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Unformatted text preview: PRACTICE EXAMINATION NUMBER 18 1. Under an insurance policy, a maximum of ﬁve claims may be ﬁled per year by a policyholder.
Let p,, be the probability that a policyholder ﬁles n claims during a given year, where n = 0,1,2,3,4,5. An actuary makes the following observations:
(i) p" 2p"+1 forn = 0,1,2,3,4.
(ii) The difference between p,‘ and PM: is the same for n = 0,1,2,3,4. (iii) Exactly 40% of policyholders ﬁle fewer than two claims during a given year.
Calculate the probability that a random policyholder will ﬁle more than three claims during a given year. A. 0.14 B. 0.16 C. 0.27 D.0.29 E.0.33 2. You are given: (i) Losses follow an exponential distribution with the same mean in all years.
(ii) The loss elimination ratio this year is 70%. (iii) The ordinary deductible for the coming year is 4/3 of the current deductible.
Compute the loss elimination ratio for the coming year. A. 70% B. 75% C. 80% D. 85% E. 90% 3. Let X1, X2, and X3 be a random sample with replacement from a discrete distribution with
probability function
0.4 for x = 3,
Pr(X = x): 0.6 forx= 6,
0, otherwise. Let X = %(X] + X2 + X3) be the sample mean. What is the probability function for .3? ? 0.064 forx = —3,
0.4 forx = —3, 0.360 for x = 0,
A. Pr(X=x)= 0.6 forx=6, B. Pr(X=x)= 0.360 forx=3,
0, otherwise. 0.216 forx = 6, 0, otherwise. ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  606  PRACTICE EXAM 18 0.25 forx = —3, 0.064 forx = 3,
0.25 forx = 0, 0.096 forx = 0,
C. Pr(X =x)= 0.25 forx= 3, D. Pr(X: x) = 0.144 forx= 3,
0.25 forx=6, 0.216 forx=6,
0, otherwise. 0, otherwise.
0.064 for x = 3,
0.288 forx = 0,
E. Pr(X =x)= 0.432 forx = 3,
0.216 forx = 6,
0, otherwise. 4. Let X1 and X2 be independent random variables, each with density function
2—2x for0<x<l, fx(x)={0
1 What is the probability that exactly 1 of the 2 variables exceeds 2? otherwise. 5. Let X be a continuous random variable with density function e'z" +le" forO <x<oo,
)= 2 fx ()6 0 otherwise.
Let Y = e'zx . Then for 0 < y < l the density function for Y is given by f, (y) = 1 1 3(2y+x/§) C. 2y3+y D. E+m E. T A. 2””; B. 2+1];
2 4y 6. Let E(XY = y) = 3y, Var(XY = y) = 2 and let Yhave density function e” for y > 0,
f, (y) — {0 otherwise. What is Var(X)? ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  607  SECTION 22
A.3 B.5 C.9 D.11 E.20 7. Let X and Y be random variables with joint density function 24xy forx>0,y>0, andx+y<1,
fX,Y(x’y)= 0 otherwise. What is the conditional probability Pr[X <;— Y = 5:]?
A. l B. E C. —1 D. 2 E. 3
4 9 2 3 4 8.LetSand The independent events, Pr(SnT)=%, and Pr(SnTC)=%. Pr((SUT)C)= A. l B. H c. l D. i E. i
10 30 15 15 10 9. Let (X ,Y) have joint density function f (x )_ 2 for0<x<y<l,
U ,y  0 otherwise. For0<x<1,whatis Var(YX=x)? _ 2 _ 3
A.— B.(1 x) c.I+—x D. 1 x
18 12 2 3(1—x) E. Cannot be determined from the given information 10. Let (X ,Y ) be the coordinates of a point randomly chosen on the xyplane, and let R = JXZ + Y2 be the distance from (X ,Y) to the origin. If X and Y are independent, normally distributed random variables, each with mean 0 and variance 0'2, what is the value of r such that
the probability that R exceeds r is 0.95? A.0.10'2 B.0.3160' c.1710 D. 2.450 E. 5.9902 11. The random variable X has density function ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20044008 by Kmysztof Ostaszewski  608  PRACTICEEXAM 18
ex” (1 — x)k for O < .7: <1, x =
fX( ) {0 otherwise,
wherec>0 and l<k<2. Whatis themode ofX? “’1 B.k+l CL DE E “2 A. .
2k+1 k+l 2k 2k+3 12. Let X have the density function e"'2 forx < 2,
fx (x) — {0 otherwise. What is the 75th percentile of X? A.2+1n3 B.21n2 C.1n(1+3e2) D.1n[1—3e2) E.2+1nl
4 4 4 4 4 13. A fair coin is tossed. If a head occurs, 1 die is rolled; if a tail occurs, 2 dice are rolled. If Yis
the total on the die or dice, then Pr(Y = 6) = B. i C. 1—1 D. 11 l 1
A. — — E. —
9 36 72 6 36 14. Let X1, X2, X3, X4, and X5 be independent normally distributed random variables each
with mean 2 and standard deviation 3. Which of the following has a chisquare distribution? 5 5 5 Agx,’ 3;in <3.%2(X,2)2 D.%2(X,—2)2 Rigor—2): i=1 i= i=1 i=1 15. Let R, S, and T are independent, equally likely events with common probability %. What is Pr(RUSUT)?
A. i B. 3 c. E D. g E. 1
‘27 3 27 27 16. Let X and Y be continuous random variables with joint density function
ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004—2008 by Knysztof Ostaszewski  609  SECTION22
0.25 for05x$2andx—2$y$x, fx.r (LY) = {0
What is E(X2Y)? otherwise. A.3 B.1 C.2 D.4 E.—
3 3 17. X is a random variable for a loss. Losses in the year 2000 have a distribution such that:
E(X A d) = —0.025d2 +1.475d — 2.25 for d = 10, 11, 12, ..., 26, where X A d = min(X,d). Losses are uniformly 10% higher in 2001. An insurance policy reimburses 100% of losses subject to a deductible of 11 up to a maximum
reimbursement of 11. Calculate the ratio of expected reimbursements in 2001 over expected
reimbursements in the year 2000. A. 110.0% B. 110.5% C. 111.0% D. 111.5% B. 112.0% 18. Total hospital claims for a health plan were previously modeled by a Pareto distribution with
a = 2 and 6 = 500. The health plan begins to provide ﬁnancial incentives by paying a bonus of
50% of the amount by which total hospital claims are less than 500. No bonus is paid if total
claims exceed 500. Total hospital claims for the health plan are now modeled by a new Pareto
distribution with a = 2 and 0: K. The expected claims plus the expected bonus under the
revised model equals expected claims under the previous model. Calculate K. A. 250 B. 300 C. 350 D. 400 E. 450 19. For a discrete probability distribution, you are given the recursion relation 2
fx(k)=2'fx(k—1)
fork: 1,2,3, Determine fK(4). A. 0.07 B. 0.08 C. 0.09 D. 0.10 E. 0.11 20. The unlimited severity distribution for claim amounts under an auto liability insurance policy
is given by the cumulative distribution Fx (x) = 1— 0.86”“ — 0.2e'°'°°”r for x 2 0. The insurance policy pays amount up to a limit 1000 per claim. Calculate the expected payment under this
policy for one claim. A. 57 B. 108 C. 166 D. 205 E. 240
ASM Study Manual for Course PI l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  610  PRACTICE EXAM 18
21. The random variable for a loss, X, has the following characteristics: Calculate the mean excess loss for a deductible of 100. A. 250 B. 300 C. 350 D. 400 E. 450 22. WidgetsRUs owns two factories. It buys insurance to protect itself against major repair costs.
Proﬁt equals revenues, less the sum of insurance premiums, retained major repair costs, and all
other expenses. WidgetsRUs will pay a dividend equal to the proﬁt, if it is positive. You are
given: (i) Combined revenue for the two factories is 3. (ii) Major repair costs at the factories are independent. (iii) The distribution of major re air costs for each factory is (iv) At each factory, the insurance policy pays the major repair costs in excess of that factory’s
ordinary deductible of l. The insurance premium is 110% of the expected claims. (v) All other expenses are 15% of revenues. Calculate the expected dividend. A. 0.43 B. 0.47 C. 0.51 D. 0.55 E. 0.59 23. Automobile losses reported to an insurance company are independent and uniformly
distributed between 0 and 20,000. The company covers each such loss subject to a deductible of
5,000. Calculate the probability that the total payout on 200 reported losses is between 1,000,000 and 1,200,000. A. 0.0803 B. 0.1051 C. 0.1799 D. 0.8201 E. 0.8575 24. Prescription drug losses, S, are modeled assuming the number of claims has a geometric
distribution with mean 4, and the amount of each prescription is 40. Calculate the expected value
of the excess of S over 100. Note: Assume the geometric distribution counts the number of
failures until the ﬁrst success in a series of independent identically distributed Bemoulli Trials. ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  611  SECTION 22
A. 60 B. 82 C. 92 D. 114 E. 146 WW) 25. An insurance agent offers his clients auto insurance, homeowners insurance and renters
insurance. The purchase of homeowners insurance and the purchase of renters insurance are
mutually exclusive. The proﬁle of the agent’s clients is as follows: i) 17% of the clients have none of these three products ii) 64% of the clients have auto insurance. iii) Twice as many of the clients have homeowners insurance as have renters insurance. iv) 35% of the clients have two of these three products. v) 11% of the clients have homeowners insurance, but not auto insurance. Calculate the percentage of the agent’s clients that have both auto and renters insurance. A. 7% B. 10% C. 16% D.25% E. 28% 26. The graph of the density function for losses is: A (x)
0.012
0.010
0.008 0.006 W.) 0.004
0.002
0.000 0 80 120
Loss amormt, 1: Calculate the loss elimination ratio for an ordinary deductible of 20. A. 0.20B. 0.24 C. 0.28 D. 0.32E. 0.36 27. Let Z1, Z2, Z3 be independent random variables each with mean 0 and variance 1, and let
X = 2Zl  Z3 and Y = 2Z2 + Z3. What is the correlation coefﬁcient for X and Y? A. —1 B. — c. — 13.0 a; 28. Let X be an exponential random variable with mean 1. Deﬁne Y = min(X, m), where m is a W; ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  612  PRACTICE EXAM 18
positive number. Find the moment generating function of Y. m(lt) A.M,(t)=l:et fort¢l,withM,(l)=m+l B. M,(t)=rnin(%,e”“) for t<1 and M,(t)=e”” for t21
em! CM 1 =— A) 1_. me" D. My(t)= t fort<m and My(t)=oo forth
m—
l—te’"('") E.M,,(t)= 1t fort<1andM,,(t)=oofort21 29. The cumulative distribution function for health care costs experienced by a policyholder is
modeled by the function X _ 'ﬁ
Fx(x)= 1 e forx>0,
0 otherWISe. The policy has a deductible of 20. An insurer reimburses the policyholder for 100% of health
care costs between 20 and 120 less the deductible. Health care costs above 120 are reimbursed at
50%. Let G be the cumulative distribution function of reimbursements given that the
reimbursement is positive. Calculate 6015). A. 0.683 B. 0.727 C. 0.741 D. 0.757 E. 0.777 30. The value of a piece of factory equipment after three years of use is 100 05" , where X is a I for t < 1. Calculate the
1 — 2t 2 expected value of this piece of equipment after three years of use. random variable having moment generating function M x (t) = A. 12.5 B. 25.0 C. 41.9 D. 70.7 E. 83.8 ASM Study Manual for Course Pll Actuarial Examination. (9 Copyright 20042008 by Knysztof Ostaszewski  613  ...
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