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Unformatted text preview: PRACTICE EXAMINATION NUMBER 19 1. You are given that the probability density function of a random variable X is fx (x) = klnx for 1 S x S 2, where k is a constant, and fx (x) = 0 otherwise. Find the expected value of this
random variable. A. 1.2500 B. 1.499 C. 1.5751 D. 1.6000 E. 1.6472 2. In the country of Freedonia, three types of insurance against government are available: war
insurance, tax insurance and bureaucracy insurance. Among the citizens of Freedonia, 55% have
war insurance. The percentage of citizens who have tax insurance and no other insurance is twice
the percentage of citizens who have bureaucracy insurance and no other insurance. 33% of all
citizens have exactly two kinds of insurance, and for any pair of two forms of government
insurance the probability of a citizen having both forms of government insurance (these two but
no all three) is the same as for any other of those pairs. Also, 20% have tax insurance but no war
insurance or bureaucracy insurance. A small group of citizens are devout socialists and have no
government insurance of any kind. What is the percentage of Freedonia’s population in that
small group of citizens that foolishly have no government insurance whatsoever? A. 4% B.5% C.6% D.7% E. 9% 3. You are given that the joint cumulative distribution function of random variables X and Yis 0 forx<00ry<0,
ny(x,y)= l—e'9‘+%e'5’(e"‘l) for0<x<y<+oo,
l+§e'°‘2e‘s‘ foery.
4 4
Find Var(E(YX)).
A. g B. 81 C. —l D. £ 13. i
16 36 81 81 4. You are given that X and Y are independent and identically distributed random variables, both 2 2
with the standard normal distribution. Find the moment generating function of X :Y . ASM Study Manual for Course Pl] Actuarial Examination. © Copyright 2004—2007 by Krzysztof Ostaszewski  635  SECTION 23 ._ 2
A.[ 0'5 )2 B.L c. 0'5 Di E. L
105: 1: 2—05: 12: 2t 5. You are given the variancecovariance matrix of random variables X and Y as [2 4]. If the coefﬁcient of variation of X is l and the coefﬁcient of variation of Y is 2, what is the coefﬁcient
of variation of X + Y? A.3 3.2 c.1.5 D.~/l—l Bin—.1 6. You are given that X follows the normal distribution with mean 1 and standard deviation of 2.
Find the probability that 2X — X 2 > 0. A. 0.2500 B. 0.3085 C. 0.3830 D. 0.6915 E. 1.0000 7. Let xpx2 ,...,x35 and y1 ,y2 ,...,y49 be independent random samples from distributions with
means ,ux = 30.4 and p, = 32.1 and with standard deviations ax =12 and 0', =14. What is the approximate value of Pr(X > Y)? A. 0.27 B. 0.34 C. 0.50 D. 0.66 E. 0.73 8. A fair die is tossed until a 2 is obtained. If X is the number of trials required to obtain the ﬁrst 2, what is the smallest value of x for which Pr(X S x) 2 %? A.2 B.3 C.4 D.5 E6 9. Let X and Yhave a bivariate normal distribution with common mean u, common variance 0'2 > 0, and correlation p, where —1 < p <1. Which of the following statements are true?
I. X and Y are independent if and only if p = 0.
11. Y — X has a normal distribution if and only if p > 0. III. Var(X+Y) < 20'2 if and only if p < 0. A. I and II only B. I and III only C. II and 111 only D. I, H, and HI ASM Study Manual for Course P11 Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  636  PRACTICE EXAM 19
E. The correct answer is not given by A, B, C, or D 10. You are given the cumulative distribution function of a random variable X 0 forx < 0,
1x for 0 S x <1,
4
Fx(x) = % forx =1,
£+—1 for l < x < 5,
l forx 2 5.
Find the coefﬁcient of variation of X.
A. 1225 B. E C. a  41255 E. 1225
1272 53 36 53 576 11. You are given that N is a Poisson random variable with mean 4. Define a new random
variable M = (N  4N 2 4). Find the mean ofM. A. 1.38 B. 1.80 C. 3.21 D.4.00 E. 8.33 12. A dart is thrown at a dartboard with radius of 7 centimeters. The point that the dart hits is
uniformly distributed on the circular dartboard. Find the expected distance, in centimeters, of that point from the center of the dartboard. A.1 3.1 cﬂ p.53 5.31
2 3 3 13. Four people wear four different hats. Each of them hands a hat to at attendant, and the
attendant mixes them up randomly and hands one hat to each one of them. What is the probability that none of the four is wearing their original hat? B A . l . . .
4 12 24 24 l
' 2 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004.2008 by Knysztof Osmszewski  637  SECTION 23 4x 14. You are given that the joint density of random variables X and Y is f” (x, y) = — for x
3’
between 0 and 1 and y between 1 and J2, and f3“, (x,y) = 0 otherwise. Find Var[Y ,X = 211]. A. 0.0025 B. 0.0202 C. 0.0297 D. 0.0349 E. 0.1869 15. Insurance company insures against ﬁre and storm in a combined policy. The probability of a
ﬁre is 0.01 and the probability of a storm is 0.03. Fires and storms are independent. Every ﬁre
produces losses of $1000 and every storm produces losses of $3000. There is a deductible of
$100. Find the coefﬁcient of variation of the amount paid by the insurance company, assuming
there can be at most one ﬁre and at most one storm per year. A. 5.24 B. 10.72 C. 126.70 D. 263.28 E. 2632.81 16. Claims follow a distribution with density function f (x) ke‘om‘ forx> 0,
x 0 otherwise. The insurance policy covering those claims has a deductible of 3. Find the net premium for that
policy. A. 98.1222 B. 97.0446 C. 97.0000 D. 96.9554 E. 96.6667
17. Claims are uniformly distributed on the interval [0, 100]. An insurance policy covering those
claims has a limit of L (meaning that losses above L are not covered), and a deductible of 10. What is that policy limit if the expected payment of the policy is 38.5? A. 68 B. 74.5 C. 80 D. 83.5 E. 90 18. You are given that X and Y are independent random variables with moment generating
2H}: functions Mx(t)=e 2 and MY(t)=e3‘2“. Find Pr(X—Y > 3). A. 0.3138 B. 0.4851 C. 0.5000 D. 0.5149 E. 0.6862 19. Let X and Y be continuous random variables with joint density function ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  638  PRACTICE EXAM 19 gal _ 2(x+y) for0<x<y<1
fx" (x,y) — {0, otherwise.
Then E(Y)=
A, i B, l c. 3 D. 1 E. 1
12 2 4 6 20. Coins K and L are weighed so the probabilities of heads are 0.3 and 0.1 , respectively. Coin K
is tossed 5 times and coin L is tossed 10 times. If all the tosses are independent, what is the
probability that coin K will result in heads 3 times and coin L will result in heads 6 times? 5 10 5 10
A. 0.33 0.72 0.1‘5 .094 B. 0.33 o.72  0.1‘5 0.9‘
(3] +16) [(3] 6 5
15 [J
C. [9 J049 O.65 D. i% E. 0.60.9 15
9
21. Let X and Y be discrete random variables with joint probability function x2 +y2
fxa' (va’)— 56 for x= 1,2,3 and y = 1,...,x. What is Pr(Y = 3Y 2 2)? A. i B. 92 c. 3 D. 3 E. 3
28 56 13 13 13 22. Which of the following are valid cumulative distribution functions? 0 forxSl
I.F(x)= x2—2x+1 for1<x.<_2
1 forx>2
0 forxSO
II. F(x)= x22x+1 forO<xSl+\/§
1 forx>l+\/§ ASM Study Manual for Course P/l Actuarial Examination. ‘9 Copyright 20042008 by Knysztof Ostaszewski  639  SECTION 23 0 forxSl
l forx=2
2
III. F(x)= x2—2x+l— for2<x51+£
2 2
1 forx>1+[[2§) A. II and II only B. I and HI only C. II and III only D. I, II, and III
E. The correct answer is not given by A, B, C or D 23. Let X1 ,X2 be a random sample from a normal distribution with mean and variance 1. If
E((:Xl — X2)=1, then c = m 1 r2: 2 f 24. Let X and Y be continuous random variables with joint density function f” (x, y) and marginal density functionst and fr, respectively, that are nonzero only on the interval (0,1).
Which one of the following statements is always true? A. E(X2Y3) =[lx2dx][ly3dy] B. E(X2) = 1x2 f“, (x,y)dx C E (X 2Y3): [lxzfxr (X»y)dx][j;y3fx.y (16.3%»)
D. E(X2) = 1x21; (x)dx E. E(X2) = if]; (y)dy 25. The number of accidents on a main street in Bloomington, Illinois, follows a Poisson
distribution with mean 0.05 per day. Occurrence of accidents on each day is independent of ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20044008 by Knysztof Ostaszewski  640  PRACTICE EXAM 19
occurrence of accidents on any other day. Accidents are being counted anew in June. What is the
probability that the ﬁrst accident in June will happen on June 6, given that there were no
accidents June 1 and June 2. A. 0.04 B. 0.05 C. 0.06 D.0.l9 E. 0.21 26. You are given that the cumulative distribution function of a random variable X is 0 x<0,
0.25+0.25x OSx<l,
0.5 +0.25x le<2,
1 x22. Find the coefﬁcient of variation of X. Fx(x)= 27 17 B 17 c 17 1 E.
3 17 27 'E 'E 27. Let X, Y, and Z be independent Poisson random variables with E (X) = 3, E (Y ) = 1, and
E(Z) = 4. What is Pr(X+ Y+Z s 1)? _1 l 1
A. 13e'12 B. 9e'8 C. 3e '2 D.9e 3 E. %e 3 28. A box contains 4 balls and 6 white balls. A sample of size 3 is drawn without replacement
from the box. What is the probability of obtaining 1 red ball and 2 white balls, given that at least
2 of the balls in the sample are white? 3.3 c3 0.3 E 54
3 4 A ,_
11 55 l
’ 2 29. You are given that the conditional distribution of a random variable Y given X, is uniform on
the interval [0, X]. Also, the marginal probability density function of X is fx (x) = 2x for 0 < x < 1, while fx (x) = 0 otherwise. Find the variance of the conditional disuibution of X ,
given Y: y. AllI): B. 12 C_(1xy)2 D ly 150202
12 l—x 12 ' 12 ' 12 ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 2004—2008 by Knysztof Ostaszewski  64] . SECTION 23
30. Glittery Property Insurance offers an insurance policy covering a loss described by a random variable X that follows a gamma distribution with parameters 0: =3 and [3 = in Note that if a gamma distribution is obtained as a sum of independent identically distributed exponential
random variables, then the parameter 0: denotes the number of those variable added, while [3 is their common hazard rate. The policy has a deductible of 10. Find the probability that Glittery
Property Insurance will not have to make any payment on this policy. A. 0.26 B. 0.33 C. 0.42 D. 0.50 E. 0.74 ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  642  r ...
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