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The questions for the m. Phil. Exam in statistical science, focusing on large deviations and queueing theory. The exam consists of four questions, with equal weight, and includes geometric random variables, large deviations principle, and queueing theory. Candidates are required to find the number of hyperspace boosts the starship voyager received, prove varadhan's integral theorem, and prove the continuity of the queue size function.
Typology: Exams
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Thursday 30 May 2002 9 to 12
Attempt THREE questions There are four questions in total
The questions carry equal weight
1 The starship Voyager is far away in the Delta quadrant, and is heading for Earth 47000 lightyears away. Every day, it may either be given a hyperspace boost by friendly aliens, bringing it 100 lightyears closer; or it may travel without incident, coming 3 lightyears closer; or it may pause to explore a strange new world, and only come 1 lightyear closer. These events occur independently: the first has probability 1/1000, and the other two are equally likely.
In fact, Voyager returns home in 4700 days, much faster than expected. How many hyperspace boosts did it get on its journey home?
State clearly any general results you use.
2 (a) Let X be a geometric random variable with mean 1/p, and let XL^ = X/L. Show that XL^ satisfies a large deviations principle with good rate function I(x) = x log(1−p)−^1.
(b) State the contraction principle. (c) Let X 1 ,... , Xn be independent geometric random variables with means 1/p 1 ,... ,
... , 1 /pn. Find a large deviations principle for X 1 + · · · + Xn.
3 State and prove Varadhan’s Integral Theorem.
4 Consider a queue operating in slotted time, with finite buffer B and constant service rate C, receiving xt units of work in timeslot t. For t > 0 let x(0, t] = x−t + · · · + x− 1. Let A be the set of input processes for which x(0, t] 6 λt eventually, and let ‖x‖ = supt> 0 |x(0, t]/t|. Prove that if λ < C then the queue size function is continuous on (A, ‖ · ‖).
