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Unformatted text preview: PRACTICE EXAMINATION NUMBER 12 1. Five different bidders are submitting independent bids for a race horse. They all believe the
value of the horse to be between $100,000 and $200,000, and as a result of that common belief,
their bids, while independent, are all be uniformly distributed between $100,000 and $200,000.
Find the expected value of the difference between the winning (i.e., the highest) bid and the lowest bid. A. 16,667 B. 33,333 C. 50,000 D. 66,667 E. 83,333 2. Customers arrive at a Post Packages Inc. stores according to a Poisson distribution with the
expected number of daily arrivals of 50. Given a random number X of customers that have
arrived in a day, each such customer will either mail one package with probability 0.60, or not
mail any packages with probability 0.40. Let Y be the random variable that describes the total
number of packages mailed in one day. Find the difference between the variance of Y and Var(E(YX)). A.0 B.6—0.24X C.8 D. 12 E.30—0.6X 3. A new supersize passenger airplane has 400 seats in coach class and 100 seats in the business
class, with these being the only two classes available. The weight of the luggage (all luggage
pieces combined) of an individual coach passenger follows a continuous probability distribution
with mean of 100 pounds and the standard deviation of 10 pounds, while the weight of the
luggage (all luggage pieces combined) of a business class passenger follows a continuous
probability distribution with mean of 80 poundsand the standard deviation of 5 pounds. The
weights of individual passengers luggages are independent. You are given that the 95th
percentile of the standard normal distribution is 1.644853627. Approximate the smallest possible
number w such that the probability that the total weight of the entire luggage on the plane is less
than w is 0.95, assuming that the airplane is full. A. 48339 B. 52500 C. 55401 D. 544531 E. 48206 4. A health insurance policy covers 80% of a doctor’s bill and 90% of a hospital bill for the
insured person. The amount of the doctor’s bill X is a continuous random variable with standard
deviation of 500, and the amount of the hospital bill Y is a continuous random variable with
stande deviation of 1000. It has been determined that the doctor’s bill amount and the hospital
bill amount have the coefﬁcient of correlation of 0.75. Find the standard deviation of the total
amount paid by the insurance company. A. 1039 B. 1118 C. 1229 D. 1441 E. 1500 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Osmszewski  433  SECTION 16
5. You are given that the random variable X is exponential with mean 1, and that the random variable Y is uniformly distributed on the interval [0,1]. Furthermore, it is known that X and Y are independent. Find the density of the joint distribution of U = XY and V = 2e"/“—“ A. foru>0,v>0,andu<v
v
eJu_v B. 2 foru>0,v>0,andu<v
v G. 2mm for u>0 and v>0
D. 2ve‘/"_" foru>0, v>0, and u<v B. we"; foru>0,.v>0, and u<v 6. You are given that a certain engineering project time is a random variable U distributed according to a gamma distribution with parameters a = 120 and B = %, where this gamma distribution can be represented as a sum of a independent, identically distributed exponential
random variables, each with hazard rate of ,3. You are also given that the 90th percentile of the standard normal distribution is 1.281551566. Find an approximate number u such that
Pr(U S u) = 0.90. A. 1495162 B. 45845 C. 1744 D. 402 E. 393 7. A box is to be constructed so that the height is 10 inches and its base is X inches by X inches.
If X has a uniform distribution over the interval (2,8) then what is the expected volume of the box in cubic inches? A. 80 B. 250 C. 252.5 D. 255 E. 280 8. Let the joint density function of X and Ybe given by kryz, for0<x<y<l,
x, = .
f” ( y) {0, otherw1se. What is the value of k? A.1 B.2 C.5 D.6 E.10 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  434  PRACTICE EXAM 12
9. Let Xl , X2 , and X3 be a random sample from a normal distribution with mean n at 0 and variance 0'2 = What are the values of a and b, respectively, in order for L = 01Xl + 4X2 + bX3 to have a standard normal distribution? A. a=2, b=2 B.a=2, b=2
C.a=l, b=3
D.a=2, b=2 E. Cannot be determined from the given information 10. Suppose that one observation at is taken on a random variable X with density function 2
fx(x(9=9)=1_1;2 for 0 S x $1, and 0 elsewhere, and suppose that the prior density function for O is
f9 (9) = 490 92) for o < e s 1, and o elsewhere. What is E(o X = x)? A; 3.1 c.2—x 9.2, 2
2 E' _
3x 15 3 x x 11. Customers at Fred’s Café win a 100 dollar prize if their cash register receipts show a star on
each of the ﬁve consecutive Monday, . . . , Friday in any one week. The cash register is
programmed to print stars on a randomly selected 10% of the receipts. If Mark cats at Fred’s
once each day for four consecutive weeks and if the appearance of the stars is an independent
process, what is the probability that Mark will win at least 100 dollars? A. 0.0004 B. 40.10” c. 40.000010.999993
4 4 20
D. Z[k]~o.00001* 0.99999“ E. [ 5 ].o.000015 .099999'5
k=l 12. Let X and Y have the joint density function f“ (x,y) = 2x for 0 < x <1 and O < y < 1. What is Pr[X+YSl XSé]? 2
A. 1 B. — C.
3 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztol' Ostaszewski  435  SECTION 16 13. Let the random variable X have the density function fx (x) = kx for 0 S x S E, and zero J5 elsewhere. If the mode of the distribution is at x = 7, then what is the median? A.— B C.[2— DJ: 1
.— .— E
6 4 4 24 l
' 2 14. Let X1," .,X,, be independently and uniformly distributed on the interval (—A,A). What is
Pr{min(X,,...,X,,) 3—12 or max(Xl,...,Xn) 2a}, where o < a < A and o <b <A? A.1—(a+b)" B.1(“+b) c.1—(a+b] D.1[“_b] E.1—[“'b)
A 2A 2A A 15. The number of trafﬁc accidents per week in a small city has Poisson distribution with mean
equal to 3. What is the probability of exactly 2 accidents in 2 weeks? A. 9e’6 B. l8e'6 c. 2555 D. 4.5a“3 E. 9.5e'3 16. A factory makes three different kinds of bolts: Bolt A, Bolt B, and Bolt C. The factory
produces millions of each bolt every year, but makes twice as many of Bolt B as it does of Bolt
A. The number of Bolt C made is twice the total of Bolts A and B combined. Four bolts made
by the factory are randomly sampled from all bolts produced by the factory in a given year.
Which of the following is most nearly equal to the probability that the sample will contain two of
Bolt B and two of Bolt C? A; Ba 0384 D32 . — . — E
243 625 2401 243 l
' 6 17. Let X be a continuous random variable with density function
be'”‘, forx > 0,
A (x) = { O, elsewhere,
where b > 0. If M X (t) is the moment—generating function of X, then M x (—6b) = l 1 1 l
A. — B. — C. — D. —
7 5 7b 5b ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004—2008 by Knysztof Ostaszewski  436 . E.+°° PRACTICE EXAM 12 18. Let each of the independent random variables XI and X2 have the density function fx (x) = e" for 0 < x < co, and O elsewhere. What is the joint density of Y1 = XI and
Y2 = 2Xl + 3X2 and the domain on which this density is positive? {w—a {we}
A.3e 3 for0<2yl<y2<oo Be 3 for0<2yl<y2<oo
1 he) C.§e for0<yl<2y2<oo D. 3e for0<y,<2y2<oo
l {was}
Ege 3 for0<2yl<y2<oo 19. Let X have the probability function we}; forx = 0,1 ,2,. . . , and zero elsewhere. What is the momentgenerating function of X? B.——— C.— D.8—— E.98e’ 20. Let X be a continuous random variable with density function 2x'2 for 1 < x < 2,
x =
fX( ) {0, elsewhere. 1
If Y = X 2 , what is the density function of onr 1 < y < J5 ? J5 A.— B.2 C
y 2 4 4
.—4 11—5 E.—3
y y 3’ y 21. X,, X2, X 3 is a random sample from the exponential distribution with mean 1.Let Y0),
Y0), and Y“) be the corresponding order statistics. Find the variance of the second order statistic. A. 25 B. E C. E D. 17 E. 2
36 36 6 9 4 ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  437  SECTION 16
22. Let events A and B be independent. What is the probability, in terms of Pr(A) and Pr(B), that exactly one of the events A and B occurs? A. Pr(A)+Pr(B)—2Pr(A).Pr(B) B. Pr(A)+Pr(B)—Pr(A)Pr(B)
C. Pr(A)+Pr(B) D. Pr(A)Pr(B) E. min(Pr(A),Pr(B)) ‘ 23. If a fair coin is tossed repeatedly, what is the probability that the third head occurs on the nth
toss? A (IIlie? B. (n:)(n—2)[%Jm c. (nn;1)(n2)‘(%]n
D. (n— 3.8.] E. 24. Let X have the Poisson distribution with mean 2 = 1. What is the probability that X 2 2,
given that X S 4? A.3 Bi CE D 17 . _ E. 3
65 41 65 41 5 25. You are given two random variables X and Y such that the conditional density of Y given that
X: x is f}, (yX = x) =1 for x < y < x +1 and zero otherwise. You are also given that fx (x) = e“ for x > 0 and is zero otherwise. Find Pr(X +Y < 2). A. 0.38 B. 0.42 C. 0.48 D. 0.52 E. 0.56 26. If X has a normal distribution with mean 1 and variance 4, then Pr(X2  2X 5 8) = ? A.0.13 B.0.43 C.0.75 D.0.86 E.0.93 27. LetX and ch random variables with Var(X) = 4, Var(Y) = 9, and Var(X— Y) = 16. What
is Cov(X,Y)?
3 1 1 3 E 13 .— .—— C.— D.— .—
A 2 B 2 2 2 16 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszcwski  438  PRACTICE EXAM 12
28. For a random sample of size 4 from the exponential distribution with mean 2 ﬁnd the
expected value of the average of the second and the third order statistics. C2 A .
12 B.2 D.2 E.2
6 3 l
' 2 29. Let X and Y be independent random variables each with density f, (t) = % for —0 < t < 9, and 0 otherwise. If Var(XY) = %, then 6 = ? A.1 3.2 C.— 4‘3/3 D. 2J5 E. 93/3 30. Two numbers are chosen independently and at random from the interval (0, 1). What is the probability that the two numbers differ by more than %? Aloe .. A. 1 B. l c. 3 D. l E. 8 4 8 2 w ASM Study Manual for Course Pl] Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  439  ...
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