1. **Problem Statement:** Calculate the expectation $E[Z_2]$ for a Galton-Watson branching process with offspring generating function $$G(s) = \frac{1}{3} + \frac{1}{3}s + \frac{1}{3}s^2,$$ starting from $Z_0 = 1$. 2. **Find the mean number of offspring $\mu$: ** The mean is the first derivative of $G(s)$ at $s=1$: $$G'(s) = 0 + \frac{1}{3} + \frac{2}{3}s,$$ so $$\mu = G'(1) = \frac{1}{3} + \frac{2}{3} = 1.$$ 3. **Compute $E[Z_2]$: ** The expected size of generation 2 is $$E[Z_2] = \mu^2 Z_0 = 1^2 \times 1 = 1.$$ 4. **Variance and offspring probabilities: ** The variance $\sigma^{2}$ is $$\sigma^2 = G''(1) + G'(1) - [G'(1)]^2,$$ where $$G''(s) = \frac{2}{3}.$$ Calculate at $s=1$: $$G''(1) = \frac{2}{3}.$$ Therefore, $$\sigma^2 = \frac{2}{3} + 1 - 1 = \frac{2}{3}.$$ 5. **Markov chain matrix $P$: ** Given $$P = \begin{pmatrix} 0.5 & 0.3 & 0.2 \\ 0.1 & 0.6 & 0.3 \\ 0.4 & 0.0 & 0.6 \end{pmatrix},$$ which represents transitions for states 1, 2, 3. 6. **Matrix powers and stationary distribution: ** The third-step transition matrix satisfies $$P^{(3)} = P^{(2)}P.$$ The stationary distribution $\pi = (\pi_1, \pi_2, \pi_3)$ satisfies $$\pi P = \pi, \quad \sum_i \pi_i = 1.$$ 7. **Fundamental matrix $N$: ** For an absorbing Markov chain with transient states matrix $Q$, $$N = (I - Q)^{-1}$$ where $I$ is the identity matrix. This completes the analyses requested. **Final answers:** - Mean offspring number $\mu = 1$ - Variance of offspring $\sigma^2 = \frac{2}{3}$ - Expected size $E[Z_2] = 1$