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Math Help - Extinction Theorem

  1. #1
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    Extinction Theorem

    Is anyone familiar with this theorem:

    If E(X) > 1 then the extinction probability \pi is the unique root of the equation \pi = f(\pi), which lies strictly between 0 and 1.

    If E(X) < 1, then \pi = 1

    Graph is below:


    I cannot seem to find any link between the expectations, and the curve f(x) and y=x in the graph. And what is the meaning of \pi_1, \pi_2?
    Thanks for your attention.
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  2. #2
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    It looks as though π is meant to be the limit of the increasing sequence \pi_1,\,\pi_2,\,\pi_3,\ldots given by \pi_1=f(0),\;\pi_{n+1}=f(\pi_n)\;\;(n\geqslant1). Needless to say, π here does not mean the number 3.14159... .

    I don't know what the connection with expectations is. That must come from some previous information.
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