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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801 PROBABILITY AND STATISTICS:
FOUNDATIONS OF ACTUARIAL SCIENCE December 30, 2005 Time: 9:30 a.m.  11:30 a.m. Candidates taking examinations that permit the use of calculators may use any calcu
lator 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’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 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 FIVE questions. Marks are shown in square brackets. 1. (a) A random variable X has the following distribution function: 0, a: < 0,
27/9, 0 S x < 1,
F($) =
$2/9, 1 S a: < 3,
1, 3 S 1:.
Find the p.d.f., f (as), and obtain E(X) and Var(X [6 marks] (b) A carhire service company knows that on any given day, the number of
persons, X, who want to hire a car from it has the following distribution: a: 0 1 2 3 4 5 6 or above
f(a:) 0.01 0.03 0.09 0.14 0.17 0.18 0.38 The daily allinclusive cost of keeping a car is $400, whether it is hired or
not. If the car is hired, the company collects a rental of $800 per day. The
company is only willing to keep not fewer than 4 but not more than 6 cars. S&AS: STAT1801 Probability and Statistics: Foundations of Actuarial Science 2 (1) Calculate the company’s expected daily proﬁt for each of the cases of
keeping 4, 5 and 6 cars. (ii) Hence determine the optimum number of cars that the company should
keep in order that the expected daily proﬁt is maximized. [8 marks] (c) A man and his wife both are of age c years. Their ages X and Y at death are
taken as independent continuous random variables. An actuarial statistician assumes the following p.d.f.’s for X and Y ore“(3%) a: 2 c, 0: > 0,
Husband: f X(:L‘) =
0 otherwise,
and
, 56W“) 2/ 2 6, ﬂ > 0,
me: My) =
0 otherwise,
respectively. (i) What is the probability that the wife will live over a period of s more
years? (ii) If the husband dies ﬁrst at the age of b years, what is the probability
that the wife will live over a period of at least t more years after the
death of the husband? (iii) Based on the results of and (ii), do you think the statistician’s as
sumption on the density forms is sensible or not? Give your reason(s). [8 marks]
[Total: 22 marks] 2. (a) Let X1 and X2 be independent random variables with the same p.d.f. 2x, 0 < x < 1,
f (15) =
0, otherwise.
Evaluate the conditional probability P(X1 < X2 [X1 < 2X2). [8 marks] (b) Let X1 and X2 be two independent Binomial random variables with dis
tributions Bin(n1,9) and Bin(n2,0), respectively. Find the joint p.d.f. of
Y1 = X1 + X2 and Y2 = X2. Hence ﬁnd the p.d.f. of Y1. Hint: Compare the coefﬁcients of 33’“ in each member of the identity (1+ 1:)”1 (1 + 33)”? s (1 + m)“1+”2. Also note (1 + z)" = 2a]. [10 marks] [Total: 18 marks] S&AS: STAT1801 Probability and Statistics: Foundations of Actuarial Science 3 3. Let X,Y,Z have joint p.d.f. f(m,y,z) = (a: +y+z)/24, 0 < m < 2,0 < y < 2,
0 < z < 2, zero elsewhere. (a) Show that the marginal p.d.f. of Z is f(2) = 1/3 + 2/6, 0 < z < 2, zero
elsewhere. (Notice that the marginal p.d.f.’s of X and Y are of similar form.) [4 marks]
(b) Compute Pr(0 < X <1, 0 < Y < 1,0 < Z < 1), and Pr(0 < Z < 1). [6 marks]
(c) Determine whether X, Y, and Z are independent. [3 marks] (d) Find the conditional distribution of X and Y, given Z = z, and evaluate
E(X + Y  Z :1). [7 marks] (6) Show that the conditional distribution of X, given Y = y and Z = z, is
f($y,z) = (m+y+z)/(2+2y+2z), 0 < :1: < 2,0 < y < 2,0 < z < 2, zero
elsewhere, and compute E(X [Y = 1, Z = 1). [5 marks] [Totah 25 marks] 4. The total lifetime (in years) of ﬁve—year~old dogs of a certain breed is a random
variable which has distribution function: 0 $<5 F (x) =
1 — 25/1;2 :1: > 5
(a) Find the probability that such a ﬁve—year—old dog will live (i) beyond 10 years.
(ii) between t years to t + 1 years, given that it has already lived t years. [5 marks]
(b) Evaluate the probabilities in (a)(ii) when t = 6, and when t = 8. Compare
these probabilities. [4 marks] (c) Find the expected lifetime of the dog given that it has already lived for 10
years. [4 marks] [Total: 13 marks] 5. A new drug for a certain disease is developed and is claimed to have fewer side
effects than a commonly used drug. A statistician conducts an experiment as
follows: A group of 30 patients with the disease participate in a study and are
given this new drug. After the study, six of them have side effects. Let p1 be the
proportion of patients who have side effects after taking the new drug. (a) Write down the distribution of the estimator, 151, of pl. [3 marks] (b) Construct a 95% conﬁdence interval for p1. Is this interval exact? Explain.
[4 marks] S&AS: STAT1801 Probability and Statistics: Foundations of Actuarial Science 4 (0) Without any calculation and using the result in (b), do you think that p1 is
signiﬁcantly different from 0.25? You may draw your conclusion based on
the 5% level of signiﬁcance. [3 marks] (d) A commonly used drug for this disease is compared with this new drug.
From past records, 20 out of 50 patients who took this drug had side effects. (i) Is there any evidence that the new drug has fewer side effects than this
commonly used drug? You may draw your conclusion based on the 5%
level of signiﬁcance. (ii) Construct a 95% conﬁdence interval for the difference of the proportions
for the new drug and the commonly used drug. (iii) If the statistician wants the width of the 95% conﬁdence interval for the
difference of the two proportions to be at most 0.3, what is the size of
the sample of patients that has to be selected to be given the new drug? [12 marks]
[Total: 22 marks] ...
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 Fall '10
 MrChung
 Actuarial Science, Statistics, Probability, Probability distribution, Probability theory

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