Download Math 318 Probability Exam - University of British Columbia, 2010 Term 2 and more Exams Probability and Statistics in PDF only on Docsity!
Be sure this exam has 13 pages including the cover
The University of British Columbia
Sessional Exams – 2010 Term 2 Mathematics 318 Probability with Physical Applications Dr. G. Slade
Last Name: First Name:
Student Number:
This exam consists of 8 questions worth 10 marks each. No aids are permitted. There are tables on the last two pages. Please show all work and calculations. Numerical answers need not be simplified.
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.
(1 points) 1. (a)Define the moment generating function of a random variable X.
(3 points) (b)Derive the moment generating function of a geometric random variable with parameter p, starting from the definition of moment generating function.
(3 points) (c)A fair coin is tossed until the nth^ Head appears. Find the moment generating function for the total number of tosses.
(3 points) (d)Determine the variance of the total number of tosses, in part (c).
(10 points) 3. A plane is missing, and it is presumed that it was equally likely to have gone down in any of 3 different regions. Due to the geographical and environmental conditions of the regions, the probability that the plane will be found upon a search of the ith^ region when the plane is, in fact, in that region is 14 in region 1, 12 in region 2, and 34 in region 3 What is the conditional probability that the plane is in the ith^ region, given that a search of region 1 is unsuccessful, i = 1, 2 , 3?
- The lifetime (in hours) of a certain electronic component is an exponential random variable with parameter λ = 14 , and thus has probability density function f (x) = 14 e−x/^4 for x ≥ 0. As soon as a component fails, it is replaced immediately, so that there is always a working component.
(3 points) (a)Find the probability that a component’s lifetime exceeds 6 hours.
(3 points) (b)Let Y denote the time that the fourth component is installed. What kind of random variable is Y , i.e. what distribution does Y have?
(4 points) (c)The first component is still working after 6 hours. What is the probability that exactly four components have failed within 24 hours of the installation of the first component?
(3 points) (c)the approximate probability that an individual reaches age 90 (use the central limit theorem).
(3 points) (d)the approximate conditional probability that an individual reaches age 100, given that the individual has reached age 90 (use the central limit theorem).
- Consider simple random walk on Z^3 , taking steps (± 1 , 0 , 0), (0, ± 1 , 0), (0, 0 , ±1) with probabilities 1/6.
(5 points) (a)Write down a formula for the probability p 2 n that the walk starting at the origin returns to the origin in 2n steps. Explain how you obtained your formula.
(5 points) (b)It is possible to show (you need not show this) that
p 2 n ≤
22 n
2 n n
3 n
n! (bn/ 3 c!)^3
where bn/ 3 c denotes the greatest integer less than or equal to n/3. Using this, show that the walk is transient. (Recall Stirling’s formula n! ∼ nne−n
2 πn.)
(3 points) (c)Find the stationary distribution.
(3 points) (d)Show that the chain is reversible.
- Each day, one of 3 possible items is requested, the ith^ one with probability 6 i (i = 1, 2 , 3). These items are at all times arranged in an ordered list which is revised as follows: The item selected is moved to the top of the list with the relative positions of the other two elements left unchanged. The state at any time is the current list ordering.
(5 points) (a)Determine the transition matrix of the above Markov chain.
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.
