Editing Branching process (section)

{{Short description|Kind of stochastic process}}
{{For|the process in representation theory|Restricted representation#Classical branching rules}}
In [[probability theory]], a '''branching process''' is a type of mathematical object known as a [[stochastic process]], which consists of collections of [[random variables]] indexed by some set, usually natural or non-negative real numbers. The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation&nbsp;<math>n</math> produces some random number of individuals in generation&nbsp;<math>n+1</math>, according, in the simplest case, to a fixed [[probability distribution]] that does not vary from individual to individual.<ref>{{Cite book | last1 = Athreya | first1 = K. B. | chapter = Branching Process | doi = 10.1002/9780470057339.vab032 | title = Encyclopedia of Environmetrics | year = 2006 | isbn = 978-0471899976 }}</ref> Branching processes are used to model reproduction; for example, the individuals might correspond to bacteria, each of which generates&nbsp;0,&nbsp;1,&nbsp;or&nbsp;2 offspring with some probability in a single time unit.  Branching processes can also be used to model other systems with similar dynamics, e.g., the spread of [[surname]]s in [[genealogy]] or the propagation of neutrons in a [[nuclear reactor]].

A central question in the theory of branching processes is the probability of '''ultimate extinction''', where no individuals exist after some finite number of generations.  Using [[Wald's equation]], it can be shown that starting with one individual in generation zero, the [[expected value|expected]] size of generation&nbsp;''n'' equals&nbsp;''μ''<sup>''n''</sup> where ''μ'' is the expected number of children of each individual.  If ''μ'' < 1, then the expected number of individuals goes rapidly to zero, which implies ultimate extinction [[with probability 1]] by [[Markov's inequality]].  Alternatively, if ''μ'' > 1, then the probability of ultimate extinction is less than&nbsp;1 (but not necessarily zero; consider a process where each individual either has 0 or 100 children with equal probability. In that case, ''μ'' = 50, but probability of ultimate extinction is greater than 0.5, since that's the probability that the first individual has 0 children).  If ''μ''&nbsp;=&nbsp;1, then ultimate extinction occurs with probability 1 unless each individual always has exactly one child.

In [[theoretical ecology]], the parameter ''μ'' of a branching process is called the [[basic reproductive rate]].