1801_0708sem1

1801_0708sem1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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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 self-contained, 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 ’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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This note was uploaded on 03/18/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

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1801_0708sem1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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