSON-STOPPED SUM DISTRIBUTION

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Type
Short Communications
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 822 - 828
Copyright
Copyright © Applied Probability Trust 1994 

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