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Unformatted text preview: Stat 150 Stochastic Processes Spring 2009 Lecture 13: Branching Processes Lecturer: Jim Pitman • Consider the following branching process: Z = 2 Z 2 = 3 Z 3 = 2 Z 4 = 2 • • • • • • • • • • • • Model : Intuitively: a random genealogical tree. Or a forest of trees, e.g. 2 trees above starting from Z = 2 individuals at time 0. Formalism captures Z n := number of individuals at n th generation. Note that Z n = 0 = ⇒ population extinct by time n . Extinction time = first n (if any) such that ( Z n = 0) = ∞ if no such n exists Z = initial number of individuals. ( Z ,Z 1 ,Z 2 ,... ) is the discrete time branching process. Mechanism : There is a fixed distribution of probabilities p ,p 1 ,p 2 ,... for the number of offspring of each individual. Generally, let X denote the number of offspring of a generic individual, so P ( X = k ) = p k . Assumptions : Informally, each individual present at time n has some number of offspring, distributed like X , independently of all past events and all other in- dividuals present at time...
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- Spring '09
- Probability theory, φ, Konrad Zuse, Zn, branching processes