Exercise 3.1. Suppose that the gambler plays boldly with initial fortune x and target fortune a. As usual, let $\mathit{X}=\left(\begin{array}{c}X_0\\ X_1\\ \mathrm{...}\end{array}\right)$ denote the fortune process for the gambler. Argue that for any $c> 0$, the random process $c\mathit{X}=\left(\begin{array}{c}cX_0\\ cX_1\\ \mathrm{...}\end{array}\right)$ is the fortune process for bold play with initial fortune $cx$ and target fortune $ca$.

Exercise 3.2. Recall that the betting function S is the function that gives the amount bet as a function of the current fortune. Show that

$$S(x)=$ \min\{x , 1-x\}$=\begin{cases}x & \text{if $0\le x\le 1/2$}\\ 1-x & \text{if $1/2\le x\le 1$}\end{cases}$$

Exercise 3.3. By conditioning on the outcome of the first game, show that F satisfies the functional equation in part (a). Show directly that F satisfies the boundary conditions in part (b):

1. $F(x)=\begin{cases}pF(2x) & \text{if $x\in \left[0 , 1/2\right]$}\\ p+qF(2x-1) & \text{if $x\in \left[1/2 , 1\right]$}\end{cases}$
2. $F(0)=0\text{, }F(1)=1$

Exercise 3.4. Show that $d(x)_i=x_{i+1}$ for $x\in \left[0 , 1\right)$

Exercise 3.5. Suppose that gambler starts with initial fortune $x\in \left(0 , 1\right)$. Show that the gambler eventually reaches the target 1 if and only if there exists a positive integer k such that $I_j=1-x_j$ for $j\in \{1, 2, \mathrm{...}, k-1\}$ and $I_k=x_k$. That is, the gambler wins if and only if when the game bit agrees with the corresponding fortune bit for the first time, that bit is 1.

Exercise 3.6. Suppose that the gambler starts with initial fortune $x\in \left(0 , 1\right)$. Use the result of the Exercise 3.5 to show that the gambler reaches the target 1 if and only if $W< x$.

Exercise 3.7. Show that W has a continuous distribution. That is, show that $P(W=x)=0$ for any $x\in \left[0 , 1\right]$.

Exercise 3.8. Show that the success function F is the unique continuous solution of the functional equation in Exercise 3.3.

1. Use mathematical induction on the rank to show that any two solutions of must agree at the binary rationals.
2. Use part (a) and continuity to show that any two continuous solutions of the functional equation must agree for all x.

Exercise 3.9. Use Exercise 3.5 to show that

$$F(x)=\sum_{n=1}^{\infty } p_{x_1}\mathrm{·\; ·\; ·}p_{x_{n-1}}px_n$$

Exercise 3.10. Show that F is strictly increasing on $\left[0 , 1\right]$. This means that the distribution of W has support $\left[0 , 1\right]$; that is, there are no subintervals of $\left[0 , 1\right]$ that have positive length, but 0 probability.

Exercise 3.11. In particular, show that

1. $F(1/8)=p^3$
2. $F(2/8)=p^2$
3. $F(3/8)=p^2+p^{2}q$
4. $F(4/8)=p$
5. $F(5/8)=p+p^{2}q$
6. $F(6/8)=p+pq$
7. $F(7/8)=p+pq+pq^2$

Exercise 3.12. Suppose that $p=1/2$. Show that $F(x)=x$ for $x\in \left[0 , 1\right]$ in two ways:

1. Using the functional equation in Exercise 3.3.
2. Using the representation in Exercise 3.9.

Exercise 3.13. Use the strong law of large numbers to show that

1. $P_p(W\in C_p)=1$ for $p\in \left(0 , 1\right)$
2. $P_p(W\in C_t)=0$ for $p\neq t$

Exercise 3.14. Show that when $p\neq 1/2$, W does not have a probability density function, even though it has a continuous distribution. The following steps outline a proof by contradiction

1. Suppose that W has probability density function f
2. Then $P_p(W\in C_p)=\int_{x\in C_p} f(x)\,d x$.
3. By the previous exercise, $P_p(W\in C_p)=1$.
4. But also $\int_{x\in C_p} 1\,d x=P_{1/2}(W\in C_p)=0$. That is, $C_p$ has Lebesgue measure 0.
5. Hence $\int_{x\in C_p} f(x)\,d x=0$.

Exercise 3.15. In the red and black experiment, select Bold Play. Vary the initial fortune, target fortune, and game win probability with the scroll bars and note how the probability of reaching the target changes. In particular, note that this probability depends only on $\frac{x}{a}$. Now for various values of the parameters, run the experiment 1000 times with an update frequency of 100 and note the apparent convergence of the relative frequency function to the probability density function.

Exercise 3.16. By conditioning on the outcome of the first game, show that G satisfies the following functional equation in (a). Show directly that G satisfies the boundary conditions (b).

1. $G(x)=\begin{cases}1+pG(2x) & \text{if $x\in \left(0 , 1/2\right]$}\\ 1+qG(2x-1) & \text{if $x\in \left[1/2 , 1\right)$}\end{cases}$
2. $G(0)=0\text{, }G(1)=0$

Exercise 3.17. Suppose that the initial fortune of the gambler is $x\in \left(0 , 1\right)$. Show that $N=\min\{I_k=x_k\lor k=r(x)\mid I_k=x_k\lor k=r(x)\}$.

1. If x is a binary rational then N takes values in the set $\{1, 2, \mathrm{...}, r(x)\}$. Play continues until the game number agrees with the rank of the fortune or a game bit agrees with the corresponding fortune bit, whichever is smaller. In the first case, the penultimate fortune is $1/2$, the only fortune for which the next game is always final.
2. If x is a binary irrational then N takes values in $\{1, 2, \mathrm{...}\}$. Play continues until a game bit agrees with a corresponding fortune bit.

Exercise 3.18. Suppose that $x\in \left(0 , 1\right)$, and recall our special notation: $p_0=p$, $p_1=q$. Show that

$$G(x)=\sum_{n=0}^{r(x)-1} p_{x_1}\mathrm{·\; ·\; ·}p_{x_n}$$

1. Note that the $n=0$ term is 1, since the product is empty.
2. The sum has a finite number of terms if x is a binary rational.
3. The sum has an infinite number of terms if x is a binary irrational.

Exercise 3.19. Use the result of the Exercise 3.18 to verify the following values:

1. $G(1/8)=1+p+p^2$
2. $G(2/8)=1+p$
3. $G(3/8)=1+p+pq$
4. $G(4/8)=1$
5. $G(5/8)=1+q+pq$
6. $G(6/8)=1+q$
7. $G(7/8)=1+q+q^2$

Exercise 3.20. Suppose that $p=1/2$. Use Exercise 3.18 to show that

$$G(x)=\begin{cases}2-\frac{1}{2^{r(x)-1}} & \text{if $x\text{ is binary rational}$}\\ 2 & \text{if $x\text{ is binary irrational}$}\end{cases}$$

Exercise 3.21. In the red and black experiment, select Bold Play. Vary x, a, and p with the scroll bars and note how the expected number of trials changes. In particular, note that the mean depends only on the ratio $\frac{x}{a}$. For selected values of the parameters, run the experiment 1000 times with an update frequency of 100 and note the apparent convergence of the sample mean to the distribution mean.

Exercise 3.22. For fixed x, show that G is continuous as a function of p.

Exercise 3.23. Show that G is discontinuous at the binary rationals in $\left[0 , 1\right]$ and continuous at the binary irrationals.