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The University of British Columbia
Sessional Exams – 2011 Term 2 Mathematics 318 Probability with Physical Applications Dr. G. Slade
Last Name: First Name:
Student Number:
This exam consists of 7 questions worth 10 marks each and 2 questions worth 5 marks each. No aids are permitted. There are tables on the last page. Please show all work and calculations. Numerical answers need not be simplified.
Problem total possible score
- 10
- 10
- 10
- 10
- 10
- 10
- 10
- 5
- 5 total 80
- Each candidate should be prepared to produce his library/AMS card upon request.
- Read and observe the following rules: No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers or memoranda, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received.
- Smoking is not permitted during examinations.
(5 points) 1. (a)Five dice are rolled simultaneously. Calculate the probability of rolling a full house (values a, a, a, b, b with a, b different).
(5 points) (b)A poker hand of five cards is dealt from a standard deck of 52 cards. Calculate the probability of a full house (face values a, a, a, b, b with a, b different).
- The occurrence of major earthquakes in a certain location can be modelled by a Poisson process with rate λ = 1001 per year. The time intervals between major earthquakes are thus given by independent exponential random variables with λ = 1001.
(3 points) (a)What is the probability that the next major earthquake occurs in less than 50 years?
(4 points) (b)Let N 500 denote the number of major earthquakes that occur during the next 500 years. What kind of random variable is N 500? What is its mean?
(3 points) (c)Compute the probability that there are three or fewer major earthquakes in the next 500 years.
(10 points) 4. A binary message—either 0 or 1—must be transmitted by wire from location A to location B. However, data sent over the wire are subject to channel noise disturbance, so to reduce the possibility of error, the value 2 is sent if the message is 1 and the value −1 is sent if the message is 0. If x is the value sent at A (x = −1 or x = 2), then the value received at B is x + N , where N represents the noise. Assume that N is a normal random variable with mean μ = 0 and variance σ^2 = 0.25. Assume also that the message to be transmitted is equally likely to be either 0 or 1. When the message is received at B the receiver decodes it according to the following rule: If R ≥ 0 .5, then 1 is concluded. If R < 0 .5, then 0 is concluded. The message concluded at B is 1. What is the probability that the message was incorrectly transmitted?
- Consider a random walk on the 2-dimensional square lattice that takes each of the four steps (+1, +1), (+1, −1), (− 1 , +1), (− 1 , −1) with equal probabilities 14.
(4 points) (a)Calculate the characteristic function φ 1 (k 1 , k 2 ) of a single step. Simplify your answer as much as possible. (Recall the trigonometric identity cos(x + y) + cos(x − y) = 2 cos x cos y.)
(3 points) (b)Identify all singularities of 1/(1 − φ 1 (k 1 , k 2 )), for k 1 , k 2 ∈ [−π, π].
(3 points) (c)Is the random walk transient or recurrent? Explain in detail.
- Two white balls and two black balls are distributed in two urns in such a way that each urn contains two balls. We say that the system is in state i, (i = 0, 1 , 2) if the first urn contains i white balls. At each step, we draw one ball from each urn, place the ball drawn from the first urn in the second urn, and place the ball drawn from the second urn in the first urn. Let Xn denote the state of the system after the nth step. This defines a Markov chain.
(4 points) (a)Calculate the one-step transition matrix for this Markov chain.
(1 points) (b)Calculate the two-step transition matrix.
- Fix a number p with 0 < p < 1. Consider the Markov chain on the non-negative integers { 0 , 1 , 2 ,.. .} whose transition probabilities are given by
Pn,n+1 = p, Pn, 0 = 1 − p, for all n ≥ 0.
Suppose that the Markov chain is initially in state 0, and let T 0 denote the time of first return to 0 (i.e., T 0 is the smallest value of n > 0 such that Xn = 0, if such a value exists, and otherwise T 0 = ∞).
(3 points) (a)Determine the probability mass function of T 0.
(2 points) (b)Determine the expected value ET 0.
(5 points) 9. We wish to use Octave to compute the integral
0 e
− 3 x/ (^2) dx via Monte Carlo integration. In general, for
∫ (^) b a f^ (x)dx, we simulate a large number of Unif(a, b) random numbers^ U^1 ,... , Un and use the approximation ∫ (^) b
a
f (x)dx ≈ (f (U 1 ) + · · · + f (Un)) b − a n
The code below runs, but gives incorrect answers:
% Script to perform the required integral
% Generate a number of uniform numbers N = 10000; uniforms = unifrnd(0,2,N,1); % generates a N x 1 matrix of % Unif[0,2] random numbers
accumulator = 0;
for i = 1:N accumulator = accumulator + uniforms(i); end % or endfor; both work
% Compute the approximate value of the integral approximation = exp((-3/2)*(accumulator/N));
The exact value is
0 e − 3 x/ (^2) dx = 2(1 − e− (^3) )/ 3 ≈ 0 .63348, but running the above code four times gave the answers 0.22423, 0.22461, 0.22568, and 0.22329. Find the problems in the code and state how to correct it. If you find it easier to rewrite all or some of the code, feel free to do so. (If you can’t remember the syntax, explain what you are trying to do in pseudocode.)
