23. Kevin and Kira are in a history competition:
(i) In each round, every child still in the contest faces one question. A child is out as
soon as he or she misses one question. The contest will last at least 5 rounds.
(ii) For each question, Kevin’s probability and Kira’s probability of answering that
question correctly are each 0.8; their answers are independent.
Calculate the conditional probability that both Kevin and Kira are out by the start of round
five, given that at least one of them participates in round 3.
(A) 0.13
(B) 0.16
(C) 0.19
(D) 0.22
(E) 0.25
Exam M: Fall 2005 -24- GO ON TO NEXT PAGE
24. For a special increasing whole life annuity-due on (40), you are given:
(i) Y is the present-value random variable.
(ii) Payments are made once every 30 years, beginning immediately.
(iii) The payment in year 1 is 10, and payments increase by 10 every 30 years.
(iv) Mortality follows DeMoivre’s law, with 110 ω= .
(v) 0.04 i =
Calculate ( ) Var Y .
(A) 10.5
(B) 11.0
(C) 11.5
(D) 12.0
(E) 12.5
Exam M: Fall 2005 -25- GO ON TO NEXT PAGE
25. For a special 3-year term insurance on ( ) x , you are given:
(i) Z is the present-value random variable for this insurance.
(ii) q k x k + = + 002 1 . ( ), k = 0, 1, 2
(iii) The following benefits are payable at the end of the year of death:
k bk+1
0 300
1 350
2 400
(iv) i = 006 .
Calculate Var Z b g .
(A) 9,600
(B) 10,000
(C) 10,400
(D) 10,800
(E) 11,200
Exam M: Fall 2005 -26- GO ON TO NEXT PAGE
26. For an insurance:
(i) Losses have density function
( ) 0.02 0 10
0 elsewhere X
x x
f x
< < ⎧
= ⎨⎩
(ii) The insurance has an ordinary deductible of 4 per loss.
(iii) P Y is the claim payment per payment random variable.
Calculate E P Y ⎡ ⎤ ⎣ ⎦ .
(A) 2.9
(B) 3.0
(C) 3.2
(D) 3.3
(E) 3.4
Exam M: Fall 2005 -27- GO ON TO NEXT PAGE
27. An actuary has created a compound claims frequency model with the following properties:
(i) The primary distribution is the negative binomial with probability generating function
( ) ( ) 2 1 3 1 P z z − = − − ⎡ ⎤ ⎣ ⎦ .
(ii) The secondary distribution is the Poisson with probability generating function
( ) ( ) 1 z P z eλ − = .
(iii) The probability of no claims equals 0.067.
Calculate λ.
(A) 0.1
(B) 0.4
(C) 1.6
(D) 2.7
(E) 3.1
Exam M: Fall 2005 -28- GO ON TO NEXT PAGE
28. In 2005 a risk has a two-parameter Pareto distribution with 2 α= and 3000 θ= . In 2006
losses inflate by 20%.
An insurance on the risk has a deductible of 600 in each year. i P , the premium in year i,
equals 1.2 times the expected claims.
The risk is reinsured with a deductible that stays the same in each year. i R , the reinsurance
premium in year i, equals 1.1 times the expected reinsured claims.
2005
2005 0.55 R
P =
Calculate 2006
2006
R
P .
(A) 0.46
(B) 0.52
(C) 0.55
(D) 0.58
(E) 0.66
Exam M: Fall 2005 -29- GO ON TO NEXT PAGE
29. For a fully discrete whole life insurance of 1000 on (60), you are given:
(i) The expenses, payable at the beginning of the year, are:
Expense Type First Year Renewal Years
% of Premium 20% 6%
Per Policy 8 2
(ii) The level expense-loaded premium is 41.20.
(iii) i = 0.05
Calculate the value of the expense augmented loss variable, 0 e L , if the insured dies in the
third policy year.
(A) 770
(B) 790
(C) 810
(D) 830
(E) 850
Exam M: Fall 2005 -30- GO ON TO NEXT PAGE

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