## Braching process and probabilities

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

Consider the branching process with braching probabilities p0=1/2, p1=a, p2 = (($$\frac{3}{8}$$)-a)), p3 = 1/8 and pn = 0 otherwise, for some number a satisfying 0 $$\leq$$a $$\leq$$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 $$\sum(p_{n}x^{n})$$

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

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

Cheers
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