2006_quiz_1 - 1 t 80 x 1 / 2 , x 80 , t 80 x. a. [ 8 marks...

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The University of New South Wales 24 august 2006 ACTl 1001 actuarial studies and commerce quiz one ALL ANSWERS MUST BE WRITTEN IN INK. EXCEPT WHERE THEY ARE EXPRESSLY REQUIRED, PENCILS MAY BE USED ONLY FOR DRAWING, SKETCHING OR GRAPHICAL WORK. 1. Time allowed—50 minutes 2. total number of questions—4 3. Marks total 100 4. Answer all questions 5. aLL QUESTIONS ARE NOT OF EQUAL VALUE 6. THE PAPER MAY BE RETAINED BY THE CANDIDATE 7. CANDIDATES MAY BRING THEIR OWN CALCULATORS OR HAND HELD COMPUTERS 8. CANDIDATES MAY BRING THE text “formulæ and tables for actuarial examinations” into the examination. it must be wholly unannotated .
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Question 1 [20 marks] Given a force of interest of 8% p.a., calculate the following: a. [ 8marks ] i (4) b. [ 6marks ] the equivalent annual rate of interest, payable monthly c. [ 1mark ] the eFective annual rate of interest. d. [ 5marks ] ±ind lim n →∞ a n . Question 2 [20 marks] The inhabitants of a country where the population is stationary experience mortality such that for all ages t p x = ±
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Unformatted text preview: 1 t 80 x 1 / 2 , x 80 , t 80 x. a. [ 8 marks ] Calculate 10 | 25 q 30 . b. [ 12 marks ] Obtain an expression for the corresponding force of mor-tality. Question 3 [20 marks] or the Poisson distributed random variable X (where Pr( X = x ) = e x /x !, for x = 0 , 1 , 2 , . . . ), derive a. [ 7 marks ] its mean, and b. [ 13 marks ] its variance. 1 Question 4 [40 marks] Let X n be the number of positive integers with n digits in which no two adjacent digits are the same. Some authors call this property redness . a. [ 8 marks ] Give an expression for X n . We are now going to derive an expression for the number of even red numbers. Let Y n be the number of even red numbers with n digits. b. [ 8 marks ] Derive a recursive equation relating Y n 1 to Y n . (A diagram may be useful.) c. [ 24 marks ] Solve your recursive equation in b above to show Y n = 9 n + ( 1) n 2 . End of paper 2...
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2006_quiz_1 - 1 t 80 x 1 / 2 , x 80 , t 80 x. a. [ 8 marks...

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