Stochastic

Since 0 is an absorbing state 0 1 2 and the sequence

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Unformatted text preview: here, we are interested in the question: Q. What is the probability the species avoids extinction? Here “extinction” means becoming absorbed state at 0. As we will now explain, whether this is possible or not can be determined by looking at the average number of o↵spring of one individual: 1 X µ= kpk k=0 If there are m individuals at time n 1, then the mean number at time n is mµ. More formally the conditional expectation given Xn 1 E (Xn |Xn 1) = µXn 1 Taking expected values of both sides gives EXn = µEXn 1. Iterating gives EXn = µn EX0 (1.32) If µ < 1, then EXn ! 0 exponentially fast. Using the inequality EXn it follows that P (Xn P (Xn 1) 1) ! 0 and we have I. If µ < 1 then extinction occurs with probability 1. To treat the cases µ 1 we will use a one-step calculation. Let ⇢ be the probability that this process dies out (i.e., reaches the absorbing state 0) starting from X0 = 1. If there are k children in the first generation, then in order for extinction to occur, the fam...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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