MTH 202 Exams

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Exams

Date: Thursday May 7, 8.30AM Location: In-class

Previous Exam

Final Exam May 3 2023

Problem 1 (14 points)

Let $S_n$ be a nearest neighbor simple random walk in $d$ dimensions. It has equal probabilities of going in each direction. In class, we stated that

$\frac{c_1}{n^{d/2}} \leq p_{2n}(0,0) = \Prob ( X_{2n} = 0 | X_0 = 0) \leq \frac{c_2}{n^{d/2}}$

(3 points) Find the periodicity of the origin in dimension $d=1,2,\ldots$. Justify your answer.
(5 points) In $d=1$, write down the equation for $\pi(x)$ and find a solution. The general theorem for irreducible chains says that if a chain is recurrent, then the stationary distribution is unique up to multiplication by constants (if $\pi$ solves $\pi P = \pi$ , then so does $(2 \pi) P = 2 \pi$ ). Based on this theorem, does there exist a stationary probability? Justify your answer.
(3 points) What is the expected time to return to the origin in $d=1$?
(3 points) Find the positive recurrence, null recurrence or transience of the random walk in $d=1,2,\ldots$. Justify your answer.

Problem 2 (10 points)

Suppose that upon striking a plate, a single electron is transformed into $N$ electrons where $\E[N] = \mu$ and $\Var(N) = \sigma^2$ . Suppose each of these electrons strike a 2nd plate and release further electrons, independently of each other and with the same distribution as $N$ . Let $Z$ be the total $\#$ of electrons emitted from the 2nd plate. Find the mean and variance of $Z$ and express it in terms of $\mu$ and $\sigma$ .

Problem 3 (13 points)

Consider the following population model in discrete time. We have a tribe on some island. At each time-step, a tribe member dies, or a new member is born. The tribe can only eat after feeding their god, and so when there are $n$ people, the probability of a death is proportional to $n + 1$ because of the fact that they always have to feed one extra person; the probability of birth is proportional to the number of people $n$ (with the same proportionality constant). A birth and a death cannot occur together in the same time-step. In return for the food, if the tribe dies out, the omnipotent being introduces a new person at the next time step.

(4 points) Find the transition probabilities for the model $p_{ij}$
(6 points) Solve for the stationary distribution $\pi$ by solving $\pi_i = \sum_{j} p_{ji} \pi_j$ . Hint: Solve for $\pi_i, i = 1,2,3$ in terms of $\pi_0$ and see if you can find a pattern that you can verify inductively
(3 points) Is this a stationary probability; i.e., do we have $\sum_i \pi_i = 1$ ?

Problem 4 (13 points)

A professor continually gives exams to her students. She can give three possible types of exams, and her class is graded as either having done well or badly. Let $p_i$ denote the probability that the class does well on a type $i$ exam, and suppose that $p_1 = 0.3, p_2 = 0.6 \AND p_3 = 0.9$. If the class does well on an exam, then the next exam is equally likely to be any of the three types. If the class does badly, then the next exam is always type $1$.

(5 pts) Draw a graph with probabilities representing the Markov chain. Alternatively, you may also just write down the transition matrix.
(8 pts) Suppose the teacher gives a type 1 exam first. What is the average number of exams she has to give before giving an exam of type 2? Hint: Can you make one of the states absorbing?

Problem 5 (19 points)

Let $X_t$ and $Y_t$ be two independent Poisson processes with rate parameters $\lambda_1$ and $\lambda_2$ , measuring the number of customers coming into stores $1$ and $2$ respectively.

(3 pts) What is the probability that a customer arrives in store 1 before any customer arrives in store 2?
(3 pts) What is the probability that in the first hour, a total of exactly four customers have arrived at the two stores?
(3 pts) Given that exactly four customers have arrived at the two stores, what is the probability that all four went to store 1?
(6 pts) Let $T$ denote the time of arrival of the first customer at store $2$ . Then $X_T$ is the number of customers in store $1$ at the time of the first customer arrival at store $2$ . Find the probability distribution of $X_T$ (i.e., for each $k$ , find $\Prob(X_T = k)$ ).
(4 pts) Can you interpret your answers to parts 3 and 4 in terms of your answer to 1? What is going on here? Explain in words.

Problem 6 (17 points)

Suppose that the probability to win a game is $p$ . You play this game repeatedly. Let $\xi_i \in \{-1,1\}$ represent the outcome of game $i$ . Suppose you bet an amount $B_i$ on the game $i$ based on how you have done in the previous games (the outcomes $\xi_1,\xi_2,\ldots,\xi_{i-1}$ ). If you win, you get $B_i$ dollarydoos and if you lose, your net fortune goes down by $B_i$ . Suppose you start with an initial fortune of $X_0$ .

(4 points) Represent $X_n$ , your fortune after $n$ games in terms of the bet amounts $B_1,\ldots,B_{n}$ and outcomes $\xi_1,\ldots,\xi_n$ .
(4 points) Compute the expected value of $B_i (\xi_n - p)$ for any $i = 1,\ldots,n$ .
(4 points) Is $X_n$ a martingale with respect to the outcomes of the games $\xi_1,\ldots,\xi_{n-1}$ . That is, do we have
$\E[ X_n | \xi_1,\ldots,\xi_{n-1} ] = X_{n-1}$
Justify your answer.
(5 points) Is $X_n - \sum_{i=1}^{n} B_i p$ a martingale? Justify your answer.

Problem 7 (16 points)

Let $\{N(t); t ≥ 0\}$ be a Poisson process of rate λ, representing the arrival process of customers entering a store. Each customer spends a duration in the store that is a random variable with exponential distribution with parameter $\mu$ . The customer durations are independent of each other and the arrival process.

(5 points) Find an expression for $\Prob( U + X \leq t)$ where $U$ is $\Uniform(0,t)$ and $X$ is $\Exponential(\mu)$ .
(8 points) Determine $\E[Y_t]$ where $Y_t$ the total number of customers who have arrived and left by time $t$. Hint: Represent $Y_t$ as a sum of indicators.
(3 points) What happens to $Y_t/t$ as $t \to \infty$ . Interpret your answer.

Arjun Krishnan