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old5 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Actuarial...

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Unformatted text preview: UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Actuarial Science Program DEPARTMENT OF MATHEMATICS Math 478 / 568 Prof. Rick Gorvett Actuarial Modeling Old Actuarial Exam Problems ’ Review [or Exam # I (l) X is a discrete random variable with a probability function which is a member of the (a,b,0) class of distributions. You are given: Pk b _ A (i) P(X=O)=P(X=1)=O.25 ---' : Q + T I k ' l, 213) . .. (ii) P(X=2) = 0.1875 PM Calculate P(X= 3). . (Sample Questions Exam C, Problem # 94) b a + ‘ :°::/.;:rl :&=/Il+L a:b$0$'=> '5' t” ' /.u' 95' A ’ .03: m5 (2) For a discrete probability distribution, you are given the recursion relation 5 , x E i i 2 E l; 8 ll 3““? “l V 2 p<k)=—,;><p(k—1), k=1,2,... ”Pom“; .- ”i: 3 Determine 19(4). (Sample Questions, Exam C, Problem # 108) .1 => pm) : e (1)74! = 0.0902 (3) The graph of the density function for losses is: Lyn: ECXAul/ECX) ECxAzo); - .2000) ).+ Wu): [8’ 0.012 0.010 00008 fix) 0.006 0.004 0.002 0.000 0 so 120 => (6730.6!) Loss amount, x ‘5‘ Calculate the loss elimination ratio for an ordinary deductible of 20. (Sample Questions, Exam C, Problem # 87) l¥:1\mmcvzrmwnw‘x(am:1xzm\ . vhumxtx‘trmvmv "M9 with”: 90’9” ) Lama) (4) You are given: / 9 : 5(1) - 50‘ A A) (i) Losses follow an exponential distribution with the same mean in a 1 years. ' i (ii) The loss elimination ratio this year is 70%. E1104 5 (iii) The ordinary deductible for the coming year is 4/3 of the current deductible. 3 ' _ a: [9’ E Compute the loss elimination ratio for the coming year. 2 {Sample Questions, Exam C, Problem # 89) Lénmwmo = I-e‘ ’9 ,5 i: ~9‘Ln63) E E l 2 (5) The random variable for a loss, X, has the following characteristics: Calculate the mean excess loss for a deductible of 100. (Sample Questions, Exam C, Problem # 10]) Em: E‘XAA)+ Elk—AL] e : E£X~A K EUX-Ioolq-3 _ E“ 40w») ’3Zfi‘q, Haw) - = M1 2—3” (6) The number of accidents follows a Poisson distribution with mean 12. Each accident generates l, 2, or 3 claimants with probabilities 1/2, 1/3, and 1/6, respectively. Calculate the variance in the total number of claimants. (Sample QuestionsZExam ,2 Problem # [1]) WM (3) = E(~)'\)An- (X) + Vm(~)~ 6(1) Fen. Poem») -‘ VAR“) = éCN)‘ 50(2) 2 '2'[’z({)*lz(’3£)+3z(%ll; :40 fl. e (7) A towing company provides all towing services to members of the City Automobile Club. 60 C- 'PMT. ""' = = You are given: ; Towing Distance Towing Cost; Frequency ' I 3 _ is . q E 09.99 miles 80 '1 L 50% I: (7‘ ' '1‘)“ 6 q E 1029.99 miles 100 a o 40% L1.) = ‘ . VIM“- % 30+ mlles 160 M 10% (i) The automobile owner must pay 10% of the cost and the remainder is paid by the City “ Automobile Club. (ii) The number of towings has a Poisson distribution with mean of 1000 per year. e(~)->VAA(N) ='— I0” (iii) The number of towings and the costs of individual towings are all mutually independent. Using the normal approximation for the distribution of aggregate towing costs, calculate the probability that the City Automobile Club pays more than 90,000 in any given year. E( fl: 3'“ 1 o o (Sample Questions Exam C, Problem # 88) 3 Van“): 7.0057400 950;: .2?” 69 Wows wax-arr) -1 .1 (8) Insurance agent Hunt N. Quotum will receive no annual bonus if the ratio of incurred losses to earned premiums for his book of business is 60% or more for the year. If the ratio is less than 60%, Hunt’s bonus will be a percentage of his earned premium equal to 15% of the difference between his ratio and 60%. Hunt’s annual earned premium is 800,000. Incurred ; losses are distributed according to the Pareto distribution, with (9: 500,000 and a = 2 . Calculate the expected value of Hunt’s bonus. 1? BL _ sigma“ .' (Sample Questions, ExamC C, Problem # 96) axis-“11> == 977°“ -o.15’ I. “a“ 1 , E(3\ = '12.,ouo- o.l.r- E ILA‘II‘oloooJ 1: 02,0001!" ”(I SEE) )3 -— 1 (9) For a certain company, losses follow a Poisson frequency distribution with mean 2 per year, 3 5; 245;] and the amount of a loss is 1, 2, or 3, each with probability 1/3. Loss amounts are independent of the number of losses, and of each other. An insurance policy covers all losses in a year, subject to an annual aggregate deductible of 2. Calculate the expected claim —2 payments for this insurance policy. ’9. = e e: 3,. Sample uestions, Exam C, Problem # 99) f, = 51-531)”) 50)— -"l: 5A2 =2ta.) “1.01%” p,(-")+2.(t- -,a.'——-)J:- -‘/- :46?! (10) The number of claims 1n a period has a geometric distribution with mean 4. The amount of- ____ each claim X follows P(X=x)= 0.25, x= l ,2 ,3 ,.4 The number of claims and the claim amounts are independent. S 1s the aggregate claim amount in the period. Calculate FS (3) ""> 0 3 3- EC»): ”F -‘I 4"- (Sample Questions, Exam C, Problem # 95) _ u /1 JELzr): _ be Fk‘tdtéfiw: a" ‘ fig’é—hzmzn—a .ur(.(15'):$_)zz;§m£(3‘ 0: 3:5 1,1-‘5 . t o 1‘“ 3 l (11) A compound Poisson distribution has A 5 and claim amount distribution as follows. i E i i i Calculate the probability that aggregate claims will be exactly 600. (Sample Questions, Exam C, Problem # 289) -55. _ _P<’)_ ‘ 2 (.r)(.1s)' E";- 100 0.80 ' 6 -s ‘ 2. +(.2>.__..—e '5’ . ma 1000 0.04 41 55—F— ...
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