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Branching Process and Markov Chain

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This is a problem on a branching process. I have come up with a solution to this problem, but have no way of telling whether it´s correct or not. Need a detailed solution, thank you.

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Solution Summary

The solution contains the determination of extinction probability of a branching processes.

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The branching process { Xn, n>=0 }starts in 1 and the number of offspring per individual, N can only be even, and has the distribution P(N = 2n ) = 3/7*(4/7)^n, n= 0, 1,.. ...

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