Let $\{Z_n : n \ge 0\}$ be a branching process such that generating function for $Z_1$ is $$G(s)=1-\frac{\sqrt{1-s}}{2}$$
(a)Find expected value of $Z_n$ and variance of $Z_1$
(b)Calculate the probability of ultimate extinction
(c)Find formula for generating function $G_n(s)$ for $Z_n$
(d)Calculate $P(T=n)$, for $T=\min\{n \ge 0 : Z_n =0\}$
(a) $E(Z_1)=G'(1)$. We have that $G'=\frac{1}{4\sqrt{1-s}}$, so $G'(1)=\frac10$. Does it mean that I can't find the expected value and also variance (because $Var (Z_1)=G''(1)+G'(1)-(G'(1))^2$), or are there any other formulas for calculating that?
(b) $s=G(s) \Rightarrow s=1-\frac{\sqrt{1-s}}{2} \Rightarrow s_1=\frac34, s_2=1$ so the probability of extinction is $P(Z_n=0)\to \frac34$.
(c) I know that $G_n(s)=\sum^{\infty}_{k=0}P(Z_n=k)s^k$, but I don't have any $P(Z_n=k), k=0,1,2,...$, so I'm not sure how to do this one.
(d) $P(T=n)=P(Z_n=0) - P(Z_{n-1}=0)=...$. Same as (c).
If anyone could check the above-mentioned, correct where I'm wrong and explain these parts where I can't proceed any further, I will be very grateful.
Expectation and variance don't exist, as you have noted. You have computed extinction probability correctly. Use induction to verify that $G_n(s)=1-\frac {(1-s)^{2^{-n}}} {a_n}$ where $a_1=2$ and $a_{n+1}=2\sqrt {a_n}$. The generating function of $Z_n$ is $G_n$ so $P\{Z_n=0\}=G_n(0)$. Hence $P\{T=0\}=G_n(0)-G_{n-1}(0)$.