Let's consider the population of parrots that come from the same female, where each female give birth to $n$ daughters with probability $p_n=\frac{2}{3^{n+1}}, \;n=0,1,...$
(a)Find the probability of ultimate extinction,
(b)Find the probability, that extinction will exactly happen in 7th generation,
(c)Find the mean amount of females in $k$-th generation,
(d)Prove the formula for generating function of random variable $Z_k$, which is the amount of females in $k$-th generation
I need help with above-mentioned exercise. According (a) and (b) I know, that I need to find a generating function $G(s)= \sum_{k=0}^{\infty} P(Z_1=k)s^k$, but I'm not sure what I have to do with that knowledge. According to (c) and (d), I don't really know what am I supposed to do. Any help will be much appreciated.
The process you are looking at is a special type of the well known Galton Watson Process and the probability of extinction is characterized by the smallest fixpoint $\eta$ of $G(s) = s$, where $s\in [0,1]$.
a) Solve $G(\eta) = \eta$.
b) By evaluating the equation of d) at $0$ we get $\Bbb P (Z_n = 0) = \underbrace{G \circ \ldots \circ G} _{n \text{ times}}(0)$.
c) Also well known : $\Bbb E [Z_k] = (\Bbb E [Z_1])^k$ (can be proved by induction)
d) Prove by induction: $\underbrace{G \circ \ldots \circ G} _{n \text{ times}} (s) = G_n(s):= \sum_{k=0}^\infty \Bbb P (Z_n = k) s^k$
The statements above are true for general Galton Watson Processes, but especially a) can be hard to solve. In your special case your can calculate $G$ explicitly:
Assuming your definition of $p_n$ is a typo (otherwise this would not be a probability distribution) you can calculate the probability generating function as follows:
$$G(s) = \sum_{k=0}^{\infty} \Bbb P (Z_1 = k) s^k = \sum_{k=0}^\infty \frac 2 {3^{k+1}} s^k = \frac 2 3 \sum_{k=0}^\infty (\frac s 3)^k = \frac 2 3 \frac 1 {1 - \frac s 3} = \frac 2 {3 -s}.$$