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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 $sin [0,1]$.

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:

$$G(s) = sum_{k=0}^{infty} Bbb P (Z_1 = k) s^k = sum_{k=0}^infty frac 2 {3^{k+1}} s^k = frac 2 3 sum_{k=0}^infty (frac s 3)^k = frac 2 3 frac 1 {1 - frac s 3} = frac 2 {3 -s}.$$