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Question

 Let 0 ≤ p ≤ 1. Consider a branching process where the number X of offspring of an

3

individual

is zero with probability p, one with probability 2p, and two with probability 1 − 3p. Initially there is one individual.

(a) (3 marks) For what values of p will the branching process become extinct after a finite number of generations with probability 1? Explain. 

(b) (5 marks) For the case p = 1, calculate the probability that the branching process 5

survives forever.

(c) (2 marks) Assume again that p = 1. Given that there are three individuals in the second generation, find the probability that the branching process survives forever. 


Below is my answer. Please verify! I'm not too sure about the my solution to the last question.


sol4.jpg

sol4.jpg

5( a)
O
1 2
P 2p 1 - 3p
M = 2p+2(1- 30)
= 2p+2-bp
= 2-4P.
If 2 - 4p SI, then the branching process will become extinct after
a finite number of generations.
- 4PSL
5 ( b) G(S ) = P+ 2ps+ ( 1 -3P)s
5 (C)
S = F TESTES ?
65= 1+25 + 25
25 -35+1 =0
2 -1
(S- D (S - 1 ) =0
S=E or S= 1.

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