TABLE OF CONTENTS. 1. Probability

Transcription

1 TABLE OF CONTENTS. Probability A. Properties of Probability B. Methods of Enumeration: Selection Without Replacement 7 C. Methods of Enumeration: Selection with Replacement 5 D. Conditional Probability E. Independent Events 5 F. Bayes Theorem 9. Discrete Distributions A. Discrete Probabilities 47 B. Density Histograms and Sample Percentiles 5 C. Moments of Discrete Variables 57 D. Skewness 3 E. Moment Generating Functions 7 F. Bernoulli Trials and the Binomial Distribution 7 G. Poisson Distribution 85 H. Negative Binomial Distribution 9 3. Continuous Distributions A. Probability Density and Distribution Functions of Continuous Variables 5 B. Modes of Continuous Distributions 3 C. Percentiles and Medians 7 D. Moments and Expectations of Continuous Distributions 3 E. Distributions of Functions of a Random Variable 39 F. Uniform Distribution 45 G. Exponential and Gamma Distributions 53 H. Normal Distribution 9 I. Normal Approximations for Discrete Distributions 75 J. Lognormal Distribution 8 K. Beta, Pareto, and Weibull Distributions Multivariate Distributions A. Joint Probability Distributions of Two Random Variables 89 B. Marginal Density Functions C. Independent Random Variables 7 D. Conditional Distributions 3 E. Conditional Moments: Discrete Cases 3 F. Conditional Moments: Continuous Cases 39 G. Transformations of Random Variables 47 H. Order Statistics 5 I. Multinomial Distribution 55

2 5. Other Topics A. Covariance 57 B. Correlation Coefficient 7 C. Bivariate Normal Distribution 7 D. Sums of Independent Random Variables 73 E. Chi-Square Distribution 83 F. Inequalities and the Central Limit Theorem 89. Anderson A. Deductibles and Limits 9 B. Inflation and Coinsurance 33 NOTES Questions and parts of some solutions have been taken from material copyrighted by the Casualty Actuarial Society and the Society of Actuaries. They are reproduced in this study manual with the permission of the CAS and SoA solely to aid students studying for the actuarial exams. Some editing of questions has been done. Students may also request past exams directly from both societies. I am very grateful to these organizations for their cooperation and permission to use this material. They are, of course, in no way responsible for the structure or accuracy of the manual. Exam questions are identified by numbers in parentheses at the end of each question. CAS questions have four numbers separated by hyphens: the year of the exam, the number of the exam, the number of the question, and the points assigned. SoA or joint exam questions usually lack the number for points assigned. W indicates a written answer question; for questions of this type, the number of points assigned are also given. A indicates a question from the afternoon part of an exam. MC indicates that a multiple choice question has been converted into a true/false question. Page references refer to Michael A. Bean, Probability: The Science of Uncertainty with Applications to Investments, Insurance, and Engineering (5); Saeed Ghahramani, Fundamentals of Probability with Stochastic Processes, (5); Matthew J. Hassett and Donald G. Stewart, Probability for Risk Management, (999); Robert V. Hogg and Elliot A. Tanis, Probability and Statistical Inference (); Irwin Miller and Marylees Miller, John E. Freund's Mathematical Statistics with Applications (4); Sheldon Ross, A First Course in Probability (); and Dennis D. Wackerly, William Mendenhall III, and Richard Scheaffer, Mathematical Statistics with Applications (). Although I have made a conscientious effort to eliminate mistakes and incorrect answers, I am certain some remain. I am very grateful to students who discovered errors in the past and encourage those of you who find others to bring them to my attention. Please check our web site for corrections subsequent to publication. I would also like to thank Jenny Carlson for doing initial solutions for most of the problems in the manual. Hanover, NH /3/ PJM

3 NOTES Questions and parts of some solutions have been taken from material copyrighted by the Casualty Actuarial Society and the Society of Actuaries. They are reproduced in this study manual with the permission of the CAS and SoA solely to aid students studying for the actuarial exams. Some editing of questions has been done. Students may also request past exams directly from both societies. I am very grateful to these organizations for their cooperation and permission to use this material. They are, of course, in no way responsible for the structure or accuracy of the manual. Exam questions are identified by numbers in parentheses at the end of each question. CAS questions have four numbers separated by hyphens: the year of the exam, the number of the exam, the number of the question, and the points assigned. SoA or joint exam questions usually lack the number for points assigned. W indicates a written answer question; for questions of this type, the number of points assigned are also given. A indicates a question from the afternoon part of an exam. MC indicates that a multiple choice question has been converted into a true/false question. Page references refer to Michael A. Bean, Probability: The Science of Uncertainty with Applications to Investments, Insurance, and Engineering (5); Saeed Ghahramani, Fundamentals of Probability with Stochastic Processes, (5); Matthew J. Hassett and Donald G. Stewart, Probability for Risk Management, (999); Robert V. Hogg and Elliot A. Tanis, Probability and Statistical Inference (); Irwin Miller and Marylees Miller, John E. Freund's Mathematical Statistics with Applications (4); Sheldon Ross, A First Course in Probability (); and Dennis D. Wackerly, William Mendenhall III, and Richard Scheaffer, Mathematical Statistics with Applications (). Although I have made a conscientious effort to eliminate mistakes and incorrect answers, I am certain some remain. I am very grateful to students who discovered errors in the past and encourage those of you who find others to bring them to my attention. Please check our web site for corrections subsequent to publication. I would also like to thank Jenny Carlson for doing initial solutions for most of the problems in the manual. Hanover, NH // PJM

4 Multivariate Distributions 89 PAST CAS AND SoA EXAMINATION QUESTIONS ON MULTIVARIATE DISTRIBUTIONS A. Joint Probability Distributions of Two Random Variables A. Let the joint density function of X and Y be given by the following: x + y for < x < and < y < otherwise What is P(X < Y)? A. 7/3 B. /4 C. 3/4 D. 9/4 E. 7/8 (79 ) A. Suppose that the joint density function of X and Y is uniform over the region R = {(x, y) x + y <, x >, y > }. What is the probability that exactly one of the two events A = {X < } and B = {Y > } occurs? A. / B. /4 C. / D. 5/8 E. 3/4 (79 4) A3. Suppose X and Y have the following joint density function: What is E[XY]? x + (/3)y for x y otherwise A. 9/ B. /3 C. /3 D. /5 E. 3/9 (8F 38) A4. Let the joint density function for X and Y be given by the following: kxy for < x < y < otherwise What is the value of k? A. B. C. 5 D. E. (83S 8) A5. Let X and Y be continuous random variables with the following joint density function: Then k = kx -3 e -y/3 for < x < and < y < otherwise A. (/3)e /3 B. e -/3 C. (/3)e /3 D. (3/)e -/3 E. e /3 (88S 5) ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

5 9 Multivariate Distributions Solutions are based on Bean, pp. 9 35, 4 9, 35 4; Ghahramani, pp. 3 5, 39 75, 378 8; Hassett, pp. 9 7, 74 78, 93 34, 3 9; Hogg, pp. 33; Miller, pp. 9 ; Ross, pp ; and Wackerly pp., 4 49, 55. A. P(X < Y) = x + y dy dx = x/ (xy + y /) x/ dx = x + / 5x /8 dx P(X < Y) = [x / + x/ (5/4)x 3 ] = 9/4 A. The region described by the inequalities is a triangle bounded by the lines x + y =, x =, and y =. This can be broken up into two triangles and a square: The upper triangle has the additional constraints X < and Y >. The lower triangle has the additional constraints X > and Y <. The square has the additional constraints of X < and Y <. Since in the upper triangle both events occur and in the lower triangle neither events occur, neither of these areas are included in the desired probability. In the square, which comprises / of the area, only one of the events occurs (X < ). A3. E[XY] = y xy[x + (/3)y ] dx dy = [x 3 y/3 + (5/3)x y 3 ] y dy = y 4 /3 + (5/3)y 5 dy E[XY] = [y 5 /5 + 5y /8] = /5 + 5/8 = 3/9 Answer: E A4. Since probability must equal one, we get: = y Answer: E kxy dx dy = kx y / dy y = ky 4 / dy = ky 5 / = k/ k = A5. Since the probabilities must integrate to unity, we get: = kx -3 e -y/3 dy dx = 3e -y/3 kx -3 dx = 3e -/3 kx -3 dx = (3/)e -/3 kx - = (3/)e -/3 k k = (/3)e /3 ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

6 Multivariate Distributions 9 A. Let X and Y be continuous random variables with the following joint density function:.5 for x and x y x otherwise What is E[X 3 Y]? A. /5 B. 4/3 C. D. 4 E. 4/5 (9S ) A7. Let X and Y be discrete random variables with the following joint probability function: / for x =,, 3; y =, 3 otherwise If U = X + Y. What is the probability function of U? A. g(u) = /4 for u = 3, 4, 5, ; otherwise B. g(u) = /5 for u = 3; /5 for u = 4, 5, ; otherwise C. g(u) = / for u = 3, ; /3 for u = 4, 5; otherwise D. g(u) = /5 for u =, 3, 4, 5, ; otherwise E. g(u) = /4 for u =, 3, 4, 5; otherwise. (9S 9) A8. Let X and Y be continuous random variables with the following joint density function: Then E[Y] = (x + y) for < x < y < otherwise A. 5/ B. / C. 3/4 D. E. 7/ (9S 5) (Sample 37) A9. Let X and Y be continuous random variables with joint density function f(x, y) and marginal density functions f X and f Y, respectively, that are nonzero only on the interval (, ). Which one of the following statements is always true? A. E[X Y 3 ] = ( C. E[X Y 3 ] = ( E. E[Y 3 ] = x dx)( x f(x, y) dx)( y 3 f(x) dx y 3 dy) B. E[X ] = (9S 38) x f(x, y) dx y 3 f(x, y) dy) D. E[X ] = x f(x) dx A. Let X and Y be continuous random variables with the following joint density function: xy for x and y otherwise What is P(X/ Y X)? A. 3/3 B. /8 C. /4 D. 3/8 E. 3/4 (9S 49) ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

7 9 Multivariate Distributions A. E[X 3 Y] = x.5x 3 y dy dx = x (/8) x 3 y dx x x- dx E[X 3 Y] = /8[x 3 (x ) x 3 (x ) ] dx = x 4 / x 3 / dx E[X 3 Y] = [x 5 / x 4 /8] = ()4 (/5 /8) = /5 Answer: A A7. Construct a joint probability table and sum the probabilities for different values of U: (X,Y) (, ) (, ) (, 3) (3, ) (3, ) (3, 3) U P(X, Y) / / / / / / g(3) = g() = / g(4) = g(5) = /3 A8. E[Y] = y y(x + y) dx dy = x y + xy y dy = 3y 3 dy = 3y 4 /4 = 3/4 A9. E[u(x)] = u(x) f(x) dx E[X ] = x f(x) dx A. P(X/ Y X) = x xy dy dx + x/ xy dy dx = x/ xy / x x/ dx + xy / x/ dx P(X/ Y X) = x 3 / x 3 /8 dx + x/ x 3 /8 dx = 3x 4 /3 + [x /4 x 4 /3] P(X/ Y X) = 3/3 + ( /) (/4 /3) = 3/8 ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

8 Multivariate Distributions 93 A. Let X and Y be discrete random variables with joint probability function y/(4x) for x =,, 4; y =, 4, 8; x y otherwise What is P(X + Y/ 5)? A. /8 B. 7/4 C. 3/8 D. 5/8 E. 7/4 (9S ) A. Let X and X be random variables with joint moment generation function M(t, t ) =.3 + (.)exp(t ) + (.)exp(t ) + (.4)exp(t + t ). What is E[X X ]? A.. B..4 C..8 D..e +.4e E..3 + (.)exp(t ) + (.)exp( t ) + (.4)exp(t t ) (9S ) A3. Let X and Y be discrete random variables with joint probability function given by the following table: (x, y) (, ) (, ) (, ) (, ) (, ) (, ) p(x, y) /5 /5 /5 /5 What is the variance of Y X? A. 4/5 B. /5 C. /5 D. 5/4 E. 7/5 (9S 4) A4. Let X and Y be discrete random variables with the following joint probability function: p(x, y) = (x+-y) /9 for x =,, and y =, otherwise Calculate E[X/Y]. A. 8/9 B. 5/4 C. 4/3 D. 5/8 E. 5/3 (9W 9) A5. Let X and Y be random losses with the following joint density function: e -(x+y) for x > and y > otherwise An insurance policy is written to reimburse (X + Y). Calculate the probability that the reimbursement is less than. A. e - B. e - C. e - D. e - E. e - (Sample 4) ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

9 94 Multivariate Distributions A. The specified outcome consists of the following five pairs: (, ), (, 4), (, 8), (, ), (, 4). Thus we get for a sum of their respective probabilities: P(Outcome) = /4 + 4/4 + 8/4 + /48 + 4/48 = 7/4 Answer: E A. E[X ] = M X '() = (.)exp() + (.4)exp() = E[X ] = M X '() = (.)exp() + (.4)exp() =. E[X X ] = E[X ] E[X ] = ()() (.) =.4 Answer: B A3. E[Y X] = ()() + ()(/5) + ( )(/5) + ()(/5) + ( )(/5) + ( )() = 3/5 E[(Y X) ] = () () + () (/5) + ( ) (/5) + () (/5) + ( ) (/5) + ( ) () = 7/5 Var(Y X) = E[(Y X) ] (E[Y X]) = 7/5 ( 3/5) = /5 A4. E[X/Y] = (/)p(, ) + (/)p(, ) + (/)p(, ) + (/)p(, ) E[X/Y] = ()() +- /9 + (/)() +- /9 + ()() +- /9 + ()() +- /9 = 5/8 A5. P(X + Y < ) = P(Y < X) = x e -(x+y) dy dx = e -(x+y) -x dx = e -x e - dx P(X + Y < ) = [ e -x xe - ] = ( e- e - ) ( ) = e - ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

10 Multivariate Distributions 95 A. Given that future lifetimes (in months) of two components of a machine have the following joint density function, what is the probability that both components are still functioning twenty months from now? (/5,)(5 x y) for < x < 5 y < 5 otherwise A. (/5,) 3 C. (/5,) 5 E. (/5,) (5 x y) dy dx B. (/5,) 5-x-y (5 x y) dy dx D. (/5,) 5-x-y 3 5-x (5 x y) dy dx 5 5-x (5 x y) dy dx ) (F ) (Sample P 89) (5 x y) dy dx A7. An insurance company insures a large number of drivers. Let X be the random variable representing the company s losses under collision insurance and let Y represent the company s losses under liability insurance. X and Y have the following joint density function: (x + y)/4 for < x < and < y < otherwise What is the probability that the total loss is at least? A..33 B..38 C..4 D..7 E..75 (F 3) (Sample P 9) A8. A device runs until either of two components fails, at which point the device stops running. The joint density function of the lifetimes of the two components, both measured in hours, is: f(x, y) = (x + y)/7 < x < 3 and < y < 3 Calculate the probability that the device fails during its first hour of operation. A..4 B..4 C..44 D. 9 E..9 (3S ) (Sample P 78) A9. Let X be the age of an insured automobile involved in an accident. Let Y be the length of time the owner has insured the automobile at the time of the accident. X and Y have joint probability density function: ( xy )/4 for x and y otherwise Calculate the expected age of an insured automobile involved in an accident. A. 4.9 B. 5. C. 5.8 D.. E..4 (3S 4) (Sample P ) A. A device runs until either of two components fails, at which point the device stops running. If the joint density function of the lifetimes of the two components, both measured in hours, is the following, calculate the probability that the device fails during its first hour of operation. f(x, y) = (x + y)/8 < x < 3 and < y < 3 A..5 B..4 C..39 D..5 E..875 (Sample P 77) ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

11 9 Multivariate Distributions A. P(Y >, X > ) = P( < Y < 5 X, < X < 3) 3 P(Y >, X > ) = 5-x P(Y >, X > ) = (/5,) Answer: B A7. P(X + Y ) = P(Y X) = P(X + Y ) = (/4) (/5,)(5 x y) dy dx 3 5-x (5 x y) dy dx (x + y) /4 dy dx = -x (4x + 4 ) [(x + )( x) ( x) /] dx [(x + )y y /]/4 -x dx P(X + Y ) = [(5/)x + 3x + /]/4 dx = [(5/)x 3 + (3/)x + x/]/4 =.7833 A8. P(X < ) = P(Y < ) = 3 P(X < ) = 3 (x + y) /7 dx dy = (/7)(/ + y) dy = (/7)(y/ + y /) 3 3 (/7)(x / + xy) dy = (/7)(3/ + 9/) = /9 P(X < ) P(Y < ) = P(X < ) P(Y < ) = (x + y) /7 dx dy = (/7)(x / + xy) dy (/7)(/ + y) dy = (/7)(y/ + y /) = (/7)(/ + /) = /7 P([X < ] and [Y < ]) = P(X < ) + P(Y < ) P(X < ) P(Y < ) P([X < ] and [Y < ]) = /9 + /9 /7 = /7 =.47 Answer: B A9. E[X] = (x/4)( xy ) dy dx = E[X] = (/4)(5x x /9) 3 (x/4)(y xy 3 /3) dx = = (/4)(5,/9 + 8/9) = 5/9 = 5.78 (x/4)( x/3) dx A. See A8. P(X < ) = P(Y < ) = (/8)(/ + y) dy = (/8)(y/ + y /) = (/8)(/ + 4/) = 3/8 P(X < ) P(Y < ) = (/8)(/ + /) = /8 P([X < ] and [Y < ]) = 3/8 + 3/8 /8 = 5/8 =.5 ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

12 Multivariate Distributions 97 A. A device has two components. The device fails if either component fails. The joint density function of the lifetimes of the components, measured in hours, is f(s, t), where < s < and < t <. What is the probability that the device fails during the first half hour of operation? A. f(s, t) ds dt B. D. f(s, t) ds dt + (Sample P 79) f(s, t) ds dt C. f(s, t) ds dt E. f(s, t) ds dt f(s, t) ds dt + f(s, t) ds dt A. Let T be the time between a car accident and reporting a claim to the insurance company. The T be the time between the report of the claim and payment of the claim. The joint density function of T and T, f(t, t ), is constant over the region < t <, < t <, t + t <, and otherwise. Determine E[T + T ], the expected time between a car accident and payment of the claim. A. 4.9 B. 5. C. 5.7 D.. E..7 (Sample P 94) A3. Let T and T represent the lifetimes in hours of two linked components in an electronic device. The joint density function for T and T is uniform over the region defined by t t L where L is a positive constant. Determine the expected value of the sum of the squares of T and T. A. L /3 B. L / C. L /3 D. 3L /4 E. L (Sample P 97) A4. Let X, X, X 3 be a random sample from a discrete distribution with probability function p() = /3 p() = /3 Determine the moment generating function M(t), of Y = X X X 3. A. 9/7 + 8e t /7 B. + e t C. (/3 + e t /3 ) 3 D. /7 + 8e 3t /7 E. /3 + e 3t /3 (Sample P 98) ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

13 98 Multivariate Distributions A. The probability the device fails in the first half hour is the sum of the probability component S fails in the first half hour and component T fails in the second half hour (case ), the probability component T fails in the first half hour and component S fails in the second half hour (case ), and the probability both fail in the first half hour (case 3). A. F This is the probability both components fail in the first half hour (case 3) but excludes the probabilities of cases and. B. F This is the probability that component S fails in the first half hour (cases and 3). It excludes the probability of case. C. F This is the probability that both components fail in the second half hour. D. F This is the sum of the probability that component T fails in the first half hour (cases and 3) and the probability that component S fails in the first half hour (cases and 3) and thus double counts the probability that both fail in the first half hour (case 3). E. T This is the sum of the probability that Component T fails in the first half hour and component S fails in the second half hour (case ) and the probability that component S fails in the first half hour (cases and 3). Answer: E A. = f(t, t ) = 4 c dt dt + 4 -t c dt dt = 4c + ct dt -t = 4 4 = 4c + c[()( 4) (3 )/] = 34c c = /34 4 E[t ] = 4 t /34 dt dt + 4 -t c( t ) dt = c[t (t /] 4 4 t /34 dt dt = E[t ] = 3t /7 dt + 4 E[t ] = 4/7 + /5 88/5 = 4/5 E[t + t ] = [t ] = ()(4)/5 = A3. = f(t, t ) = L L t t t /34 dt + 4 t t /34 dt -t t ( t )/34 dt = 3t / (5 t )(t )/ 4 L c dt dt = ct dt = ct / L = cl / c = /L t L (/L )(t + t ) dt dt = (/L 3 )(t /3 + t t ) dt t L = E[t t + t ] = L /3 A4. P(Y = ) = (/3) 3 = 8/7 P(Y = ) = P(Y = ) = 8/7 = 9/7 M(t) = P(Y = ) + P(Y = )e t = 9/7 + 8e t /7 Answer: A [8/(3L )]t 3 dt = [/(3L )]t 4 L ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

14 Multivariate Distributions 99 A5. An auto insurance policy will pay for damage to both the policyholder's car and the other driver's car in the event that the policyholder is responsible for an accident. The size of the payment for damage to the policyholder's car (X) has a marginal density function of for < x <. Given X = x, the size of the payment for damage to the other driver's car (Y) has conditional density of for x < y < x +. If the policyholder is responsible for an accident, what is the probability that the payment for damage to the other driver's car will be greater than? A. 3/8 B. / C. 3/4 D. 7/8 E. 5/ (Sample P 9) A. Let X and Y be identically distributed independent random variables such that the moment generating function of (X + Y) is M(t) =.9e -t +.4e -t e t +.9e t < t <. Calculate P[X ]. A..33 B..34 C. D..7 E..7 (Sample P 37) A7. A machine consists of two components, whose lifetimes have the joint density function /5 for x >, y >, and x + y < otherwise The machine operates until both components fail. Calculate the expected operational time of the machine. A..7 B. C. 3.3 D. 5. E..7 (Sample P 38) A8. A client spends X minutes in an insurance agent s waiting room and Y minutes meeting the agent. The joint function of X and Y can be modeled by f(x, y) = 8 e -x/4 e -y/, for x >, y > f(x, y) =, otherwise. Which of the following expressions represents the probability that a client spends less than minutes at the agent s office? A. B. C. D. E x e -x/4 e -y/ dy dx e -x/4 e -y/ dy dx 4 4-x e -x/4 e -y/ dy dx e -x/4 e -y/ dy dx -x 8 (Sample P 44) e -x/4 e -y/ dy dx ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza

15 Multivariate Distributions A5. P(Y > ) = P(Y > ) = x+ x dy dx + dx + x+ dy dx = y dx x+ x + y dx x+ (x + ) dx = + (x / + x) =.875 A. Since X and Y are identical, the given m.g.f. is the square of the m.g.f. of one random variable. Thus we get: M X (t) =.3e -t e t This is the moment generating function of a discrete random variable with probability distribution: p( ) =.3 p() =.4 p() =.3 P[X ] = =.7 Answer: E A7. (, ) (5, 5) (, ) (, ) This diagram describes the area of f(x, y). The lower two triangles contain points where x > y and the upper two triangles describe points where y > x. Since their distributions are uniform, and the area of each set of two triangles equals 5, f(x) = f(y) = /5. The expected operational time when x > y is calculated as follows: 5 x E(X X > Y) = x/5 dy dx + 5 -x 5 x/5 dy dx = x /5 dx + 5 (x x )/5 dx E(X X > Y) = x 3 / (5x x 3 /3)/5 5 E(X X > Y) = {/5}{5/3 + [(5)() (,)/3] [(5)(5) 5/3]} = (/5)(5) = 5 An identical results occurs in the case where y > x. A8. The upper limit for the second integral is the greatest amount of time that can be spent in the meeting. This equals minutes less the amount of time spent in the waiting room, i.e., x. Answer: E ACTEX Publications, Inc. SOA Exam P and CAS Exam Peter J. Murdza