Braching process and probabilities

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1. The problem statement, all variables and given/known data

Consider the branching process with braching probabilities p0=1/2, p1=a, p2 = (([tex]\frac{3}{8}[/tex])-a)), p3 = 1/8 and pn = 0 otherwise, for some number a satisfying 0 [tex]\leq[/tex]a [tex]\leq[/tex]3/8

a) Find the probability generating function G(x).

b) Find the expected number of branches at a node.

c) Find those values of a for which G has two fixed points in the interval [0,1].

d) Find the probability of long-term survival when a = 0.


3. The attempt at a solution
(This attempt may be flawed and any correction would be appreciated)

The generating function is produced by the rule [tex]\sum(p_{n}x^{n})[/tex]

There for G(x) = (1/2)+(ax)+(([tex]\frac{3}{8}[/tex])-a))x[tex]^{2}[/tex]+[tex]\frac{x^{3}}{8}[/tex]

The remaining questions i am unsure with what to do any help would be appreciated.


Cheers