Suppose that in a branching process the number of offspring of an initial particle has a...

Question

Suppose that in a branching process the number of offspring of an initial particle has a distribution whose generating function is f .s/. Each member of the first generation has a number of offspring whose distribution has generating function g.s/. The next generation has generating function f , the next has g, and the distributions continue to alternate in this way from generation to generation.

(a) Determine the extinction probability of the process in terms of f .s/ and g.s/.

(b) Determine the mean population size at generation n.

(c) Would any of these quantities change if the process started with the g.s/ process and then continued to alternate?

Aditi · Accepted Answer

ANSWER
a) Let Xn be the number of individuals in generation n then X0 = 1 let un = P(extinction in step n) then 
u1 = f(0) 
u2 = f(g(0)) 
u3 = f(g(f(0))) 
	.
	.
	.
. u2n = f(g(...f(g(0))...)) 
u2n+1 =f(g(f(...g(f(0))...))) 
Using what we learned in class,

Suppose that in a branching process the number of offspring of an initial particle has a distribution whose generating function is f .s/. Each member of the first generation has a number of offspring...

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