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Unformatted text preview: PRACTICE EXAMINATION NUMBER 6 1. An insurance company examines its pool of auto insurance customers and gathers the
following information: (i) All customers insure at least one car. (ii) 64% of the customers insure more than one car. (iii) 20% of the customers insure a sports car. (iv) Of those customers who insure more than one car, 15% insure a sports car. Calculate the probability that a randomly selected customer insures exactly one car and that car
is not a sports car. A. 0.16 B. 0.19 C. 0.26 D.0.29 E.0.31 2. The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function fx, where fx (x) is proportional to (10 + x)‘2 . Calculate the
probability that the lifetime of the machine part is less than 5. A. 0.03 B. 0.13 C. 0.42 D. 0.58 E. 0.97 3. An insurer’s annual weatherrelated loss, X, is a random variable with density function
2.5  20025
fx (x) = x 35 ’
0, otherwise.
Calculate the difference between the 25th and 75th percentiles of X. forx > 200, A. 124 B. 148 C. 167 D. 224 E. 298 4. A device runs until either of two components fails, at which point the device stops running.
The joint density function of the lifetimes of the two components, both measured in hours, is fxy (x.y) = x+ y during its ﬁrst hour of operation? for 0 < x < 2 and 0 < y < 2. What is the probability that the device fails A. 0.125 B.0.l4l C.0.391 D.0.625 E.0.875 5. Let X(1)'X(2)" . . ,Xm be the order statistics from a random sample of size 6 from a distribution
with density function ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  256  PRACTICE EXAMINATION N0. 6
2x, for0<x<l, fx(X)={0' What is E(X(6))? otherwise. A RE C.§ DE E.—
3 6 7 l
' 2
6. Let X be a normal random variable with mean 0 and variance a > 0. Calculate Pr(X2 < a). A. 0.34 B. 0.42 C. 0.68 D. 0.84 E. 0.90 7. An urn contains 100 lottery tickets. There is one ticket that wins $50, three tickets that win
$25, six tickets that win $10, and ﬁfteen tickets that win $3. The remaining tickets win nothing.
Two tickets are chosen at random, with each ticket having the same probability of being chosen.
Let X be the amount won by the one of the two tickets that gives the smaller amount won (if both
tickets win the same amount, then X is equal to that amount). Find the expected value of X. A. 0.1348 B. 0.0414 C. 0.2636 D. 0.7922 E. Does not exist 8. (X1,X2 ,X3) is a random vector with a multivariate distribution with the expected value
(0,0,0) and the variance/covariance matrix: 4 1.5 1
1.5 1 0.5 .
l 0.5 1 If a random variable W is deﬁned by the equation XI = aX2 + bX3 + W and it is uncorrelated
with the variables X2 and X3 then the coefﬁcient a must equal: A.l B.i C.5 D.2 El
3 3 3 9. A random variable X has the exponential distribution with mean 1. Let [x]] be the greatest A
integer function, denoting the greatest integer among those not exceeding x. Which of the following is the correct expression for the expected value of N = [[X]? ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20042008 by Knysztof OstaszeWSki  257  PRACTICE EXAMINATION NO. 6
1 el 1 1
B. —  C.——— D. E.
"Au 2 [[11]] 2 e*—1 e*—1
10. X and Y are independent and both distributed uniformly from 0 to 20. Find the probability
density function of Z = 25X 10Y. A. fz (z) = 1.5 where nonzero
B. Stepwise formula: 200—25, 200$z<0,
100000
1
= —, 0 s < 300,
fl (z) 300 7'
500 + z
, 300 s s 500.
100000 2 C. f2 (z) = ﬁ where nonzero 1 ———z D. f2 (z) = 200a 20° , for z > 0, zero Otherwise
E. Stepwise formula: 200”, —200 52 <0,
100000
1
= —, 0 s < 300,
fz (z) 500 z
500 — z
, 300 s s 500.
100000 2 11. Let X(1)'X(2)""’X(8) be the order statistics from a random sample X1,X2,...,X8 of size 8
from a continuous probability distribution. What is the probability that the median of the distribution under consideration lies in the interval [X(2),Xm]? “0 B 112 C 12 D. 2‘} E. Cannot be determined A. —— . —— .
128 128 128 128 12. In a block of car insurance business you are considering, there is a 50% chance that a claim
will be made during the upcoming year. Once a claim is submitted, the claim size has the Pareto
distribution with parameters a = 3 and 0 = 1000. Only orgeglaim will happen during the year. ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  258  PRACTICE EXAMINATION N0. 6
Determine the variance of the unconditional distribution of the claim size. A. 62500 B. 437500 C. 500000 D. 750000 E. 1000000 13. A random vector (X ,Y) has the bivariate normal distribution with mean (0,0) and the 1 0
variancecovariance matrix [0 I]. Find the probability that (X ,Y) is in the unit circle centered at the origin. A. 0.2212 B. 0.3679 C. 0.3935 D. 0.6321 E. 0.7788 14. You are given that X and Y both have the same uniform distribution on [0, l], and are
X X+Y independent. U = X + Y and V = . Find the joint probability density function of (U, V) evaluated at the point (1,1).
2 2 A.O B 13.1 ml— 1
' 4 15. Three fair dice are rolled and X is the smallest number of the three values resulting (if more
than one value is the smallest one, we still use that value). Find Pr(X = 3). A_ i B. 3_7 c. i D_ E E, ﬂ
216 216 216 216 216 16. The momentgenerating function of a random variable X is M x (t) = 3 J5 Calculate the excess of Pr(X > Z + 7] over its bound given by the Chebyshev’s Inequality. 1+3.;’ for t>l.
4 4 l—t A. 0.35 B.0.22 C.0 D.O.15 E.—0.20 17. The time to failure X of an MP3 player follows a Weibull distribution. It is known that ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  259  PRACTICE EXAMINATION N 0. 6
Pr(X > 3) = l , and that Pr(X > 6) = —14. Find the probability that this MP3 player is still
e e functional after 4 years. A. 0.0498 B. 0.0821 C. 0.1353 D. 0.1690 E. 0.2231 1 18. You are given that the hazard rate for a random variable X is A (x) = 5x 2 for x > 0, and zero otherwise. Find the mean of X. A.1 B.2 C.2.5 D.3 E.3.5 19. Let P be the probability that an MP3 player produced in a certain factory is defective, with P
assumed a priori to have the uniform distribution on [0, 1]. In a sample of one hundred MP3 players, 1 is found to be defective. Based on this experience, determine the posterior expected
value of P. A; 11—2 cl mi El
100 101 99 50 51 20. You are given that Pr(A)=%, Pr(AuB)= %, Pr(BA)= %, Pr(C{B) =%,and
Pr(CAnB)=%. Find Pr(ABnC).
A. 1 B. 3 c. i D. l E. l 3 5 10 2 4 21. Let X0) ,...,X(n) be the order statistics from the uniform distribution on [0, 1]. Find the correlation coefﬁcient of X0) and X“). A. —l B. —; C.0 D. L E. l n n+1 n+1 n 22. A random variable X has the lognormal distribution with the density ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  260  PRACTICE EXAMINATION NO. 6 forx > 0, and 0 otherwise, where p is a constant. You are given that Pr(X S 2) = 0.4. Find
E(X). A. 4.25 B. 4.66 C. 4.75 D. 5.00 E. Cannot be determined
23. You are given a continuous random variable with the density fx (x) = %x +% for —1 S x S l,
and 0 otherwise. Find the density of Y = X 2 , for all points Where that density is nonzero. 3
13.2 c. — D. — 2 E. —
y 2y 3y yln2 1
A. —
25 24. An insurer has 10 independent oneyear term life insurance policies. The face amount of each
policy is 1000. The probability of a claim occurring in the year under consideration is 0.1 . Find
the probability that the insurer will pay more than the total expected claim for the year. A.0.01 B.0.10 C.0.l6 D.0.26 E. 0.31 25. An insurance policy is being issued for a loss with the following discrete distribution:
_ 2, with probability 0.4,
‘ 20, with probability 0.6. Your job as the actuary is to set up a deductible d for this policy so that the expected payment by
the insurer is 6. Find the deductible. A. l B. 5 C. 7 D. 10 E.15 26. For a Poisson random variable N with mean ’1 ﬁnd lim E (N IN 2 1). 11)0 A. 00 B. 0 C. l D. e E. Cannot be determined 27.X is a normal random variable with mean zero and variance % and Y is distributed ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof 0staszewski  26]  PRACTICE EXAMINATION N 0. 6
exponentially with mean 1.X and Y are independent. Find the probability Pr(Y > X 2). 1 J5 A.— B. 3 0—1— D E—
75 1
JZ J5 ‘2 '2 28. X (1): X (2), X (400) are order statistics from a continuous probability distribution with a ﬁnite mean, median m and variance 0’. Let (I) be the cumulative distribution function of the
standard normal distribution. Which of the following 18 the best approximation of Pr ((11220) < m) using the Central Limit Theorem? A. @(0050‘) B. 0.0049 C. 0.0532 D.0.0256 E. ¢[:—0] 29. N is a Poisson random variable such that Pr(N $1): 2  Pr(N = 2). Find the variance of N. A.0.512 B. 1.121 C. 1.618 D.3.250 E.5.000 30. There are two bowls with play chips. The chips in the ﬁrst bowl are numbered 1, 2, 3 , . . . , 10, while the chips in the second bowl are numbered 6, 7, 8, . . . , 25. One chip is chosen randomly
from each bowl, and the numbers on the two chips so obtained are compared. What is the
probability that the two numbers are equal? 1 l 1 1 1
A. — B . — C. — D. — E. —
2 5 10 40 50
ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  262  ...
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