30 Consider the branching process of Example 12 23 In that process, before death an organism produces j (j 0) offspring with probability j Let X0 1, and let , the expected number of offspring of an organism, be greater than 1 Let p be the probability that extinction will occur Show that p is the sm...

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30. Consider the branching process of Example 12.23. In that process, before death an organism produces j

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30. Consider the branching process of Example 12.23. In that process, before death an organism produces j (j ≥ 0) offspring with probability αj . Let X0 = 1, and let μ, the expected number of offspring of an organism, be greater than 1. Let p be the probability that extinction will occur. Show that p is the smallest positive root of the equation x =

∞ i

=0 αixi.

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