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Unformatted text preview: PRACTICE EXAMINATION NUMBER 9 1. Let X be the random outcome of tossing a fair sixfaced die, and Y0) = rnin(Xl ,X2 ,X3) , Ym
(where Y“) S Ya) S 113)), and Y“) = max(Xl ,X2 ,X,) be the order statistics from a random sample
Xl ,X2,X3 from the distribution of X. Find Pro/(l) = 3). A, i B, 3—7 C_ i 13.2 E, i
216 216 216 216 216 2. It has been determined that the probability distribution of the loss X for an automobile insurance policy has the density fx (x) = ge" + gem, where x > 0 is the amount of the loss in thousands of dollars. Find the probability that the actual loss is more than twice the expected
value of the loss. A. 0.05 B. 0.10 C. 0.14 D. 0.20 E. 0.25 3. Let X and Ybe independent Poisson random variables with Var(X) = 112 and Var(Y) = 42.2 .
What is Pr(X+Y 2 2)? 4
A. l—e’S": (1+523) B. 1e'“’ [1+5].2 + 25)” J c. l—e'3"(l+32.)
2 2
D.1—e“[1+3,1+%] E. 1—e31[1+32+25’1] 4. Let Y1 ,Y2 ,Y3,Y4 ,Ys be the order statistics of a random sample of size 5 from a distribution having PDF fx (x) = e" for x 2 0, and fx (x) = 0 elsewhere. Calculate the probability that
Ys >1. A. Less than 0.55 B. At least 0.55, but less than 0.65
C. At least 0.65, but less than 0.75
D. At least 0.75, but less than 0.85
B. At least 0.85 ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  343  PRACTICE EXAMINATION NO. 9 5. Let X and Y be random variables. The density function of X is 1 .
24x2, for0<x<—,
f (x) = 2
0, otherwise.
The conditional density of Y, given X = x, is
y
— , for 0 < < 2x,
g( y) = 2.7:2 y
0, otherwise. What is the form of the marginal density function of Y where it is nonzero? A. ky,for0<y<1
B. ky(l—y) for0<y<l
C. ky(l2y) for 0<y<% D. ky(2y) for0<y<2
E. ky2 for 0<y<l 6. You are given a random variable X with probability density function fx (x) = xe" for x > 0
and 0 otherwise. Find the ratio of the first and the second moment of this probability distribution. A.2 B. 1.5 C] D E l l
’2 '3 7. You are given the following graph of the cumulative distribution function of a random
variable X: Find the expected value of this random variable. A. l B. 1.1875 C. 1.25 D. 1.375 E. 1.5 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  344  SECTION 13
8. You are given that the joint PDF of random variables X and Y is f“ (x,y) = x + y for 0<x<l, 0<y<1, andOotherwise.Find Var(X—Y). A.l B.l C.1 D E
2 3 l l
'4 '6 9. A cookie jar has 3 red marbles and 1 white marble. A shoebox has 1 red marble and 1 white
marble. Three marbles are chosen at random without replacement from the cookie jar and placed
in the shoebox. Then 2 marbles are chosen at random and without replacement from the
shoebox. What is the probability that both marbles chosen from the shoebox are red? 3 3
10 4 10 8 4 40 10. Claim size X follows a twoparameter Pareto distribution with parameters a and 0, for
which the density and the cumulative distribution function are given below: a9“ 0"
fx(x)=W’ FX(x)=1—(x+9)a' l
A transformed distribution Y is created by Y = X ' . Which of the following is the probability
density function of Y? _ 10y"‘ = a0“ry"' = 9a"
A' fl’ " (y+0)r+l Bf fl’ (yr +6)a+l C' fY (y+9)9+1
we a
D. fy (y) = —9— E fr (y) = a6 11. A Mars probe has two batteries. Once a battery is activated, its future lifetime is exponential
with mean 1 year. The ﬁrst battery is activated when the probe lands on Mars. The second
battery is activated when the ﬁrst fails. Battery lifetimes after activation are independent. The
probe transmits data until both batteries have failed. Calculate the probability that the probe is
transmitting data three years after landing. A. 0.05 B.0.10 C.0.15 D.0.20 E.0.25 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  345  PRACTICE EXAMINATION NO. 9
12. For an insurance: (i) Losses have density function
0.02x, 0 < x <10, fx (x) = {0, elsewhere. (ii) The insurance has an ordinary deductible of 4 per loss.
(iii) Y P is the claim payment per payment random variable.
Calculate E(YP A. 2.9 B. 3.0 C.3.2 D.3.3 E. 3.4 13. The length of time, in years, that a person will remember an actuarial statistic is modeled by
1 an exponential distribution with mean In a certain population, Yhas a gamma distribution
with a = 2 and ﬂ = Calculate the probability that a person drawn at random from this . . . . . 1
population Will remember an actuarial statistic less than 2 year. A. 0.125 B. 0.250 C. 0.500 D. 0.750 E. 0.875 800 )3
x+ 800 ' The annual inﬂation rate is 8%. A franchise deductible of 300 will be implemented in 2006.
Calculate the loss elimination ratio of the franchise deductible. 14. In year 2005, claim amounts have the following Pareto distribution Fx (x) = l—( A. Less than 0.15 B. At least 0.15, but less than 0.20
C. At least 0.20, but less than 0.25
D. At least 0.25, but less than 0.30
B. At least 0.30 15. You are given that X and Y have a joint distribution with density f,” (x, y) = e“"‘”’ for x > 0 and y > 0, and f“, (x,y) = 0 otherwise. A new random variable Z is deﬁned as Z = (WY). Find
the density of Z. , l
C. ze“ D. ln— E. lnz Z ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski o 346  SECTION 13 16. Suppose X and Yhave the joint density function f (x, y) = x + 1—39 y2 for 0 S x S y S 1. What
is E(XY)? a 3.1 I_1 D, 2 E, a 60 3 30 5 90 17. Compute the expected value of the random number of coin tosses until a run of k successive heads occurs when the tosses are independent and each lands on heads with a probability A. 2""1 B. 2" + 2 C. 2'“1 — 2 D. 2"“ E. Does not exist 18. An automobile manufacturing company produces three different car models. The table
below presents sales data and average gasoline consumption data for these three models. What is
the mean miles per gallon (mpg) for the cars sold by the company, assuming each car uses the
same number of allons of asoline? I Number of cars sold m
. 2000 _ I_ 4000
A. 25 B. 21 C. 20 D. 15 E. Cannot be determined from the
given information 19. You are given that for a certain insurance policy, loss X in 2005 followed the Pareto
2000 x + 2000 5% due to inﬂation, and an ordinary deductible of 100 is applied. Find the expected cost per loss
(unconditional) in the year 2006. 2
distribution with the survival function sx (x) =( j for x > 0. For 2006, losses increase A. 2000 B. 2005 C. 2010 D. 2050 E. 2100 20. Let N l and N2 be two independent Poisson random variables with expected values
E(N,)= 2 and MM) = 3. Find Var(Nl Nl + N2 = 5). A. 0.4 B. 1.0 C. 1.2 D. 2.4 E. 4.8 ASM Study Manual for Course Pll Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  347  PRACTICE EXAMINATION N O. 9
21. An um contains 12 black balls and n white balls. Three balls are chosen from the urn at random and without replacement. What is the value of n if the probability is % that all three balls are white? A.4 B.5 C.8 D. 10 E. 12 22. The mean and variance of X are p ¢ 0 and 0'2 > 0, respectively. If the third moment of X
about the mean is —,u3, what is E(X3)? A. [402 B. 302 c. 3i“;2 D. 3p0'2 — 2m E. 3,1(02 — [12) 23. Let X and Y have the joint bivariate normal distribution, where X and Y have common mean 0, common variance 1, and covariance Which of the following is equal to Pr(X + Y 5 J3)? A. 0.11 B.0.16 C.0.84 D.0.89 E.0.96 24. Suppose Y“) S Ym S Y“) denote the order statistics of a random sample X1 ,X2 ,X3 from a uniform distribution over the interval (0, 1). Which of the following is true of the conditional
distribution of Ym, given Y0) = y‘, Y“) = y3? A. It is uniform over the interval (y1 ,y3)
B. It is uniform over the interval (0, 1) 6(3’2 ‘yt)()’3 — 3’2) C. It has density function f ( y2 I yl , y3) = for yl S y2 S y3 (3’3 _ 3’1)2
D. It has density function f(y2 y,,y3) = W157 for yl s y2 s y3
’ 3 l
E. It has density function f(y2 y1 ,y3) = % for y, s y2 s y3
3 1 25. A uniform density function for X over an interval of unit length is such that P[% < X < = What is the lefthand endpoint of that interval of unit length? ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  348  SECTION 13 A. 0 B. D. g E. Cannot be determined from the given information 26. Suppose X} =Zj —Zj_1, wherej: 1,2, ...,n and Z0,Z,,Z2,...,Z,. are independent and identically distributed with common variance 0'2 . What is Var[12X I]? n 1:]
0'2 0' 262 20"
A. — B. 0'2 C. — D. — E. —
n n J; n2 n 27. Suppose X is a binomial random variable based upon n independent trials, with p being the
probability of success on each trial. If Pr(X = n) = 0.00032 and Pr(X = n — l) = 0.0012821 what is p? A. 0.20 B. 0.25 C. 0.40 D. 0.80 E. — 28. An insurance agent will receive a bonus if his loss ratio is less than 70%. You are given:
(i) His loss ratio is calculated as incurred losses divided by earned premium on his block of business. (ii) The agent will receive a percentage of earned premium equal to g of the difference between 70% and his loss ratio. (iii) The agent receives no bonus if his loss ratio is greater than 70%. (iv) His earned premium is 500,000. (v) His incurred losses are distributed according to the Pareto distribution with the cumulative 600,000 3
— for x > 0. Calculate the expected value of his
x + 600,000 distribution function F (x) = 1—(
bonus. A. 16,700 B. 31,500 C. 48,300 D. 50,000 E. 56,600 29.x and Yhave a bivariate normal distribution with E(X) = 0, E(Y) = 1, and E(XY)=1.
If E(YX = 2): 1, and E(XY = 0)=%, ﬁnd Var(YX=—2). ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Osmszewski  349   osc  nsmnmso 30mm)! Kq soozvooz tuﬁuﬂdoa ® 'uoswugumxa Immav I/d asmoo 10; [WWW Rpms wsv OOI 'EI SCI 'CI 0” '3 SI! '8 OZI 'V '003 01 pas'eejou; s; elqponpap sq: J} (spew s; mamﬁed
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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
 Staff

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