” 辅导STAT0005程序、 写作Java，c++，Python程序Examination Paper for STAT0005 Page 1STAT0005: Probability and Inference2019/20, Level 5Answer ALL questions.You may submit only one answer to each question.The relative weights attached to each section are Section A (39 marks),Section B (59 marks), Section C (38 marks).The numbers in square brackets indicate the relative weights attachedto each part question.Marks are awarded not only for the final result but also for the clarityof your answer.Asking questions during the coursework periodYou may not email the course organizer directly during the period setfor this coursework.If you need to clarify any part of the coursework, you may post tothe course Moodle forum within the first two working days of thecourseworks release. No clarification questions will be answered afterthis.You may not ask questions other than clarifications at any pointduring the period set for the coursework.Turn OverExamination Paper for STAT0005 Page 2Formatting your solutions for submissionSome part-questions require you to type your answers instead of handwritingthem. These questions state [Type] at the start of the partquestion.You must follow this instruction. Failure to do so mayresult in marks being deducted. For questions without the [Type]instruction, you may choose to type or hand-write your answer.Some part-questions ask you for an explanation using only words andno formulae. If you use formulae anyway, these may be entirely ignoredin the marking process. 辅导STAT0005作业、 写作Java，c++，Python程序设计作业、 辅导data语言作业Where a word limit has been set for a part question, this has beenchosen to be at least three times the length of the expected answer.Hence you should view the word limit as a strict upper limit ratherthan as the number of words to achieve.You should submit ONE document that contains your solutions for allquestions/ part-questions. Please follow UCLs guidance on combiningtext and photographed/ scanned work.Make sure that your handwritten solutions are clear and are readablein the document you submit. You are encouraged to write out solutionsneatly once you are happy with them.Plagiarism and collusionYou must work alone. In particular, any discussion of the courseworkwith anyone else is not acceptable. You are encouraged to read theDepartment of Statistical Sciences advice on collusion and plagiarism,which you can find here.Parts of your submission will be screened via Turnitin to check forplagiarism and collusion.If there is any doubt as to whether the solutions you submit are entirelyyour own work you may be required to participate in an investigatoryviva to establish authorship.ContinuedExamination Paper for STAT0005 Page 3Section AA1 Let X U(1, 1) and let Y = X4.(a) Compute E[Y ]. [3](b) Compute Var(Y ). [3](c) Compute the pdf fY of Y . [5]A2 Let the joint distribution of X and Y be given by the following two-waytable:XY -1 0 1-1 b 0 a0 0 1-2a-2b 01 a 0 bHere, a, b are unknown constants such that the above table is a validtwo-way table. For parts (a)-(d) you may leave your results in termsof a and b where necessary.(a) Compute the marginal pmfs pX and pY . [3](b) Compute E[X] and E[Y ]. [3](c) Compute Var(X) and Var(Y ). [3](d) Compute Corr(X, Y ) in the case (a, b) 6= (0, 0). [5](e) What constraints must the pair (a, b) satisfy to ensure that thetable above is valid? [2](f) What is the smallest value Corr(X, Y ) can take in this case? Givea value of (a, b) for which the smallest possible value of Corr(X, Y )is attained. [2]A3 Let the joint cdf of X and Y be given byFX,Y (x, y) =0 if x 0 or y 0min{x, y} if x, y 0 and (x 1 or y 1)1 if x, y 1.(a) Compute P(0 X 1, 0 Y 1). [2](b) Compute the marginal cdf FY of Y . [3](c) Compute P(X 1/2 | Y 1/2).[Type] Using only words and no formulae, decide whether X andY are independent and justify your decision. Maximum length:150 words. [5]Turn OverExamination Paper for STAT0005 Page 4Section BB1 For i {1, . . . , n} with n N and n 2, let Xii.i.d. N(, 2) where and 2 0 are both unknown. To estimate the variance 2, considerthe estimator. In this expression, the number (, 2) isused to obtain different estimators.(a) Name the estimator in the case = 1. What is the expectedvalue of T1? (You do not need to compute the expected value ifyou know it.) [2](b) Compute the bias of T. [2](c) Compute the sampling variance of T. Hint: Start from thevariance of T1. [5](d) Show that the mse for T is given bymse(T; 2) = 4(n )22 2 1 + 2n.[4](e) Given the sample size n, which value (n) of results in thesmallest mean square error of T? You need to provide a derivationand justification of your result and while you may omit checkingthe second order condition you should check the boundaries. [8](f) Hence provide a formula for an estimator of the form T withsmallest mse.[Type] Using only words and no formulae, give one reason why T1is often used in practice in spite of your result. Maximum length:300 words. [5]ContinuedExamination Paper for STAT0005 Page 5B2 For n N, consider the regression model of the formY = x + , N(0, ),where the covariate x Rn with x 6= 0 and the positive definite symmetricmatrix Rnn are fixed and known whereas the parameter R is unknown. The sample consists of a single observation y fromthis model.(a) [Type] Using only words and no formulae, explain how a maximumlikelihood estimator is obtained in general. Maximum length:150 words. [4](b) In the setting described above, obtain the log-likelihood for theparameter given the one observation y Rn. [4](c) Show that the MLE of is given bybMLE =xT1yxT1x.You need to check all applicable conditions for a maximum.Hint: Note that is a number (not a matrix or a vector) andnote the dimension of xT1y as well as that of xT1x. [8](d) Compute E[bMLE]. [5](e) Compute the Cramer-Rao lower bound for unbiased estimators of. What is the interpretation of this bound? [4](f) Compute the sampling variance of bMLE. Hence decide whetheror not bMLE achieves the Cramer-Rao lower bound.Hint: First show that bMLE is of the form bMLE = c + aT forsome constant c R and some constant vector a Rn which youshould specify.[8]Turn OverExamination Paper for STAT0005 Page 6Section CLet X N(, ) where Rn and Rnnfor dimension n 2.Also assume that is positive definite symmetric. Let A Rkn withk {1, . . . , n}. You may use the facts that, firstly, AATis positive definitesymmetric if A has full rank and that, secondly, A has full rank if and onlyif its row vectors are linearly independent.(a) Compute the mgf of Y = AX and thus show that Y follows a normaldistribution and specify its mean vector and covariance matrix. [4](b) Under the condition that A has full rank, write down the pdf of Y .Explain why it is impossible to write down the pdf of Y if A does nothave full rank. [4](c) Using the mgf or otherwise, prove that Cov(Xi, Xj ) = 0 if and only ifXi and Xj are independent. Note that during the course we have onlyestablished this in the case of bivariate normal distributions. [6](d) Suppose that Cov(X1, Xj ) = 0 holds for all j {2, . . . , d}. Show thatX1 is independent of X2 + X3 + . . . + Xd. [5](e) Consider three discrete random variables U, V, W each taking values in{1, 1} and such that U is independent of V and U is independent ofW. For each of the statements (i) and (ii) below, decide whether itis true in general or whether it may be false. If the statement is truein general, provide a proof. Otherwise, find a joint pmf for U, V andW that provides a counterexample (i.e. a joint distribution for whichthe statement does not hold), explaining your reasoning and presentingyour joint pmf in a table as follows:u v w pUV W (u, v, w)(i) U and V + W are uncorrelated. [4](ii) U and V + W are independent. [5](f) [Type] Using only words and no formulae, write a short essay on threeways in which the Gaussian distribution is special among probabilitydistributions. You need to make clear how the ways are directly relatedto this Section C and/or to results in the lecture notes. Maximumlength: 300 words. [10]End of Paper如有需要，请加QQ：99515681 或邮箱：99515681@qq.com
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