Discriminating between Weibull distributions and log-normal distributions emerging in branching processes

We consider the Yule-type multiplicative growth and division process, and describe the ubiquitous emergence of Weibull and log-normal distributions in a single framework. With the help of the integral transform and series expansion, we show that both distributions serve as asymptotic solutions of the time evolution equation for the branching process. In particular, the maximum likelihood method is employed to discriminate between the emergence of the Weibull distribution and that of the log-normal distribution. Further, the detailed conditions for the distinguished emergence of the Weibull distribution are probed. It is observed that the emergence depends on the manner of the division process for the two different types of distribution. Numerical simulations are also carried out, confirming the results obtained analytically.