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(V (14%;)? ACTSC 431/831  FINAL EXAMINATION
Spring 2004 — University of Waterloo Total Marks = 85 Aids: Non—programmable scientiﬁc calculator Instructor: Steve Drekic
Duration: 3 hours . Formula sheets are attached 1. Suppose the cumulative distribution function for individual losses X is given by 1+ 233/45 “Manama {i 3,
(1+ .CC/90)4 The number of losses has a binomial distribution with parameters m = 6 and q = 3/5.
Suppose that a loss limit of 160 and an ordinary deductible of 45 are applied to each
individual loss. M " 1&0 . 0‘» [1‘5 F(:r)=1— for $20.. [3] Xa) Prove that the probability density function of X is given by 8
f(:c)=—LOOO$ for :r>0. (90 + m5 w/ , a
» \
[ lw ’etermine the failure rate /\(:r). What does the failure rate tell you about this par /
‘ ticular distribution? [tit/(c) Determine the expected value of the aggregate payments on a @. .YL . £1,ch ”Z [3) (1) Determine the distribution of the number of (positive) payments. [4] ) Show that the cumulative distribution function of the per payment random variable I Yp is given by g ‘35
1—%—1 for ogy<115
FYP (y) = 135+y)
1 for y 2 115. ‘
[lg/6 the severity distributlon from part (9) usmg the method of rounding and a __
late (to 5 decimal places of accuracy) all values up to a discretized k a 35 ‘
. ' f...“ [5] ‘/(g) Calculate (to 5 decimal places of accuracy) the discretized distribution of aggregate
payments up to a discretized amount paid of 105. [3] J‘) Apply the piecewise uniform procedure to approximate the probability that aggregate
payments exceed 90. 2. Suppose that A is a random variable having the gamma probability density function 1 ...
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