632-seppalainen-07spfin

# 632-seppalainen-07spfin - 632 Introduction to Stochastic...

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Unformatted text preview: 632 Introduction to Stochastic Processes Spring 2007 Final Exam Instructions: Show calculations and give concise justifications for full credit . Points add up to 200. 1. Born again branching process. Let { p k } k&lt; be the offspring distribution of a branching process. Assume 0 &lt; p &lt; 1 so that interesting behavior is possible and let m be the mean of the offspring distribution. Let the process begin with one progenitor, and let X n be the population size of the n th generation with X = 1. The Galton-Watson branching process we studied in Chapter 1 was ab- sorbed at zero if it ever dies out. Now we give the process a chance to come to life again after extinction. Take another parameter 0 &lt; &lt; 1. Here is the transition rule for X n : (i) If X n &gt; 0 the next state X n +1 is determined by the branching process transition with offspring distribution { p k } . (ii) If X n = 0 we put X n +1 = 1 with probability and put X n +1 = 0 with probability 1- ....
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## 632-seppalainen-07spfin - 632 Introduction to Stochastic...

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