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The University of British Columbia
Sessional Exams – 2006 Term 2 Mathematics 318 Probability with Physical Applications, All sections Dr. A.E. Holroyd, Dr. G. Slade
Last Name: First Name:
Student Number:
Circle section number: 201 (Slade), 202 (Holroyd)
This exam consists of 8 questions worth 10 marks each. No aids are permitted.
Problem total possible score
- 10
- 10
- 10
- 10
- 10
- 10
- 10
- 10 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.
Tables on last page.
- A closet contains 10 different pairs of shoes (each pair consists of a left shoe and a right shoe, so there are 20 shoes in total). 6 shoes are chosen at random. Find (do NOT simplify) the probability that:
(3 points) (a)3 complete pairs are chosen;
(3 points) (b) exactly 1 complete pair is chosen;
(4 points) (c)at least one left shoe and at least one right shoe are chosen.
- Let X 1 , X 2 ,... be i.i.d. continuous random variables with probability density function
f (y) =
cy−^4 if |y| ≥ 1 0 otherwise,
where c is a constant. Let S 0 = 0 and let Sn = X 1 + · · · + Xn for n ≥ 1. Then Sn denotes the position, after n steps, of a random walk performed on the real line (not just the integers), with steps Xi.
(2 points) (a)Determine the value of the constant c.
(2 points) (b) Determine the variance of a single step Xi.
(6 points) (c)Calculate the approximate probability that the walker is at least distance 60 from the origin after 300 steps, i.e., find P (|S 300 | ≥ 60).
- Pizza orders for meat and vegetarian pizzas arrive according to independent Poisson processes with rates μ and ν, respectively. Let Ms denote the number of meat orders, and Vs the number of vegetarian orders, received in the time interval [0, s]. Let Xs = Ms + Vs denote the total number of orders received during [0, s].
(2 points) (a)Write down the characteristic functions of Ms and Vs.
(2 points) (b) Hence find the characteristic function of Xs = Ms + Vs.
(1 points) (c)What is the distribution of Xs (give the name and parameter(s))?
(4 points) (d) Suppose that M 1 = 1, and let T denote the arrival time of the unique meat order during [0, 1]. Calculate the conditional probability P (T ≤ s|M 1 = 1), for s ∈ [0, 1].
(1 points) (e)What is the conditional probability density function of T in part (a), given that M 1 = 1.
- Consider the Markov chain with state space { 0 , 1 , 2 , 3 , 4 , 5 } and transition matrix
P =
(2 points) (a)Draw the transition diagram showing the six states with arrows indicating possible transitions and their probabilities.
(3 points) (b) Which states are aperiodic and which are periodic?
(3 points) (c)Which states are recurrent and which are transient?
(2 points) (d) Determine the probability, starting from state 0, of ever hitting state 5.
- Two white balls and two black balls are divided between two urns, with each urn containing two balls. At each step, a ball is chosen at random from each urn, and the two balls are interchanged. (I.e., the ball from the first urn is placed in the second urn, and the ball from the second urn is placed in the first.) Let Xn denote the number of white balls in urn 1 after the nth step.
(1 points) (a)Explain briefly why (Xn) is a Markov chain.
(2 points) (b) Find the transition matrix P.
(2 points) (c)Find the matrix of two-step transition probabilities.
(3 points) (d) Find the stationary distribution.
(1 points) (e)In the long run, what fraction of the time does urn 1 contain no white balls?
(1 points) (f)If there were m black and m white balls, with m balls per urn, and the same procedure was performed, guess the stationary distribution. (You do not need to verify that your guess is correct.)
Table 1: Common Distributions
Distribution Mean Variance Characteristic function
Binomial (n, p) np np(1 − p) (1 − p + peit)n Geometric (p) 1 /p 1 − p p^2
peit 1 − (1 − p)eit Poisson (λ) λ λ eλ(e it−1)
Uniform (a, b) a + b 2
(b − a)^2 12
eita^ − eitb it(b − a) Exponential (λ) 1 /λ 1 /λ^2
λ λ − it Normal (μ, σ^2 ) μ σ^2 eiμt−σ (^2) t (^2) / 2
Table 2: Cumulative distribution function Φ(x) of standard Normal distribution
x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0. 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0. 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0. 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0. 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0. 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0. 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0. 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0. 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0. 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0. 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0. 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0. 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0. 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0. 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0. 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0. 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0. 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0. 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0. 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0. 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0. 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0. 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0. 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0. 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0. 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0. 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0. 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0. 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0. 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0. 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0. 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.
