1 Workshop Exercises 1

Let $T$ be the time at which three successes in a row have occurred for the first time in a Bernoulli process with probability $p$ of success. Using conditional expectations show that

(2003 B1) Consider the random walk, $X_t$, specified by

Show that $\text{P}(\text{Ruin}|X_0=1)$ is the same for this random walk as it was for the simple random walk in lectures. (You may like to calculate the probabilities of two successive upward moves, an upward move followed by a downward move, and a downward move for the random walk studied in lectures; can you relate the two random walks?)

Consider a simple random walk starting at $X_0=0$, with

Define $T$ to be the next time that $X_t=0$.

By conditioning on the $X_1$, show that $\text{E}(T)$ is infinite regardless of the value of $p$. [You may use without proof the results about expected time to ruin in the lecture notes.]

Calculate $\text{P}(T=2m)$, for $m=1,2,\mathrm{\dots }$.

Assume that you have just thrown two consecutive successes. What is the expected further time until you next have two consecutive successes in your Bernoulli process? (Note that if the next outcome is a success, then your further time is 1).

Repeat the question, but this time for the time until a success follows a failure i.e. what is the further time to obtaining FS given you have just observed FS? (You may want to use the result of Exercise 2.1.4 from lectures.)

How do these expected times relate to the probability of 2 consecutive successes ($p^2$) and a success followed by a failure ($pq$) respectively?

Consider a random walk, $X_t$, satisfying $X_0=1$, and

Calculate the probability of ruin ($X_s=0$ for some $s>0$). How does this relate to the probability of ruin of the simple random walk considered in lectures, and why?

Consider the following game. You start with one token. You repeatedly toss a fair coin: if it is a head you gain a token, otherwise you lose a token. The game finishes if you ever have no tokens (you lose the game) or if you ever have three tokens (you win the game).

Let $X_t$ be the number of tokens you have after the $t$th coin toss. If, before the $t$th coin toss you have either won or lost the game, then $X_t=3$ or $X_t=0$ respectively.

• (i)

  Show $\text{E}(X_t|X_{t-1}=i)=i$. (It is a fair game.)

• (ii)

  By using $\text{E}(X_t)=\text{E}(\text{E}(X_t|X_{t-1}))$, or otherwise, show that $\text{E}(X_t)=\text{E}(X_{t-1})$. Hence show that $\text{E}(X_t)=1$ for all $t$.

• (iii)

  What are the two potential, eventual outcomes of the game, and their values for $X_t$?

• (iv)

  Let $p$ be the probability that you eventually win the game. Use (ii) and (iii) to show that $p=1/3$.

• (v)

  Now assume that you start with $i$ tokens, and to win the game you need to accrue $n$ tokens. Show that the probability of winning the game is $i/n$.