1301_1011sem1_ans - .___.1 THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: .___.1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STATI301 PROBABILITY AND STATISTICS I December 14, 2010 Time : 2:30pm — 4:30pm. Only approved calculators as announced by the Examinations Secretary can he used in this examination. It is candidates' responsibility to ensure that their calculator operates satisfactorily, and candidates must record the name and type of the calculator used on the front page of the examination script. Answer ALL FOUR Questions. Marks are shown in square brackets. 1. An amplifier circuit must be designed to achieve a gain of 100. The “minimum” circuit, which will achieve this gain, is shown in figure (i) below. It consists of two small amplifiers; each small amplifier magnifies the volume of sound by a factor of 10. Two alternative circuits are shown in figures (ii) and (iii) that will achieve the desired gain. Each circuit can function as long as signals can pass from A to B. Suppose each individual small amplifier has a reliability (chance of functioning properly) of 0.9. Assume that they are operating independently. {i} A Amplifier 1 Amplifier 2 B Amplifier 1 Amplifier 2 Amplifier 3 Amplifier 4 (iii A (iii) p I Amplifierl I Am lifier3 (a) Compute the reliability (chance of functioning properly) of each of the three circuits. Which circuit is the mo st reliable? (TEE l , DAM} Ci, 0. Cl 3’0, [Smarks] (b) Let X be the number of amplifiers functioning properly in circuit (iii). Write down the distribution, expected value and variance of X. lo {4,6 I, (336 [6marks] S&AS: STAT1301 Probability and Statistics I 2 (c) If only two of the amplifiers in circuit (iii) are functioning, What is the probability that this circuit can still function properly? a / 3 [6 marks] [Total: 20 marks] 2. A point (X , Y ) is chosen uniformly at random from within a unit circle. The joint probability density function (pdf) of (X , Y ) is given by I _ — if x2 + y2 S l f(x3y)= 7r 0 otherwise (a) Are X and Y independent? Why? O/(C/ [2 marks] (13) Find the conditional pdf of Y given X = x . Which family of distributions does it belong to? U ('JWZ, W1 ) [5 marks] (c) Find C0v(X,Y). 0 [5 marks] Y . (d) Let Z = End the pdf of Z. l , . 1D; (gt): fir],1L-§'L) ,. -'7°<?"9“ [71113er (e) Let R = 4X2 + Y 2 . Find the hazard rate function of R. ‘ 9. r _ [6 marks] Fl E { Y“) = f GS V“ ] S&AS: STAT1301 Probability and Statistics I 3 (f) To choose a point uniformly at random from within the unit circle, we can first randomly draw a point from the square bounded by the dashed line and accept it only if it falls within the circle, i.e. R £1. For 100 points uniformly and independently drawn at random from the square, what is the probability that more than one—third of the points are accepted? [5 marks] 1 [Total: 30 marks] 3. (a) Suppose a random variable Y given p follows a binomial distribution, Y l p w Binomial (n, p). If we specify a beta distribution for p, p ~ Bera(a, fl), derive the conditional distribution of p given Y = y. {.dfl 45% my) [10 marks] (13) Suppose a random variable X given p also follows a binomial distribution, X I p w Binomial (N a rap), and is conditionally independent of Y given the value of p. Derive the probability mass fiinction of X given Y z y , which is known as the predictive distribution. “ i 15 marks] g7. - {A Ax-” r i P -' ,r i 6 E ( 1 364(5!’ H J > [Total: 25 marks] 4. (a) Suppose X1,...,X” are i.i.d. Bernoulli random variables with a success probability p. Derive the distribution of S” 2 X i. and find the mean and variance of S“. For what value of p does the variance of Sn achieve its maximum? fit, [7) I K F (F WI [5 marks] (b) As 11 —> co and p —> 0, while np —> A a constant, what distribution can S" be approximated as? Show the derivations. MCCQA/Ufi (. a > [10 marks] (0) For p not close to 0 or 1, as n —> 00, use the Central Limit Theorem to derive the limiting distribution of 1/1"” 2 (Sn — Hp). i [10 marks] N r .- ' l “5' l m T i 3 [Total: 25 marks] *********=’=** END OF PAPER ...
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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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1301_1011sem1_ans - .___.1 THE UNIVERSITY OF HONG KONG...

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