12F (Probability)

Conditioning on generation

n

and using independence gives

F_{n + 1} (s) = F_{n} (F_{1} (s))

ii

Differentiating at

s = 1

gives

E X_{n + 1} = μ E X_{n}

, hence

E X_{n} = μ^{n}

The extinction probability is the smallest fixed point of the convex generating function

F_{1}

[0, 1]

. Since

F_{1}' (1) = μ \leq 1

and

p_{0}, p_{2} > 0

make strict convexity relevant, the only fixed point is

1

Here

F_{1} (s) = 1/ (2 - s)

and

μ = 1

, so extinction is certain. Iteration gives

F_{n} (s) = \frac{n - ( n - 1 ) s}{n + 1 - n s},

and

P (X_{n} = 0) = F_{n} (0) = n / (n + 1)

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