ST102 2014.pdf

ST102 2014.pdf - Summer 2014 examination ST102 Elementary...

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Summer 2014 examination ST102 Elementary Statistical Theory 2013/14 syllabus only – not for resit candidates Instructions to candidates Time allowed: 3 hours This paper contains seven questions. Answer BOTH questions from Section A, and THREE questions from Section B. All questions carry equal numbers of marks. If you answer more than three questions from Section B, only your BEST 3 answers will count towards the final mark. You are supplied with: Murdoch and Barnes Statistical Tables Formula sheet (at the end of the examination paper) You may also use: Scientific calculators are permitted in the examination, as prescribed by the School’s regulations. If you have a programmable calculator, you must delete anything stored in the memory in the presence of an invigilator at the start of the examination. c LSE 2014/ST102 Page 1 of 9
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SECTION A Answer BOTH questions from Section A. Both questions carry equal numbers of marks. 1. (a) The random variable X has a Poisson distribution with P ( X = 1) = P ( X = 2). i. Determine the mean and variance of X . ii. Find P ( X = 4). . (4 marks) (b) Suppose that commercial aeroplane crashes in a certain country occur at the rate of 2.5 per year. i. Is it reasonable to assume that such crashes are Poisson events? Briefly explain. ii. What is the probability that four or more crashes will occur next year? iii. What is the probability that the next two crashes will occur within three months of one another? . (8 marks) (c) Let X be a continuous random variable with the following probability density function: f ( x ) = x e - x : x 0 0 : otherwise . i. Show that its moment generating function is M X ( t ) = 1 / (1 - t ) 2 . ii. Use M X ( t ) to find the expected value of X . . (8 marks) [TURN TO NEXT PAGE] c LSE 2014/ST102 Page 2 of 9
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2. (a) A random sample { X 1 , X 2 , . . . , X n } is drawn from the following distribution. p ( x ; λ ) = λ 2 x e - λ 2 x ! . i. Find the maximum likelihood estimator for λ . ii. State the maximum likelihood estimator for = λ 3 .
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