FinalexamACTL2003s22008 - THE UNIVERSITY OF NEW SOUTH WALES...

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THE UNIVERSITY OF NEW SOUTH WALES MONTH OF EXAMINATION - NOVEMBER 2008 Final Examination ACTL2003 STOCHASTIC MODELS FOR ACTUARIAL APPLICATIONS INSTRUCTIONS: 1. TIME ALLOWED - 3 HOURS. 2. TOTAL NUMBER OF QUESTIONS - 9. 3. TOTAL MARKS - 100. 4. THERE ARE 2 SECTIONS. EACH SECTION SHOULD BE ANSWERED IN A SEPARATE EXAMINATION BOOK. 5. SECTION I HAS FIVE (5) QUESTIONS. USE A SEPARATE EXAMINATION BOOK AND INDICATE THE SECTION NUMBER ON THE FRONT PAGE. AN- SWER EACH QUESTION STARTING ON A NEW PAGE. 6. SECTION II HAS FOUR (4) QUESTIONS. USE A SEPARATE EXAMINA- TION BOOK AND INDICATE THE SECTION NUMBER ON THE FRONT PAGE. ANSWER EACH QUESTION STARTING ON A 7. QUESTIONS ARE NOT OF EQUAL VALUE. 8. CANDIDATES MAY BRING THE "FORMULAE AND TABLES FOR ACTUARIAL EXAMINATIONS" BOOK (ANY EDITION) INTO THE EXAMINATION. IT MUST BE WHOLLY UNANNOTATED. 9. CANDIDATES MAY BRING THEIR OWN CALCULATORS PROVIDED THEY DO NOT HAVE A "QWERTY" KEYBOARD. ALL ANSWERS MUST BE WRITTEN IN INK. EXCEPT WHERE THEY ARE EXPRESSLY REQUIRED, PENCILS MAY BE USED ONLY FOR DRAW- ING, SKETCHING OR GRAPHICAL WORK. Answer each question starting on a new page 1
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SECTION I [50 MARKS] START A NEW EXAMINATION BOOK. ANSWER ALL QUESTIONS. START EACH QUESTION ON A NEW PAGE. Question 1 (10 marks) a) Consider a Markov chain { X n ,n =0 , 1 , ···} with state space { 1 , 2 , ··· , 5 } .S u p p o s e its probability transition matrix is given as below: 1 5 000 4 5 1 3 1 3 0 1 3 0 00 1 2 0 1 2 1 4 1 2 0 1 4 0 1 2 0 1 2 i) Classify the states, and determine whether they are recurrent or transient [1 marks] . ii) Find the expected number of visits in each transient state starting in any transient state. [2 marks] iii) Given that X 0 is equally likely to be in any state, calculate the probability that two steps later, the process is in state 3 . [2 marks] b) Consider a matrix μ x 1 y 2 1 xy i) Find all the possible values of x and y such that the above matrix is a probability transition matrix? [2 marks] ii) Show in detail when the above matrix will be a probability transition matrix of a Markov chain whose limiting probabilities exist, and when it will not. [3 marks] 2
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Question 2 (14 marks) A no claims discount system for car insurance has 3 discount levels: 0% , 20% and 30% .
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This note was uploaded on 06/12/2011 for the course ASB 2003 taught by Professor Kim during the Three '11 term at University of New South Wales.

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FinalexamACTL2003s22008 - THE UNIVERSITY OF NEW SOUTH WALES...

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