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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 10, 2007 Time: 9:30 a.m.  11:30 a.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,
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 ’3 responsibility to ensure that the calculator operates satisfacto—
rily 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 candi—
date to ensure that the calculator used will not be in violation of the criteria listed
above. Answer ALL questions. Marks are shown in square brackets. 1. Let X and Y have the joint probability density function (pdf) 6y, 0<y<x<1, flab?!) :{ O, elsewhere. (a) Show that the marginal pdf of X is f ( ) 39:2, 0 < :I: < 1,
a: =
1 0, elsewhere. (b) Obtain the conditional pdf of Y given X = 11:. Hence ﬁnd E(YIX = (c) Find the marginal pdf of Y, f2 (y), and obtain E (Y) and var(Y).
((1) Based on f2 (y) obtained in (c), derive the conditional pdf of X given Y = y. (8) Evaluate E(XIY = 0.5) and E(YIX = 0.5). Comment on Why the ﬁrst expectation is greater than the second one.
[Total: 20 marks] S&AS: STAT1801 Probability and Statistics: Foundations of Actuarial Science 2. A certain machine consists of 3 components, A, B and C. In 10,000 routine
examinations of this machine defects in A occurred 400 times, defects in B 300
times, defects in C 200 times, while in both A and B but not C occurred 12
times. Suppose that the defects of components are pairwise independent. Let P(A), P(B) and P(C) be the probabilities of defects in A, B and C,
respectively. (a) Write down the values for P(A), P(B) and P(C). (b) Compute the probability that the machine has defects in all three
components in a routine examination. Are the defects of three components
independent? ((3) Compute the probability that the machine has defects in only two
components in a routine examination, i.e. P(A H B n C") + P(A 0 B’ D C)+
P(A’ D B F] 0'). Note A’ is A complement. (d) Compute the probability that the machine has no defect in a routine
examination. (e) If it is known that the machine has defects in either component A or
component B, or both, what is the probability that it has also a defect
in component O? (f) Suppose that the replacement costs of components A, B and C are $300,
$500 and $1,000, respectively. What is the expected replacement cost of
components in a routine examination of the machine? [Total: 26 marks] S&AS: STAT1801 Probability and Statistics: Foundations of Actuarial Science 3. (a) Suppose that X and Y are continuous random variables with joint density 1, for OSxSI,OSy§1,
f($,y)= Determine the distribution of the sum Z = X + Y. 0, elsewhere. (b) Let X be a continuous random variable with density function 6
J21): _9 S m .<_ 0)
= 0 — 02$, 0<m$0,
0, elsewhere. where 0 > O. (i) Consider Y = ]X Obtain the probability density function (pdf) of Y. (ii) Find P(Y < 6/2).
[Total: 16 marks] 4. (a) The chemistry examination scores for random samples of Mathematics
students and Biology students are recorded as follows: Mathematics students 65 72 54 70 63
Biology students 72 68 65 76 68 Construct a 95% conﬁdence interval for n M — p3, where ,u M and MB
are the population mean chemistry scores for Mathematics students and Biology students respectively. (ii) If we want the width of the 95% conﬁdence interval to be less than 2.2,
what additional numbers of students are needed? State your assump tions. (b) Let X1,X2, . . . ,Xn be a random sample from a Bernoulli distribution with
f(x) = 675(1 — (9)1‘x, a: = 0, 1. Find the value of the constant k such that k[X1+X12+X2+X§++Xn+X§]
is an unbiased estimator of 6. (G) Let X1, X2, . .. ,Xn be a random sample from a uniform distribution over
the interval [1, 6]. Suggest a moment estimator for 0 and ﬁnd its variance. [Total: 22 marks] S&AS: STAT1801 Probability and Statistics: Foundations of Actuarial Science 5. The Environmental Protection Agency sets a limit of 5 parts per million on
PCB (a dangerous substance) in water. A major manufacturing ﬁrm producing
PCB for electrical insulation discharges small amounts from the plant. The
company management, attempting to control the PCB in its discharge, has
given instructions to halt production if the mean amount of PCB in the efﬂuent
exceeds 3 parts per million. A random sample of 36 water specimens produced
the following statistics: :2 = 3.2 parts per million and s = 0.6 part per million. (a) Do these statistics provide sufﬁcient evidence to halt the production pro
cess? Conduct a hypothesis test on the mean amount of PCB in the efﬂuent
with the level of signiﬁcance of 0.01. State clearly the null and alternative
hypotheses and your assumptions. (b) If you were the plant manager, would you want to use a large ‘or a small
value for the level of signiﬁcance, a for the test in part (a)? Justify your
choice. (0) Calculate [3, the type II error for the test described in part (a) supposing
that the true mean is p. = 3.3 parts per million. State your assumptions. (d) What is the power of the test to detect the efﬂuent’s departure from the
standard of 3.0 parts per million when the mean is 3.3 parts per million? (e) Repeat part ((1) supposing that the true mean is 3.4 parts per million. What
happens to the power of the test as the mean PCB of the manufacturing
ﬁrm departs further from the standard? [Total: 16 marks] ...
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 Fall '10
 MrChung
 Actuarial Science, Statistics, Probability, Probability distribution, Probability theory, probability density function, stat1801 probability

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