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Unformatted text preview: UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
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 _> {MFmtcAL 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 ‘
Ansn: PMﬂoﬂmoH A.» .} _ [than ‘63)
(A)0.377 X 83:!) 7V0 >q° Q"‘°p° )
(B) 0.400 g} 25“ . zr .01.?
(C) 0.417 m“ zrvr 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 VIsr' I as ' ° 7 r
‘ ' ' °° ’7. 424’ (2) From a population having distribution function F, you are given the following sample:
e‘nﬂmmc ﬂeas, 3F (Acu 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° 1.5535! 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 Firn ‘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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 Spring '08
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