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The University of British Columbia
Sessional Exams – 2010 Winter Term 1 Mathematics 418 Probability
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This exam consists of 7 questions worth 10 marks each. No aids other than nonprogrammabble calculators are permitted. Answers must be accompanied by proofs/explanations unless the question says otherwise.
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- 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.
- Let X, Y be random variables on (Ω, F, P ).
(a) Define “random variable”.
(b) Prove that Z = |X| is a r.v.
(c) Let N be an integer, let Ω = R and let FN be the smallest σ-field that contains all the intervals ( i− N^1 , (^) Ni ], where i ∈ Z. Describe, without proof, the random variables in this case.
- Let Y be a continuous random variable with density fY (y) =
2 y for 0 < y < 1 0 else.
(a) What is the distribution of Y
(b) Let X have the uniform distribution on [0, 1]. Find a function g such that g(X) equals Y in distribution. Does g have to be defined on all of R?
- Suppose that the number X of cars that cross the border into the USA in a given hour is a Poisson random variable with parameter λ > 0. Independently, each of these cars has probability p of being searched. Let Y be the number of cars that are searched in the specified hour.
(a) What is E(Y |X)?
(b) Find P (Y = k).
- Let En(b) be the event that random walk hits b ∈ Z for the first time at step n. The hitting time theorem for symmetric random walk says that P (En(b)) = | nb| P (Sn = b), where the walk starts from the origin and n ≥ 1. Let T = {n ≥ 1 : Sn = 0} be the time of first return to the starting point. Find P (T = 2n) in a simple form.
(10 points) 7. Alice and Bob are playing a series of games numbered k = 1, 2 ,.... In the kth game, Alice either wins $k from Bob or loses $k to B, both with probability 1/2. Let Sn be the total amount of money that Alice has won from Bob in the first n games including game n. If she has lost money then Sn is negative.
(a) What does it mean to say that a sequence Zn of random variables converges in distribution to a random variable Z?
(b) If Zn converges in distribution to Z, when does it also converge in probability? (No proof).
(c) Let Zn = n−pSn. Find p such that the variance of Zn has a non-zero finite limit as n → ∞.
(d) Prove that the sequence Zn converges in distribution to a random variable Z and find the distribution of Z. Limits of sums can sometimes be evaluated by recognising them to be Riemann sums.
