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Unformatted text preview: PRACTICE EXAMINATION NUMBER 8 1. Two types of vehicles, cars or trucks, enter a certain tunnel. The probability that a next vehicle
to enter the tunnel is a car is 10 times the probability that the next vehicle is a truck. Determine
the probability that among the ﬁrst 23 vehicles to enter the tunnel, there are at least 20 cars. A. Less than 72% B. At least 72%, but less than 77%
C. At least 77%, but less than 82%
D. At least 82%, but less than 87%
E. 87% or more 2. The probability density function of the k—th order statistic of size n is: "1 !(F(y))k"(1F(y))"'kf(y). (k — 1)!(n — k)
where F is the cumulative distribution function of the original distribution, and f is the density of
it. Samples are selected from a uniform distribution on [0, 10]. Determine the expected value of
the fourth order statistic for a sample of size ﬁve. A. Less than 6.5 B. At least 6.5, but less than 7.0
C. At least 7.0, but less than 7.5
D. At least 7.5, but less than 8.0
E. 8.0 or more 3. XYZ Insurance issues lyear policies.
0 The probability that a new insured has no accidents last year is 0.70.
0 The probability that an insured who was accidentfree last year will be accidentfree this year is 0.80.
0 The probability that an insured who was not accidentfree last year will be accidentfree this year is 0.60.
What is the probability that a new insured with an unknown accident history will be accidentfree in the second year of coverage? A. Less than 71% B. At least 71%, but less than 72%
C. At least 72%, but less than 73%
D. At least 73%, but less than 74%
E. 74% or more ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  315  SECTION 12
4. Let X1 ,X2 ,X3 be a random sample from a continuous probability distribution. Find Pr(xl s X2 5 X,). A. i B. i C. :1 D. i E. Not enough information to find 5. Let X be a discrete random variable with moment generating function A. l B. % C. % D. 28 : E. E(X) does not exist 6. The distribution of Y, given X, is uniform on the interval [0, X]. The marginal density of X is
2x, for 0 < x < 1 fx(x)={0, Determine the conditional density of X, given Y = y, where positive. otherwise. E; A.1 3.2 C.?.x D.l
y 1y 7. Let X be a discrete randomvariable with moment generating function Mx(t)=i(l+e‘°')+—lw£ 2 "non! for oo < t < +oo. Find Pr(X 2 3). A B C D. % E. Cannot be determined 1 l l
.8 .4 .2 8. Mr. Flowers plants 10 rosebushes in a row. Eight of the bushes are white and two are red, and
he plants them in a random order. What is the probability that he will consecutively plant seven
or more white bushes? ASM Study Manual for Course Pl] Actuarial Examination. © Copyright 2004—2008 by Knysztof Ostaszewski  316  PRACTICE EXAMINATION NO. 8
l 2 7 1
. — B. — C. — D. — E. —
A 10 9 15 45 5 9. You are given that N has Poisson distribution with mean 4. Find Var(N IN 2 4). A. 2.10 B. 3.57 C. 4.00 D. 4.67 E. 5.33 10. The claim amount on a certain insurance contract has a normal distribution with mean $1,000
and standard deviation $250. Given 10 independent claims, what is the probability that the
number of claims less than $1 ,050 is less than or equal to 2? A. 0.9026 B. 0.6025 C. 0.5793 D. 0.3356 E. 0.0174 11. A random variable has the probability density function fx (x) = 2L , for l S x S e, and 0
x e otherwise. What is the probability that among of four independent observations of X there are
three less than 1 and one that is greater than 1? A.— B .1 c 13.3 El
16 4 4 3 l
' 2 12. A machine consists of two components. The lifetimes of the two components are identically distributed with the following common density function fx (x) = %e‘3‘ + ge'z’ for x > 0, and 0 otherwise. The machine breaks down if any of the two components fails. Find the expected
lifetime of the device. A. 0.1124 B. 0.2260 C. 0.298] D. 0.3267 E. 0.3999 13. Let X be an exponential random variable with mean 2. Deﬁne a new random variable
Y = min(X2,2). Find F, (1). A.0 B.l Cl E.l—1— l ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  317  SECTION 12
14. X is a random variable uniformly distributed on the interval [1, 2]. Find the probability density function of Y = lnX. l ey
A. — B. 1n2 C. lny D. — E. e”
y ln2 15. You are given that X1 and X2 are two independent and identically distributed random
variables with a Poisson distribution with mean 2. Let Y = max(XI ,Xz). Find Pr(Y = l). A. 0.1201 B. 0.1465 C. 0.2578 D. 0.3381 E. 0.4255 16. Let Xl and X2 be a random sample from the uniform distribution on [0, 1] and let
Y1 = min(X,,X2), Y2 = max(X,,X2). Find fyl (yIIY2 =y2). B.— C.— D.2 E] A. l
2 2Y2 Y2 17. Michiko is performing a Bernoulli Trial experiment with probability of success of P, where P
is distributed on the interval [0, 1] according to a probability distribution with density f,. (p) = 3 p2 for 0 S p S 1, and 0 otherwise. She performs the experiment 10 times and has two successes. Find the mean of the posterior probability distribution of P, i.e., E (PI X = 2), where
X is the number of successes in 10 Bernoulli Trials with a probability of success P. A.1 3.1 Ci 9.1 5.?—
2 5 5 10 14
2
18. Let X have the density function f (x;9) = 36:3 for 0 < x < 0, and 0 otherwise. If
7 Pr(X >1): 8, what is the value of 6?   1
B. (1)3 c. [gs D. 23 E. 2
8 7 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004—2008 by Knysztof Ostaszewski  318  PRACTICE EXAMINATION N0. 8
19. Telephone numbers in a certain area all start with 7 and then 6. The third digit can be 4, 5, or
6. Each of the last four digits can be any number from 0 to 9. There are 18,243 telephone
numbers assigned in the area. How many phone numbers are still unassigned? A. 18,243 B. 30,000 C. 321,757 D. 11,757 E. None 20. A random sample of size 16 is to be taken from a normal population having mean 100 and
variance 4. What is the 90th percentile of the distribution of the sample mean )7 ? A. 97.44 B. 100.08 C. 100.32 D. 100.64 B. 102.56 21. Suppose that the joint density function of X and Y is defined by
10xy2, 0<x<y<1,
x, =
f( y) {0, otherwise. What is 13(X2 Y = y)? 5 5 y2 2y 2y
A.— 13.—7‘—41 .— D.— E.—
14 42( y y ) C 2 3 3
22. Events Sand Thave probabilities Pr(S)=Pr(T)=% and Pr(ST)=%. What is
Pr(SC nTc)?
1 1 7 4 1
A. — B. — c. — D. — E. —
6 3 18 9 2 23. Let Yand Zbe two random variables. If Var(Y) = 4, Var(Z)=16, and Cov(Y,Z) = 2,
what is Var(3Z — 2Y)? A. 160 B. 136 C. 56 D.32 E. 16 24. Suppose X is a random variable with Pr(X =1): p and Pr(X = k +1) =(l p)Pr(X = k)
for k =1,2,..., and 0 < p <1, and Pr(X = x) = O for any x that is not a positive integer. What is
E(X)? ASM Study Manual for Course Pll Actuarial Examination. (9 Copyright 20042008 by Knysztol’ Ostaszewski  319  SECTION 12 A.p B.1 C.1p D.1——p 3.1
p p 25. Five urns are numbered 3, 4, 5, 6, and 7, respectively. Inside each urn is N 2 dollars where N
is the number on the urn. The following experiment is performed: An urn is selected at random.
If its number is a prime number the experimenter receives the amount in the um and the
experiment is over. If its number is not a prime number, a second urn is selected from the
remaining four and the experimenter receives the total amount in the two urns selected. What is
the probability that the experimenter ends up with exactly twentyﬁve dollars? A. l B. 5"— c. l D. l E. i
25 11 5 4 10
1 26. Let X have a distribution with the 75th percentile equal to 3 and density function equal to fx(x)={ )Le‘“, forx > O, 0, otherwise.
What is A?
A. ln6—‘l B. ilng C. ln12 D. 1n64 E. ln81
27 3 2 27. A continuous random variable X has density function f as shown below. f (x) What is Pr(0 s X51)? 1 D. _
10 20 40 ‘ 100
E. Cannot be determined from the information given A.i B.i oi ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 2004—2008 by Knysztof Ostaszewski  320  I ' PRACTICE EXAMINATION N0. 8 28. Suppose X and Y have the joint density function 8xy, for0<x<y<1,
fxx (x’y) = {0 otherwise. What is the joint density function of W =% and Z = Y? A. 6wz2 for 0 < w < 1,0 < z <1; zero otherwise B. 8wz2 for 0 < w < 2, <1; zero otherwise C. 8wz2 for 0 < w <1, 0 < 2 <1; zero otherwise
D. 8wz3 for 0 < w <1, 0 < 2 <1; zero otherwise
E. lsz2 for 0 < w < z <1; zero otherwise 1
29. Let X and Y be independent random variables with common density function f (t) = 0 for
0<t<9. Whatis Pr(£S—l]?
Y 3
A. 2 B. L: C. 9 D. —l E. l
4 6 6 4 6 30. Let X be a random variable with density function ‘4! ae , forx>0,
x =
fX( ) {0, otherwise. where a > 0. If M x (t) denotes the momentgenerating function of X, what is M x (3a)? A. e‘3“ B.  C 1
.— D E.+°°
30 3 l
' 4 (a ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  321  ...
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This note was uploaded on 10/27/2010 for the course PSTAT 172a taught by Professor Staff during the Winter '08 term at UCSB.
 Winter '08
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