To assist with the mathematical questions in the image, I'll solve them step-by-step:

1. Branching Process

(i) Generating Function $G_X(t)$

The generating function for $X$, given $X \in \{0, 1, 2, 3\}$ with the specified probabilities, is:

$G_X(t) = \mathbb{E}[t^X] = \sum_{k=0}^3 P(X=k)t^k$

Substituting the given probabilities:

$G_X(t) = \frac{1}{8}t^0 + \frac{1}{2}t^1 + \frac{1}{4}t^2 + \frac{1}{8}t^3$

$G_X(t) = \frac{1}{8} + \frac{1}{2}t + \frac{1}{4}t^2 + \frac{1}{8}t^3$

(ii) Probability of Extinction Using Total Probability Law

Let $\eta$ be the probability of extinction. Using the law of total probability:

$\eta = \sum_{j=0}^3 P(Z_1 = j) \eta^j$

The generating function $G_X(\eta)$ gives the simplification:

$\eta = G_X(\eta) = \frac{1}{8} + \frac{1}{2}\eta + \frac{1}{4}\eta^2 + \frac{1}{8}\eta^3$

The equation $\eta = G_X(\eta)$ must be solved to find $\eta$, which represents the probability of extinction.

(iii) Probability of Extinction

This corresponds to finding the smallest non-negative root of the equation $\eta = G_X(\eta)$. Solve numerically or algebraically:

$\eta = \frac{1}{8} + \frac{1}{2}\eta + \frac{1}{4}\eta^2 + \frac{1}{8}\eta^3$

Rearranging:

$\frac{1}{8}\eta^3 + \frac{1}{4}\eta^2 - \frac{1}{2}\eta + \frac{1}{8} = 0$

This cubic equation gives $\eta = 1$ as a trivial root, which corresponds to the extinction probability if there are no further individuals. Other roots may also need verification depending on the process context.

2. Simple Random Walk

(i) Using Total Probability for $P(T_0 = n)$

Let $T_0$ be the first hitting time of zero for $S_n$. Using total probability and the fact that $X_1 = \pm 1$ with equal probabilities:

$P(T_0 = n) = \frac{1}{2}P(T_0 = n-1) + \frac{1}{2}P(T_1 = n-1)$

Given symmetry, $P(T_0 = n) = P(T_1 = n-1)$, so:

$P(T_0 = n) = P(T_1 = n-1)$

(ii) Showing $G_0(t) = tG_1(t)$

The generating function for $T_0$ satisfies the recurrence:

$G_0(t) = \sum_{n=1}^\infty P(T_0 = n)t^n$

From part (i), $P(T_0 = n) = P(T_1 = n-1)$, so:

$G_0(t) = tG_1(t)$

(iii) Generating Function for $T_a$ (for $a > 1$)

Define $T_a$ as the first hitting time of $a$. The recurrence relation for $T_a$ depends on the state and follows similar logic. The generating function can be derived as a function of $a$ and analyzed for specific values of $a$.

Would you like detailed numerical solutions or expansions for these steps? Here are five related questions you might want to explore:

1. How can we numerically solve the cubic equation for extinction probability?
2. Can we generalize $G_X(t)$ for other branching processes?
3. What are the conditions under which extinction is certain?
4. How do we interpret $G_0(t) = tG_1(t)$ in the context of first hitting times?
5. How does the random walk's behavior change as $a$ increases?

Tip: Generating functions are powerful tools for solving recurrence relations and analyzing probabilities systematically.