1301_1011sem1_ans

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

### Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online