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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 Spring, 2011 Old Actuarial Exam Problems Kernel Estimators (1) You are given the following ages at time of death for 10 individuals: 4 _> {MFmtcA-L 90MB. 0? 25 30 55 35 37 39 45 47 49 55 - you: 931.: 7’0. Using a uniform kernel with bandwidth b = 10, determine the kernel density estimate of the probability of survR/al to age 40. '0 8 ‘ Ans-n: PMfloflmoH A.» .} _ [than ‘63) (A)0.377 X 83:!) 7V0 >q° -Q"‘°p° ) (B) 0.400 g} 25--“ . zr .01.? (C) 0.417 m“ zr-vr nu" .efl" (D) 0439 3? 2.9—4? .3! .031" @0485 39 2.4 “(a .qr .o'u" ‘16!“ 3r—J'f . or (Sample Questions, Exam C, Problem # 268) 4 394') , n- MA“ ” 3" “"7 . 9r «:3:- rr VI-sr' I as ' ° 7 r ‘ ' ' °° ’7. 424’ (2) From a population having distribution function F, you are given the following sample: e‘nflmmc fleas, 3F (Ac-u 2.0, 3.3, 3.3, 4.0, 4.0, 4.7, 4.7, 4.7 _ ¢ — /: Calculate the kernel density estimate of F (4), using the uniform kernel with bandwidth 1.4 ' “”6‘ Hormonal r“ 3' it 7": 7.4:. X t 5 .o o B: _ (A) 0.31 ———-— M ___f___‘!‘:—9———- 5’ 14:1. (B) 0.41 2.0 94"“ Lee .12.: l .ur (C) 0.50 3,3 [.9 - ‘0) 6.95“ ‘3‘; .m’lf 7- .137! .53 4° 1.5-535! are .062! Z .214“ )0.63 4“, 3.3 - 6.1 0.2.r .oanr 3 .oqaqr ' (Sample Questions, Exam C, Problem # 1 4 7) fl (3) You study five lives to estimate the time from the onset of a disease to death. The times to death are: ‘ 2 3 3 3 7 }) ‘9 PMB.“ EACH. J' Using a triangular kernel with bandwidth 2, estimate the density function at 2.5. b = 2. """' (A) 8/40 2/40 2‘5: 2 = (0.53 4.5-) ) 14/40 (D) 16/40 (B) 17/40 =7 Fir-n ‘7’ or f 08:. mm. cwmisure (Sample Questions, Exam C, Problem # 3) To P093- @ 1.5‘. I HG‘IGH‘?’ = lzJ Junce’ 1 .1 :- 35 (4)0451“) ...
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