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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II May 23, 2008 Time: 6:30 p.m. — 8:30 p.m. Candidates taking ezaminations that permit the use of calculators may use any cal—
culator which fulﬁls the following criteria: (a) it should be selfcontained, silent,
batteryoperated and pocketsized and (b) it should have numeraldisplay facilities
only and should be used only for the purposes of calculation. It is the candidate’s 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 ofpermitted/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 FIVE questions. Marks are shown in square brackets. 1. Let X1,X2, . . . ,X71 be independent and identically distributed random vari—
ables. The cumulative distribution and density functions of X1 are, respec tively,
6‘1e*(x“9)/9, :c > 6, 0, 5656, 1— e’lz‘ew, CE > 6,
O, a: S 6, F(:t[6) ={ and f(a:[6) :{ for an unknown parameter 6 > O. Denote by X (1) and X the sample minimum and sample mean respectively. (a) Show that the statistic T = (Xu) , X) is sufﬁcient for 6. [6 marks] (b) By noting that X Z Xm always, show that the maximum likelihood
estimator of 6 is X“). [4 marks] (0) Show that the density function of Xm is n6'1e“”(3‘9)/9. :1: > 6
:1: 6 = ‘ 7
g( l J 0, x39. [4 marks]
(d) Show that as an estimator of 6, X“) has a bias equal to 6/n. [4 marks] (e) Using (d) or otherwise, ﬁnd a constant factor c such that cXm is an
unbiased estimator of 6. [4 marks] S&AS: STAT1302 Probability and Statistics II 2 2 Let X], . . . ,Xn, Y1, . . , ,Yn be independent random variables such that
X1~N(a,02) and K~N(0,a2), 2:1,. ,n, for some unknown a 6 (~00, 00) and a > 0. It is known that the N(a,, b2) distribution has the density function f(z[a,b) = \/217r_b2exp{—(E?_b2a—)}, :r E (—oo,oo). (a) Show that the loglikelihood function of (a, a) is 1 n
S(,LL,0) = —2nlna— ﬁgﬁXl—MQ—tyf}. [4 marks]
(b) Find the maXimum likelihood estimator ([i, d) of (an). [4 marks]
(C) Show that the Fisher information matrix of (a) a) is
0‘2 0
z ,0 2 n ' .
(M ) [ 0 404 ]
[6 marks] (d) Quoting the largesample properties of the maXirnuIn likelihood estimator
(i1, (3), find the approximate distributions of [L and (7 separately.
[6 marks] (e) Suggest a method to calculate the standard errors of the maximum like—
lihood estimates [i and [7 [2 marks] 3 Let X1, . ,Xn be independent and identically distributed random variables.
The density function of X1 is O, :5 g (0, 1),
for some unknown parameter 6 > 0 (a) Show that the above statistical model for X1, . , ,Xn possesses the mono—
tone likelihood ratio property [6 marks] (b) Let 60 be a ﬁxed positive constant Consider the hypotheses
HQIQSQO and H116>90. Show that if X1 is observed to be 2:1, for z z 1, i . . ,n, then the uniformly
most powerful size a test of HO against H1 18 to reject H0 whenever 1— Fn(:c1:z:2    Inldo) < a, where Fn(9) is the cumulative distribution function of I]; Xz
[4 marks] S&AS: 100 095 025 020 015 ’ 010 005 0.00 STAT1302 Probability and Statistics II 3 (c) Show that X1 has the cumulative distribution function F1($6’) : 339(1+ 6 — 6:5), for a: E (0,1). [2 marks] ((1) It is known that F1(0.96) = (0.9)6(1 + 0.16) is a decreasing function in
6 > O. The ﬁgure below plots it against 9. Plots of 0.99(1+0.1e) Suppose that n = 1 and X1 = 0.9 is observed. Deduce from (b), (c) and
the above ﬁgure (i) a 95% lower conﬁdence bound for 6,
(ii) a 95% upper conﬁdence bound for 6,
(iii) a 95% equal—tailed conﬁdence interval for 6.
[8 marks] 4. A FIFA World Cup tournament is an international football competition con—
tested by national teams of the world. A total of 18 such tournaments had
been held since 1930. The following contingency table shows how the win
ners, the runners—up and the thirdplaces were distributed between countries in Europe and outside Europe. S&AS: STAT1302 Probability and Statistics II 4 Countries \ Title Winner Runner—up Third—place European
non—European Total 18 1 8 18 We wish to test the null hypothesis H0 that the three titles (Winner, runner—up
and thirdplace) all have the same probability to go to Europe, against the
general hypothesis H1 that the three probabilities have no special relationships. (a) Set up an appropriate statistical model under the general hypothesis H1. [4 marks]
(b) Calculate the estimated counts in the contingency table under H0.
[5 marks]
(c) Based on the estimated counts obtained in (b) conduct a 5% Pearson
chisquared test of HO against H1. [7 marks]
f 1 2 3 4 5 6 upperquantile 3.8415 5.9915 7.8147 9.4877 11.0705 12.5916 5 The data below compare the personhours lost in a month due to accidents at
each of eight different industrial plants before and after a safety program was
established. For 2' = 1, 2, . . . ,8, deﬁne D22B1_Aza where B1 and A. denote the numbers of personhours lost? before and after
the program was instituted, respectively. It is found that D 2 2le D1/8 =
59.1/8 : 7.3875. Plant Before program After program Difference
i B. AZ D2 (D1 — DP
1 51.2 45.8 5.4 3.9502
2 46.5 41.3 5.2 4.7852
3 24.1 15.8 8.3 0.8327
4 10.2 11.1 0.9 68.6827
5 65.3 58.5 6.8 0.3452
6 92.1 70.3 21.8 207.7202
7 30.3 31.6 —1.3 75.4727
8 49.2 35.4 13.8 41.1202 Total 59.1 402.9088 8&AS: STAT1302 Probability and Statistics II 5 Assume that (D1, . . . ,DS) constitutes a normal random sample with unknown mean u and unknown variance 02. We wish to test the null hypothesis H0: 11 = 0 against the alternative hypothesis H1: u > 0 using a one—sided t test.
(a) Calculate an unbiased estimate of 02. [4 marks] (b) Calculate the t test statistic. [6 marks] (c) Determine whether you would reject H0 at the 5% signiﬁcance level.
[4 marks] I 5% and 2.5% upper quantzles of tf 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 (d) It is suggested that a two—sample t test can be used to test HO: #3 2 MA
against H1: 113 > uA, taking (B1, . . . , Ba) and (A1, . . . , A8) as two normal
random samples drawn from N(/,LB,0§) and N(MA,01?) respectively, for
unknown MB, MA and of. Do you agree with this suggestion? Explain. [6 marks] ...
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 Spring '10
 SMSLee
 Statistics

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