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LANCASTER UNIVERSITY
2012 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS 2 Hours
MATH 332/MATH 432: Stochastic Processes
You should answer ALL Section A questions and TWO Section B questions.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40.
SECTION A
A1. (a) Define the probability generating function (pgf) of a random variable^ X^ which takes values in the non-negative integers. [2] (b) Let G(z) be the pgf of a random variable X. If a random variable Y has pgf G(z)^3 , then calculate its mean and variance in terms of the mean and variance of X. [4] (c) Let T be the time at which three successes in a row have occurred for the first time in a sequence of Bernoulli trials with success probability p. By using a suitable conditional expectation result (which you should state clearly if you use it), calculate E(T ). Please define clearly your notation. [6]
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SECTION A continued
A2. A tennis game has reached deuce. The server has probability p of winning for each subsequent point (otherwise the receiver wins the point). Winning of each point is independent of winning any other, and to win the game either the server or receiver needs to win two consequetive points. If one player wins the first point and the other player wins the second point then the score is again deuce. Let N be the number of points played from now until the game finishes.
(a) Show that the pgf of N is
GN (z) = (p
(^2) + q (^2) )z 2 1 − 2 pqz^2
. [5]
(b) Let p = 1/3 and denote pi = P(N = i), (i ≥ 0). (i) Calculate E(N ). [4] (ii) By expanding the pgf as GN (z) = p 0 + p 1 z + p 2 z^2 + · · · show that for i > 2
9 pi − 4 pi− 2 = 0
[4] (iii) Hence calculate the probability mass function for N , i.e. calculate pi for i = 0, 1 , 2 ,... [5]
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SECTION B
B1. (^) (a) A gambler’s wealth Xt is modeled by the following modified simple random walk
Xt =
Xt− 1 + 1 with probability p, Xt− 1 with probability r, Xt− 1 − 1 with probability q,
where p + q + r = 1. The gambler starts out with £k, where k is a positive integer, and only stops playing if she is ruined, i.e. loses all this capital. Let the random variable Tk be the number of bets played when ruin occurs. (i) Explain how Tk can be expressed as the sum of k independent random variables T (1), T (2),... , T (k), each of which has the same distribution as the random variable T = T 1. [4] (ii) By conditioning on the outcome of the first bet when k = 1, show that the probability generating function G(z) of T satisfies
G(z) = z[q + rG(z) + pG(z)^2 ].
[5] (iii) Hence obtain G(z) explaining the reason for your particular choice of the root of this equation. [3] (iv) By considering G(1) show that the probability of ruin is
P (Ruin) =
1 if p ≤ q q/p if p ≥ q
(Hint: Recall that
x^2 = |x|) [5]
Question B1 continued over the page
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SECTION B continued
B1 continued
(b) Consider a simple random walk satisfying
P(Xt = Xt− 1 + 1) = p, P(Xt = Xt− 1 − 1) = q = 1 − p.
Assume that the walk starts at X 0 = k, where k is a positive integer. Define Nt(k, b) to be the number of paths that end at Xt = b. Further define for b > 0, N (^) t^0 (k, b), the number of paths that start at X 0 = k and end at Xt = b, for which Xs = 0 for some s = 1, 2 ,... , t − 1. (i) For what values of t and k is Nt(k, 0) = 0? Derive an expression for Nt(k, 0) for these cases. [5] (ii) Using the reflection principle, calculate the number of paths that hit 0 for the first time at time t. Carefully explain the steps you take. [5] (iii) Hence calculate the probability mass function of the time to ruin T , i.e. the first time the walk hits the value 0. [3]
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SECTION B continued
B3. A discrete time Markov chain with four states, 1,2, 3 and 4, has one-step transition probability matrix (TPM)
P =
The eigenvalues of P are 1, 3 / 4 , 1 / 2 , and 1/4.
(a) For a general discrete-time Markov chain prove that the asymptotic distribution, if it exists, is also the unique invariant distribution. [6] (b) Now consider the specific 4-state chain described at the start of the question. Which state(s) of the chain are persistent? What is the asymptotic distribution of the chain? What is the geometric rate of convergence to this distribution? [4] (c) Consider a Markov chain with TPM P which starts in state X 0 = 1. (i) Write down P (X 0 = 4), P (X 1 = 4), P (X 2 = 4), and limn→∞ P (Xn = 4). [4] (ii) Use the eigenvalues of the TPM to write down the general form for the n step probability P 14 (n ). [2] (iii) Use your answers to Parts (i) and (ii) to write down 4 equations in the four unknowns and solve these to find P 14 n. [10] (iv) Consider the probability that a chain with X 0 = 1 is not in state 4 at time t. Show that the earliest time t, such that this probability is less than 10−^3 is approximately
t = 3 log 10 + log 3 log 4 − log 3
Hint: for large t, (3/4)t^ is much greater than (1/2)t. [4]
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