MTHM002 方法统计学

除非您提交了缓解案例，否则逾期工作将被打0%（零）的分数缓解委员会接受的。请问我，你的私人导师或哈里森枢纽，如果你需要建议。这是一门单独的课程，请注意学院和大学关于合作和剽窃的指导方针，可从Fac获得-ulty网站。这意味着，你必须独自完成这项任务与任何其他人讨论，Methods for Stochastics and Finance MTHM002 Term 1Coursework 2

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Please attempt all questions and hand them in by 12pm MIDDAY Friday 16th De-cember 2022 via BART. LATE WORK will be given a mark of 0% (ZERO) unless you submit a case for mitigationthat is accepted by the Mitigation Committee. Please ask me, your personal tutor or theHarrison Hub if you need advice.This is an individual coursework and your attention is drawn to the Faculty andUniversity guidelines on collaboration and plagiarism, which are available from the Fac-ulty website. This means, you must do this assignment yourself,alone, withoutdiscussing it with any other persons, as if in an invigilated exam. Suspected cases ofacademic misconduct will be dealt with according to University regulations. This is an “open book” assessment. You may use this module’s lecture notes and problemsheets as a guide to appropriate methods. You also may use other resources, such asmethods described in textbooks or in the internet, and software to assist your computa-tions, however in that case the onus is on you to demonstrate full understandingof what you are doing. If you used a method adopted from an outside source, pleaseidentify that source in your submission. If you used software to assist your calculations,please state that in your submission and provide details of how you did it, e.g. by at-taching a listing of the relevant computer code. Results presented without appropriatereferences and explanations may be interpreted as guesswork and then earn no credit, oras evidence of plagiarism or collusion and then incur disciplinaryaction.

Question 1. Suppose the function u(x, t) satisfies the partial differential equationin the domain x ∈ R, t ∈ [0, 1]. Let u(x, 0) = f(x) and u(x, 1) = g(x), and assume that bothfunctions f(x) and g(x) quickly decay as x→ ±∞.(i) Use the Fourier transform method for solving the PDE, to show that the two functionsare related by convolution,g(x) =∞∫∞f(s)K(x s) dswhere you need to identify the convolution kernel K(·).(ii) Determine whether it is possible to relate the two functions in the opposite direction,that is,for some function L(·). That is, either find L(·) or show why it is nopossible. [25]1November 23, 2022Question 2. Let Z be a normally distributed random variable with mean μ and variance σ2.Consider the random variable X = eZ .(i) Using the definition of the probability density function (PDF), find the PDF f(x) for X.(ii) Using the PDF found in part (i) and the definition of expectation, find the moments
E(Xn), n ∈ N. In particular, find the mean E(X) and the varianceVar(X).(iii) Using the definition of a moment generating function (MGF), find the MGF MX(s) ofX, and state where it is defined. The result will have the form of an improper integralwhich cannot be evaluated in a closed form, but you are expected to (iv) Suppose that MX(s) is analytic at s = 0, i.e. equal to the sumof its Taylor series, andfind this function using the results of part (ii). Compare the answer with the result youobtained in part (iii).
Question 3. Two brothers, Nick and Dan, are enthusiastic gamblers with a taste for mathe-matics. One day, they walk into a casino where tickets for an unusual lottery are sold. Thelottery has the following rules. A fair coin is tossed, and in the event of heads, the lottery stopsand the holder of the ticket gets the prize of ￡1. Otherwise, the coin is tossed again, and in theevent of heads, the lottery stops and the holder gets ￡2. Otherwise the coin is tossed again,and in the event of heads, the holder gets ￡4, and so on, with thesum of the prize doublingafter each toss, and the lottery stopping on the first heads.(i) In order to decide upon the reasonable price PN worth paying for the ticket, Nick enthu-siastically calculates the expectation of the prize in this lottery. Assuming he does hiscalculations right, what result will he get?(ii) Nick is perplexed by the result of his calculations, and asks for Dan’s opinion. Dan
thinks it is wrong to estimate the ticket’s value by the mathematicalexpectation of theprize, particularly when very high prizes are possible. He says that,for instance, a manwith a capital of ￡1bn is not necessarily 1000 times happier than aman with a capital￡1m, as it is physically not possible to be that happy. So, argues Dan, the price of theticket should be obtained not from the mathematical expectation of the prize, but fromthe mathematical expectation of the ‘happiness’, which is some monotonically increasingfunction of the prize, h(w). Give an expression for the fair price PD of the ticket accordingto Dan’s criterion. In particular, calculate the price of the ticket, rounded to the nearest
(iii) Generalise these results for the case when the coin is not fair and the probability of headsis p ∈ (0, 1), the prize increases after each toss by a factor of q ∈ (1,∞), and h(w) = wα,α ∈ (0, 1). Determine the condition on p, q and α that the fair price Pg of the
November 23, 2022Question 4. A phone customer service system supports a queue which may hold from 0 upto N customers waiting to be served. The service provided may take an integer number ofminutes, and the system checks the state of the queue every minute. The length of the queue,X(t), is an integer-valued function depending on the discrete time t. An analyst hired by thecompany develops a probabilistic model of the queue. The model is in the form of a Markovchain, with the following rules: every minute the length of the queue may increase by1 withprobability p ∈ [0, 1], or decrease by 1 with probability q ∈ [0, 1], or otherwise stays unchanged,with probability r = 1 ? p ? q ∈ [0, 1]. The exceptions are: when the queue is empty, X = 0,then its length cannot decrease, and when it is at the maximal length, X =N , then its lengthcannot increase.(i) Sketch a diagram of this chain for N = 3. Write down the corresponding transitionmatrix. Classify this chain: Is this chain irreducible, and if not, what are the subchains? Does this chain have any absorbing states and if yes,what are those? Are any states in the chain periodic, and if yes, what are their least periods?(ii) Still for N = 3, find the steady-state probability vector (P (j), j = 0, . . . N) for this chain(present your answer for each component of the vector as anirreducible rational functionof p and q).(iii) Generalise the result of (ii) for arbitrary N > 1.
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