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Unformatted text preview: PRACTICE EXAM 20
PRACTICE EXAMINATION NUMBER 20
SOLUTIONS 1. May 1990 Course 110 Examination, Problem No. 49
Let X and Y be continuous random variables with joint density function xy forOSxSZandOSySl,
fx.r(x’y)= 0 otherwise. What is Pr[§ SY S X)? A. B C 1 ,1 ,1 D2 53
32 8 4 8 4 Solution.
The joint density is positive in the rectangle deﬁned by the conditions 0 S x S 2 and O S y S l, but the region of probability sought is deﬁned by the condition is y S x, as shown below. l 2 The probability sought is calculated as the integral of the joint density over the region between the lines y =25 and y = x, marked in the ﬁgure above, either as ASM Study Manual for Course Pll Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  673  SECTION 24
2. May 1990 Course 110 Examination, Problem No. 50 A coin is twice as likely to turn up tails as heads. If the coin is tossed independently, what is the
probability that the third head occurs on the fifth trial? A.—8— 3.39. c.19 13.31”— E.3
81 243 81 243 5
Solution.
Because this coin is twice as likely to turn up tails as heads, we have Pr(Head) =% and Pr(Tail) = Therefore, Pr(3rd head on 5th toss) = Pr({2 heads in ﬁrst 4 tosses} 0 {Head on 5th toss}) = . 4 12 2 21 8
=Pr(2heads1nﬁrst4tosses)Pr(Headon5thtoss)= 2  3  E We can also express the probability sought in terms of a negative binomial distribution. Let X be the number of tails that are tossed that are tossed until the third head occurs. Then X has a negative binomial distribution with p =% and r = 3, and we are looking for Pr(X = 2). This also gives us Pr<x=2>=[:J(%T(aer Answer A. 3. You are given that the time T until a major engine failure in a new car is exponentially
distributed with mean of 8 years. You are given that if the ﬁrst major engine failure occurs
during the ﬁrst three years of the car’s life, the cost of the repair to the car owner is $0, because
the repair is covered by the manufacturer’s warranty. If the ﬁrst major engine failure occurs at
any later time, the cost to the repair is $8,000 (we disregard any time value of money). Calculate
the expected cost of the ﬁrst major engine repair that is paid by the car owner. A. $5125 B. $5498 C. $5617 D. $8000 E. $43497 Solution. The expected value sought is
3 $0 « Pr(O S T S 3) + $8000 ' Pr(T > 3) = $8000ei = $5498.31.
Answer B. 4. You are given a continuous random variable X, whose density is
ASM Study Manual for Course P/1 Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  674  PRACTICE EXAM 20 f(x)={l—le for—1<x<1, 0 otherwise.
If 21:0_75 is the 75th percentile of this random variable, ﬁnd E(X X > xo_7s A. l—l B.O.5284 C. l—ﬂ D. E E.0.75
Ji 3 3
Solution.
First, we find the 75th percentile of X. We must have
30.1: 30.15 0 x1175
0.75 = fo(x)dx = j (1 x)dx = {(1 (—x))dx+ I (1 —x)dx =
_.. 1 1 o 3=Io.7s = 0.5 + (x—05x2) = 0.5 + xm O.5x:..,5. x=0 Therefore,
0.5x§_.,5 — x0” + 0.25 = 0, resulting in x035 = l i J; Because the 75th percentile must lie between 0 and 1, we have x015, = l J; Furthermore, fx (3‘)
f
l , (x)=———i
“'ng Pr[X>1—J;] for lJ;<x<l, and jixx>l_‘g](x) X = 0 otherwise. This gives us l l
E XX>1—\/T = wa—ll—dx: jxl—xdx=
2 , 1 I l
in]; Pr X>1 E H]; 4
I x=l .
=4 j x (1—x)dx=4[lx21x3] =1—£.
x=l l 3 Answer C. 5. You are given that X and Y are independent, identically distributed, normal random variables
with mean 4 and variance 12. Find Pr(]X—Y <1). 6%“ ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004—2008 by Knysztof Ostaszewski  675  SECTION 24
A. 0.5832 B. 0.5793 C. 0.5000 D. 0.3333 E. 0.1618 Solution.
We have E(X—Y)= E(X)E(Y)= 4—4 =0,
and
Var(X Y) = Var(X)+Var(Y)—2Cov(X,Y) = 12+ 12— o = 24,
and therefore
(X—Y) ~ N(0,Jﬁ) = N(0,2JE).
Let us write Z for a standard normal random variable, and (I) for the CDF of Z. We conclude that pr(lx_yl<1)=Pr(1<XY<l)=Pr(:Jgo ((XZ/gO <%) = Pr[% <z <ﬁ): ¢[ﬁ)—¢(—E§EJ= 1 l l
=<I) —  l—q) —— =24) — 1==2<I) 0.2041 —l=
(2J6) [ (NED [2J6] ‘ )
= 2 '(0.410.5832+0.59 ~0.5793)— 1 = 0.1618. using linear hitupolan'on Answer E. 6. For cell phones produced by a certain manufacturer: (i) Lifetimes follow a Pareto distribution with a >1 and 0 = 4. (ii) The expected lifetime of a cell phone is 2 years. Calculate the probability that the lifetime of a cell phone is at least 6 years. A. 0.04 B. 0.06 C. 0.08 D. 0.10 E. 0.12
Solution.
The expected value of a variable following the Pareto distribution is a6 l, and in this case 9 = 4 and the expected value is 2. This gives a = 3. The probability that the lifetime of a cell
phone exceeds 6 years is 9 a 4 3
6 = — = —— =0.43=0.064.
SA) (6+6) [6+4] 7. The amount of damage X in a car accident is given by the exponential distribution with mean
10,000. The insurance policy covering that damage has a deductible of 500 and a policy limit of ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  676  PRACTICE EXAM 20
100,000. Find the coefﬁcient of variation of the amount paid per payment (i.e., amount paid given that a payment is made by the insurance company).
A. 13.43 B. 13.47 C. 14.25 D. 15.11 E. 16.00 Solution.
Let us write Y for the amount paid per payment. We have Y = (x— 500 500 < x < 100000).
Note that for 0 S y S 99500, s, (y): Pr(Y > y) = Pr(X—500 > yl500 < X <100000)=
Pr({X > 500 + y}n{500 < x <100000}) _ 91(500 + y < X < 100000) _ Pr(500 < x < 100000) Pr(500 < X < 100000)
1 500» 1 m y
_ 10000 e W ' 10000 e W _ em 'em  e"°
1 e_%  l e_% eODS _ eIO
10000 10000 and for y > 99500, 3,, (y) = 0. This implies that )l
0.05 
e , e fr (3')  W
for 0 < y < 100000 and zero otherwise. Based on this, y w 995m ..
eons.e loooo_e l0 E(Y)=I5y(y)dy= I Wdy= )7 e _— o.05 _ 1o " e e )‘=0 "”’°°] _ 99500.24"10 _ _10000 e'°”5 99500e°"° _ . (1 _ 8995 ) _ _
3005 _ eIO eons _ eIO —0.lO
933% z 9995 .2509. e = 10000 —
8 Also, ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  677  SECTION 24
y 005 _ ‘lomo
’ u = 3’2 V = —%
99500 o,os 'm __ _
2 _ 2 e e _ e e _ )— I y .10000(e0.05 e—lo)dy L —
o  eODS . e 10000 (1 =2 (1 dv=———d
u y y 10000(e‘°°5e"°) y ——,—_——1 Inteynﬁonbypam
y “99500 y
2 8095 _ e'm 99500 eoos . e'm
= — — + J. 2y—dy =
y eODS _ 6—10 0 eODS _ eIO
y=0
_.."_.
u _ 2y v _ 10000e4’05 e m
— . elO + eODS _ eIO _
" — eoos _ eIO , '
eons . e‘m
du = Zdy dv= mdy
e  e
———,——_—1
Intcgmtion by pans in the second integral
y=99500 y . e—lO e0.05 _ em 99500 eODS . e_mooo
= ———————+ —20000y— + j 20000———dy=
e—oos _ elO eoos _ e1o 0 8055 _ e—IO
y=0
y=99500
_ 995002 ~e"° 9950020000e‘lo + 2 100002 em” e '°°°° _
eons _ eIO eoos _ eIO eons _ eIO
r=0 e"° 1o'° (—9.952 — 19.9 — 2)+ 2 . 10l0 ~e'°”5
= —— z 18129180349. eoos _ elo
Based on this
Var(Y) = E(Y2) (19(1))2 z 18029275309.
The coefﬁcient of variation is
Var(Y)
E(Y) Answer A. = 13.4337. 8. You are given that X and Y are jointly uniformly in the region deﬁned by the conditions
0<x<2 and x2 <y<4. Find Var(XY=3). ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  678  PRACTICE EXAM 20 1 1
A c. — D. — .
2 J2 12 ul— 1
'4 Solution.
The relationship x2 < y < 4 can be rewritten as x < J; < 2. Therefore, for a fixed value of y, x varies between 0 and J}. As the joint distribution is uniform, the conditional distribution of X given that Y: 3 is also uniform. Being uniform on [0,6] = [Op/3], this conditional
distribution has its variance equal to (iior l
12 4 '
Answer A. 9. Four actuarial students are taking Course P examination at the same time. For two of them, the
probability of passing the examination is 0.6, while for the other two, that probability is 0.9.
Their performances are independent of each other. Find the probability that at least one of them
passes the examination. A. 0.9999 B. 0.9984 C. 0.9084 D. 0.8667 E. 0.8298 Solution.
Let El, E2, E3, and E4 be the events of passing Course P examination by students 1, 2, 3, and 4, respectively. The probability that at least one of them passes the examination is
Pr(E, uEz uE, UE4)=1— Pr((E, UEZ uE, UE4)C)=
=1— Pr(E,cr1Efn Ef n Ef)=1Pr(Ef)Pr(Ef)Pr(E§)Pr(Ef)= =1—(10.6)(1—0.6)(1—0.9)(l—0.9)= 0.9984.
Answer B. 10. The joint density of X and Y is
«(yx)
xe forOSxSlandy>x,
fxy (xvy) = .
0 othermse. Find Pr(Y > E(YX = x)X = x).
A. 0.25 B. e" c. 0.75 D. (1+ e)" E. 2(1+ e)" Solution.
Note that
ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  679  SECTION 24 2 +03 M x
fx (x) = l “""""dy = l xe"""’dy = xe" I e“"’dy = — xe (3'8?
ally x x x
so that X is uniform on the interval [0, 1], Based on this f)’ = X) = fx'y (x’y) = xe'XUx) yHeo _2
)=exze x =1,
i=3 fx ('7‘)
for y > x. Furthermore
+°° ‘“ u  v  le"’
E(YX = x) = Ixye‘("’)dy = Jcex2 Iye'”dy = y _ x =
x X du = dy dv = e'wdy
\__v____—J
integrationbypam
y—Hw m
= xe“2 ~ (—ze’”] + are“2  Ilery =
x F: x x
)H«7 _xz
=xe‘1e"2+£e‘2(—le"’) =x+e‘2£——=x+l.
x x y“ x x
This results in
My > E(YX= x)x=x)= Pr[[Y >x+l X “D:
x
+0 1_x
= I xeXUxldy = x .(_l)e‘x(f'1) P": = ( x ] = 31
1 x F"; X+
1 Answer B. 11. Study Note P0908, Problem No. 131
Let N1 and N2 represent the numbers of claims submitted to a life insurance company in April and May, respectively. The joint probability function of N1 and N2 is Ill—l
N,.N1 ’ 2 — 0 otherwise. Calculate the expected number of claims that will be submitted to the company in May if exactly
2 claims were submitted in April. A. 3(e2 l) B. 3e2 C. i D. e2 1 E. e2
16 16 4—e
Solution. We begin by ﬁnding the conditional probability function of N2 given N I = nI , i.e., ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Ktzysztof Ostaszewski  680  PRACTICE EXAM 20 AW(MMJ
Nl = =.._'_L—.
fN, "1) fN‘ (n!)
We have
~ 3 1”ﬂ ” _  M4
fN. ("1): mew, ("19”2)='Z[Z) ' 2e "l (l—e "1) =
"2:1 '11:! geometric series with ratio l:"" and ﬁrst term 2"“ 3(1)“
‘4 4 1—0—ew)'4 4 ' Therefore
fN N ("1312) _ _ nzl
f1n2N=nl =—'*—’—=e"‘l—e"' .
N ( I l ) le (n1) ( )
This is actually the probability function of a geometric random variable with parameter p = e'"'.
The mean of this distribution is i = 117 = e"' , and that equals e2 when n1 = 2.
P e ‘ Answer E. 12. Study Note P0908, Problem No. 132 A store has 80 modems in its inventory, 30 coming from Source A and the remainder from
Source B. Of the modems from Source A, 20% are defective. Of the modems from Source B,
8% are defective. Calculate the probability that exactly two out of a random sample of ﬁve
modems from the store’s inventory are defective. A. 0.010 B. 0.078 C. 0.102 D. 0.105 E. 0.125 Solution:
The number of defective modems is 20%  30 + 8%  50 = 10. The probability that exactly two of
a random sample of ﬁve are defective is (85°) Answer C. 13. Study Note P0908, Problem No. 133
A man purchases a life insurance policy on his 40th birthday. The policy will pay 5000 only if he dies before his 50th birthday and will pay 0 otherwise. The length of lifetime, in years, of a
male born the same year as the insured has the cumulative distribution function ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  681  SECTION 24 0 for t S 0
[31(1): 11.1' l em for t > 0.
Calculate the expected payment to the man under this policy. A. 333 B. 348 C. 421 D. 549 E. 574
Solution. H! "
We have Pr(Man dies before age 50) = Pr(T < 50T > 40) = w = Pr(T < 40)
( ) ( ) ll.l‘° l—l.l’° (I I” n”)
F50F40 e‘°°°e'°°° ''
=W=T=l—e woo e IMO Therefore, the expected beneﬁt is
5000 Pr(Man dies before age 50) = 5000 00696 = 347.96. Answer B. 14. Study Note P0908, Problem No. 134 A mattress store sells only king, queen and twinsize mattresses. Sales records at the store
indicate that onefourth as many queensize mattresses are sold as king and twinsize mattresses
combined. Records also indicate that three times as many kingsize mattresses are sold as twin
size mattresses. Calculate the probability that the next mattress sold is either king or queensize. A.0.12 B.0.15 C.0.80 D.0.85 E. 0.95 Solution.
Let t denote the relative frequency at which twinsized mattresses are sold. We know that the relative frequency at which kingsized mattresses are sold is 3t and the relative frequency at
3t +t which queensized mattresses are sold is = t. Since the sum of all relative frequencies must be 1, we have 1+ 3t + t: 1, and therefore t = 0.2. The probability we seek is 3t + t = 0.80.
Answer C. 15. Fall 1979 Part 2 Examination, Problem No. 7 Sixty percent of new drivers have had driver education. During their ﬁrst year, new drivers
without driver education have probability 0.08 of having an accident, but new drivers with driver
education have only a 0.05 probability of an accident. What is the probability a new driver has
had driver education, given that the driver has had no accidents the ﬁrst year? ASM Study Manual for Course P/l Actuarial Examination. ‘9 Copyright 20042008 by Knysztof Ostaszewski  682  PRACTICE EXAM 20 A. 0.05 0.6 . 0.92.0.4 C. 0.95 0.4 +092 .04
0.05 +0.6 0.6 +0.920.4 0.95 ~0.6
0.95  0.4 0.95  0.6 D. —————— E. ——
0.95  0.4 + 0.92  0.4 +0.92  0.6 0.95 0.6 + 0.92  0.4 Solution.
Let E be the event of a driver having had driver’s education, and let A be the event of having an accident. We are given that Pr(E) = 0.60, and that Pr(AE) = 0.05 and Pr(AEc) = 0.08. Using
the Bayes’ Theorem, we get Pr(A°E)  Pr(E)
Pr(AcE) . Pr(E)+ Pr(ACEC)Pr(Ec)
_ (1 — Pr(AE))~Pr(E) _ 0.95 060
_ (1 Pr(AE))Pr(E)+(l — Pr(AE‘)). 13456) ' 0.95 . 0.60 +0.92  0.40 ' Pr(EAC) = Answer E. 16. May 1979 Part 2 Examination, Problem No. 2
Suppose the random variable X has mean [.1 and variance 0'2 > 0. What are the values of the
numbers a and b such that a + bX has mean zero and variance 1? A.a=—£ and b=i
o 0'
13.a=—i anal):ﬂ
a a C. a=—[.t and b=0'
D.a=/1 and 12:0 E. a=/,l and b=i
0' Solution.
We have E(a+bX)=a+bE(X)=a+bp=0
so that p = —g. Also, Var(a+bX)= bZVar(X) =b20'2 =1
1 :1 sothatb=— and a=.
O' 0' Answer A. ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  683  SECTION 24
17. Study Note P0908, Problem No. 135 The number of workplace injuries, N, occurring in a factory on any given day is Poisson
distributed with mean ,1. The parameter ,1 is a random variable that is determined by the level
of activity in the factory, and is uniformly distributed on the interval [0, 3]. Calculate Var(N). A. it B. 2 )1 C. 0.75 D. 1.50 E. 2.25 Solution.
We have Var(N) = E(Var(1v,t))+ Var(E(N/l)) = E(A)+ Var(zi.) = Answer E. 18. A dental insurance policy provides reimbursements for up to three ﬁllings and one crown per
policy period. The numbers of ﬁllings needed by the insured in a policy period follows a Poisson
distribution with mean 0.7, and is independent of the number of crowns needed by the insured,
which in turn follows a Poisson distribution with mean 0.2. Calculate the probability that there is
maximum possible reimbursement under this policy in a single policy period. A. 0.0046 B. 0.0062 C. 0.0279 D. 0.0332 E. 0.0569 Solution. Let N be the number of ﬁllings needed by the insured in a single policy period, and M be the
number of crowns needed by the insured in the same single policy period. Note that the random
variables N and M are independent. The reimbursement for under this policy is maximized if the
insured simultaneously needs at least three ﬁllings and at least one crown, i.e., if N 2 3 and M 21. The probability of that is Pr({1‘123}o{M 21})=Pr(N23)Pr(M 21)=
=(1'P1'(N=0)PI(N=l)—Pr(N=2))(l—Pr(M=O))= =[l_e.u,0.7°_e..1,ﬂ_e.o.7.o.7z]. 1_e..2,0.7°]= o! 1! 2! 0!
= (1 e'°'7 0.7e'°'7 —0.245e‘°‘7)(1—e'°2)= 0.0062. 19. 1986 Casualty Actuarial Society Part 4 Examination, Problem No. 31 Which of the following statements are true about the distribution of a random variable X?
1. If X is discrete, the value of X, which occurs most frequently is the mode. 2. If X is continuous, the expected value of X is equal to the mode of X. ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  684  PRACTICE EXAM 20
3.The median of X is the value of X , which divides the distribution in half. A.I B.l,2 C.l,3 D.2,3 E.1,2,3 Solution.
Statement 1 is true, as this is the deﬁnition of the mode for a discrete distribution. Statement 2 is not true: for example, exponential distribution’s mode is 0, while its mean is positive. Statement
3 is a form of the deﬁnition of the median, so it is true.
Answer C. 20. November 1986 Course 110 Examination, Problem No. 13
The following table re resents the relative fre uenc of accidents er day in a city: EI
mm Which of the following statements are true? I. The mean and modal number of accidents are equal.
II. The mean and median number of accidents are equal.
III. The median and modal number of accidents are equal. A. I B. II C. 111 D. I, II, IH E. None of these answers are correct Solution.
The mean of the distribution is greater than 0055 +10.20+2 0.10+ 30.10 = 0.70.
The mode is zero. The median is also zero because Pr(X = 0) = 0.55. Thus only the median and the mode are equal.
Answer C. 21. 1988 Casualty Actuarial Society Part 4 Examination, Problem No. 31
The following table represents data observed for a certain class of insureds. The regional claims ofﬁce is being set u  to service a grou  of 10,000 olicyholders from this class. Number oflaims (n) ' ' El Probabili of a olicyholder makin n claims in a car If each claims examiner can service a maximum of 500 claims in a year, and you want to staff
the ofﬁce so that you can handle a number of claims equal to two standard deviations more than
the mean, how many examiners do you need? A.S5 B.6 C.7 D.8 E.29 Solution.
Let X be the random individual claim. The mean of that distribution is ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Oslaszewski  685  SECTION 24 00.84+10.07+20.05+30.04 =0.29.
The second moment is E(X2) = o2 084 +12 .007 + 22 .0.05 + 31004 = 0.63.
The variance is
Var(X) = E(X2) (13(X))2 = 0.63 — 0.292 = 0.5459.
For 10000 policyholders, the aggregate claim mean is 10000  0.29 = 2900, and the aggregate variance is 10000 05459 = 5459. The aggregate standard deviation is J5459 = 73.89. The
needed number of examiners is Aggregate Mean + 2  (Aggregate standard deviation) _ 2900 + 2  73.89 H 500 500
Because the number of examiners must be an integer, the smallest number is 7.
Answer C. 6.1. 22. February 1996 Course 110 Examination, Problem No. 25
The sum of the sample mean and median for ten distinct data points is equal to 20. The largest data point is equal to 15. Calculate the sum of the sample mean and median if the largest data
point were replaced by 25. A. 20 B.21 C. 22 D. 30 E.31 Solution
Since both the current largest value and 25 are greater than the median, replacing one by the other does not change the median. But increasing the largest value by 10 increases the sample mean by —10— = 1, and increases the sum of the sample mean and median by 1. Number of points Thus the new sum equals 20 + 1 = 21.
Answer B. 23. Let N be the number of rolls of a fair die before getting a l, and let M be the number of rolls
before getting the ﬁrst odd number. Find E (N  M = 4) . A.6 3% CS DE Solution.
The event {M = 4} means that the die was rolled four times before the first odd number appeared, and that ﬁrst odd number appeared on the ﬁfth roll. Given that, l was rolled on the
ﬁfth ro...
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