Question

9. Consider the Branching Process {Xn,n = 0,1,2,3,...} where Xn is the population size at the nth generation. Assume P(Xo = 1) = 1 and that the probability generating function of the offspring distribution is common A(z) (z3322z + 4) 10 (а) If gn 3 P(X, — 0) for n %3D 0, 1,..., write down an equation relating ^n+1 and qn. 0,1,2 Hence otherwise, evaluate qn for n= or (b) Find the extinction probability q = lim00 n 6 marks]

Answer 1

9)

a)

If
A
generating function of $X_n$.

So,
A
(0) =
Чn

and
An+1
= A(
A
)

So,
n+1
= (
Чn
).

b)  

(minimum non-negative solution of A(3) = 3)

lim qn= q n-too

So, Z =

3/10)(3 32/10)(2* 3/10)+4/10

z has solutions 3/10 and 1/5.

So, q = 1/5