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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I December 9, 2008 Time: 2:30 p.m. — 4:30 p.m. Candidates taking examinations that permit the use of calculators may use any
calculator which fulﬁls the following criteria: (a) it should be selfcontained, silent,
battery—operated and pocket—sized and (b) it should have numeral—display facilities only
and should be used only for the purposes of calculation. It is the candidate ’5 responsibility to ensure that the calculator operates satisfactorily
and the candidate must record the name and type of the calculator on the front page
of the examination scripts. Lists of permitted/prohibited calculators will not be made
available to candidates for reference, and the onus will be on the candidate to ensure
that the calculator used will not be in violation of the criteria listed above. Answer ALL FOUR questions. Marks are shown in square brackets. 1. Three candidates, A, B and C, participate in an election in which eight voters
will cast their votes. The candidate who receives the absolute majority, that is
at least ﬁve, of the votes will win the election. All voters must vote for one and
only one candidate and cannot abstain. (a) Suppose the voting is done by a secret ballot such that the voters’ identities
are kept conﬁdential. (i) Show that there are 45 distinct ways to distribute the eight votes among
the three candidates. [5 marks] (ii) Show that there are 10 distinct ways to distribute the eight votes among
the three candidates so that A will win the election. [6 marks] (iii) How many distinct ways are there to distribute the eight votes among
the three candidates so that nobody will win the election? [6 marks] (b) Suppose that the eight voters all vote independently such that each voter
votes for each candidate with probability 1/3. Show that the probability 1610
. k
2187 [8 mar S] >1: Hint: the mass function f of binomial (8,1/3) is tabulated below. Illlﬂl 0 1 2 3 4 5 6 7 8
f(£L') 256 1024 1792 1792 1120 448 112 16 1
561 65 1 6561 6561 6561 6561 6561 6561 6561 [Total: 25 marks] that nobody will win the election is S&:AS: STAT1301 Probability and Statistics I 2 2. On Doomsday every married couple in a community is brought to the entrances
of two tunnels, one leading to Heaven and the other to Hell. The tunnels look
identical so that nobody knows the exact destinations of the two tunnels. For
each couple, 0 if the husband and wife love each other, the wife will pick one tunnel
randomly (with equal probabilities to pick either tunnel) and pass through
it together with her husband; 0 if the husband and Wife have no feeling for each other, each of them will
pick a tunnel randomly (with equal probabilities to pick either tunnel) and
pass through it; their choices of tunnels are independent of each other so
that they may or may not pass through the same tunnel; o if the husband and wife hate each other, they Will be punished and deported
directly to Hell without having to pass through either tunnel. Assume that all married couples behave independently. For each couple, let oz be
the probability that the husband and wife love each other, H be the probability that they hate each other, and 1 — oz — H be the probability that they have no
feeling for each other. (a) Show that a wife has probability (1 + ﬂ)/ 2 to end up in Hell. [5 marks] (b) Show that for a wife in Hell, the conditional probability that her husband l—a—B [6 marks] is in Heaven is ———.
2(1 + ﬂ) (c) Show that for a wife in Hell, the conditional probability that she has not 1 2:53 I [6 marks] passed through a tunnel is (d) A survey indicates that among all married women in Hell, 50% have not
passed through a tunnel and 25% have their husbands in Heaven. Suppose
that the population of married women in Hell is so big that the above
percentages can be interpreted as probabilities. Show, using (b) and (c),
that (i) husbands and wives hate each other in onethird of the married couples
in the community; [4 marks] (ii) there is no love between husbands and wives in the community.
[4 marks] [Total: 25 marks] S&AS: STAT1301 Probability and Statistics I 3 3. (a) The density and distribution functions of the exponential distribution with
mean 6 > O are given respectively by 6—1 mas/9, > 0, 1 _ —z/9
f(xl6) = e ”3 and PW) = e ’ m > 0’
O, :c S 0, 0, x _<_ 0. Let X ,Y be independent random variables, both of which follow the
exponential distribution with mean 2. (i) Show that the minimum of X and Y, that is min(X,Y), follows the
exponential distribution with mean 1. [5 marks] (ii) Show that the conditional distribution of X given that X < 1 has the
density function ‘ e—zz/2 g(xX < 1) = 2(1 — 6—1/2)’
0, otherwise. 0<x<1, [5 marks] Donald and Daisy are two ducks born on the same day. They love each other
so much that if one dies, the other will drink a deadly poison immediately
to die too. Assume that the natural lifetimes of ducks are independent random variables
following the exponential distribution with mean 2 years. Use the results in
part (a) to answer the following questions. You may quote the memoryless
property of the exponential distribution without proof. (i) If Donald and Daisy are to live for a further period of T years before
they die together, ﬁnd the expected value of T. [5 marks] (ii) A biologist kidnaps Donald and hides him in a laboratory for an
experiment on bird flu. A year later Donald escapes from the
laboratory, returns home but cannot ﬁnd Daisy there. Believing that
Daisy must have been dead during the past year, Donald drinks the
poison and dies. If Donald’s belief is correct, then he must have lived
longer than Daisy, by some U years say. Show that the expected value
of U is approximately 0.5415. [5 marks] (iii) In fact, after Donald has been kidnapped, Daisy leaves home and looks
for Donald everywhere. Two years later when Daisy dies naturally, she
still believes that Donald is alive somewhere in the world. If Daisy’s
belief is correct, then Donald must have lived longer than Daisy, by
some V years say. Find the expected value of V. [5 marks] [Total: 25 marks] S&AS: STAT1301 Probability and Statistics I 4 4. (a) Let X ,Y be two continuous random variables having the joint density function
g(22+2y) xyE[01]
3 7 a a a
{I}, =
ﬂ 3/) { 0, otherwise.
(i) Show that E [X] = 5/9. [4 marks]
(ii) Show that E [Y] = 11/18. [4 marks]
(iii) Show that Cov(X, Y) = —1 / 162. [4 marks] (b) Let X , Y be two discrete random variables having the joint mass function %($+2y), 06,7; E {071}, 0, otherwise. ﬂay) ={ (i) Show that X has mean 2/3 and variance 2/9. [4 marks]
(ii) Show that Y has mean 5/6 and variance 5/36. [4 marks] (iii) Show that the correlation coefﬁcient between X and Y is ~1/\/16.
[5 marks] [Total: 25 marks] ...
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 Spring '08
 SMSLee

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