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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.
 Fall '08
 SMSLee
 Statistics, Probability

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