Probability | Part IA, 2003

A laboratory keeps a population of aphids. The probability of an aphid passing a day uneventfully is q<1q<1. Given that a day is not uneventful, there is probability rr that the aphid will have one offspring, probability ss that it will have two offspring and probability tt that it will die, where r+s+t=1r+s+t=1. Offspring are ready to reproduce the next day. The fates of different aphids are independent, as are the events of different days. The laboratory starts out with one aphid.

Let X1X_{1} be the number of aphids at the end of the first day. What is the expected value of X1X_{1} ? Determine an expression for the probability generating function of X1X_{1}.

Show that the probability of extinction does not depend on qq, and that if 2r+3s12 r+3 s \leqslant 1 then the aphids will certainly die out. Find the probability of extinction if r=1/5,s=2/5r=1 / 5, s=2 / 5 and t=2/5t=2 / 5.

[Standard results on branching processes may be used without proof, provided that they are clearly stated.]

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