ACTEX Seminar Exam P

Transcription

1 ACTEX Seminar Exam P Written & Presented by Matt Hassett, ASA, PhD 1 Remember: 1 This is a review seminar. It assumes that you have already studied probability. This is an actuarial exam seminar. We will focus more on problem solving than proofs. 3 This is an eight hour seminar. You may want to study more material.

2 Other Study Materials: Probability for Risk Management (Text and Solutions manual) Matt Hassett & Donald Stewart ACTEX Publications ACTEX Study Guide (SOA Exam P/CAS Exam 1) Sam Broverman ACTEX Publications 3 Exam Strategy: 1 Maximize the number of questions answered correctly. Do the easier problems first. 3 Don t spend too much time on one question. 4

3 Points to Remember: A TV screen holds less content than a blackboard; use your handout pages for overview. Algebra and calculus skills are assumed and required. Expect multiple skill problems. Calculators? 5 Probability Rules: Negation Rule: PE ( ) = P(~ E) = 1- PE ( ) Disjunction Rule: PA ( B ) = PA ( ) + PB ( )- PA ( B ) 6

4 Probability Rules, cont.: Definition: Two events A and B are called mutually exclusive if A B= Addition Rule for Mutually Exclusive Events: If A B=, P(A B) = P(A) + P(B) 7 Exercise: The probability that a visit to a primary care physician s (PCP) office results in neither lab work nor referral to a specialist is 35%. Of those coming to a PCP s office, 30% are referred to specialists and 40% require lab work. Determine the probability that a visit to a PCP s office results in both lab work and referral to a specialist. (A) 0.05 (B) 0.1 (C) 0.18 (D) 0.5 (E)

5 Solution: Let L be lab work and S be a visit to a specialist. ( ) = 0.35 = 1 ( ) P L S P L S ( S) = 0.65 PL PS ( ) = 0.30 and PL ( ) = 0.40 ( ) = 0.65 = ( ) + ( ) ( ) = PL ( S) ( S) 0.05 PL S PL PS PL S PL = Answer A 9 Venn Diagrams Can Help: You are given: ( B) = 0.7 and P( A B ) = 0.9. P A Determine PA [ ]. (A) 0. (B) 0.3 (C) 0.4 (D) 0.6 (E)

6 Venn Diagrams Can Help: A 0.10 B Unshaded region P A B = ( ) 0.9. Area of the shaded region must be 0.10 The total area of the two circles represents: P A B = ( ) 0.7 Subtracting the area of the shaded region: PA ( ) = = 0.6 Answer D 11 A More Complicated Venn Diagram: An insurance company has 10,000 policyholders. Each policyholder is classified as young/old; male/female; and married/single. Of these, 3,000 are young, 4,600 are male, and 7,000 are married. They can also be classified as 1,30 young males, 3,010 married males, and 1,400 young married persons. 600 are young married males. How many policyholders are young, female, and single? 1 (A) 80 (B) 43 (C) 486 (D) 880 (E) 896

7 A More Complicated Venn Diagram: Y=3, M Y = young M = male H = married H 13 A More Complicated Venn Diagram: Y=3, M Y = young H M = male 3, = 880 H = married Answer D 14

8 Some Problems are Trickier: An insurer offers a health plan to the employees of a company. As part of this plan, each employee may choose exactly two of the supplementary coverages A, B, and C, or may choose no supplementary coverage. The proportions of the employees that choose coverages A, B, and C are 1/4, 1/3, and 5/1, respectively. Determine the probability that a randomly chosen employee will choose no coverage. 15 (A) 0 (B) 47/144 (C) 1/ (D) 97/144 (E) 7/9 Trickier Problem, cont.: A 0 x y 0 z 0 B Find ( ) ( x y z) P A B C = C 0 This is a linear system for x, y, z. 16

9 Trickier Problem, cont.: A B 1 PA ( ) = x+ y+ 0= 0 y x z PB () = 0+ y+ z= C PC ( ) = x+ 0+ z= 1 Solution: x = /1, y = 1/1, z = 3/1 6 1 P ( A B C) = 1 ( x+ y+ z) = 1 = 1 17 Answer C More Probability Rules: Conditional probability by counting for equally likely outcomes n A PA ( B) = ( B) nb ( ) Definition: For any two events A and B, the conditional probability of A given B is defined by PA ( B) = ( B) PB ( ) P A 18

10 More Probability Rules: Multiplication Rule for Probability = PA ( B) P A B P B ( ) ( ) 19 Exercise: A researcher examines the medical records of 937 men and finds that 10 of the men died from causes related to heart disease. 31 of the 937 men had at least one parent who suffered from heart disease, and, of these 31 men, 10 died from causes related to heart disease. 0

11 Exercise, cont.: Find the probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease. (A) (B) (C) 0.4 (D) 0.37 (E) Solution: A: 31 H: 10 A = At least one parent with heart disease H = Died of causes related to heart disease Find ( A) PH = ( A) n( A) nh 937

12 Solution: A: 31 H: A = At least one parent with heart disease H = Died of causes related to heart disease n( A ) = 31 n( ~ A ) = = 65 n( A H) = 10 nh ( A) = nh ( ) 10 = Solution: A: 31 H: A = At least one parent with heart disease H = Died of causes related to heart disease ( A) n( A) Answer B 937 nh 108 PH ( A) = = =

13 A Harder Conditional Problem: An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman in the population has only this risk factor (and no others). For any two of the three factors, the probability is 0.1 that she has exactly these two risk factors (but not the other). The probability that a woman has all three risk factors, given that she has A 5 and B, is 1/3. A Harder Conditional Problem, cont.: What is the probability that a woman has none of the three risk factors, given that she does not have risk factor A? (A) 0.80 (B) (C) (D) (E)

14 DeMorgan s Laws: ( ) ( ) A B= A B A B= A B 7 Harder Problem Solution: We want to find P( A B C A) = A B x C P( A B C) P ( A) P[ ( A B C)] = P ( A) But, ( ) P A B C = x is not given. 8

15 Harder Problem Solution, cont.: Fill in 0.1 in each of the areas representing exactly two risk factors, and fill in 0.10 in each of the areas representing exactly one risk factor. A x.1.10 B C.10 9 Harder Problem Solution, cont.: Probability of a woman having all three risk factors given that she has A and B is 1/3. A.10.1 x.1 C B ( ) P( A B C) = = P( A B) ( ) = ( ) = P A B C A B P A B C x 1 3 P A B x x 1 = x = 0.06 x

16 Harder Problem Solution, cont.: ( ) = 0.40 P( A) = 0.60 ( ) ( ) ( ) P A = P A B C = = 0.7 A B ( ) P A B C = = 0.8 C Harder Problem Solution, cont.: P( A B C A) = A.10 C B = = P[ ( A B C)] P ( A) Answer C 3

17 More Probability Rules: Definition: Two events A and B, are independent if PA ( B) = PA ( ) Multiplication Rule for Independent Events If A and B, are independent, P A B = P A P B ( ) ( ) ( ) 33 Exercise: An actuary studying insurance preferences makes the following conclusions: (i) A car owner is twice as likely to purchase collision coverage as disability coverage. (ii) The event that a car owner purchases collision coverage is independent of the event that he or she purchases disability coverage. (iii) The probability that a car owner purchases both collision and disability coverages is

18 Exercise, cont.: What is the probability that an automobile owner purchases neither collision nor disability coverage? (A) 0.18 (B) 0.33 (C) 0.48 (D) 0.67 (E) Solution: Let C be collision insurance and D be disability insurance. We need to find P ( C D) = 1 P( C D). i) PC ( ) = PD ( ) ii) P( C D) = P( C) P( D) iii) PC ( D) =

19 Solution, cont.: ( ) 0.15 = P C D = P( C) P( D) = P( D) ( ) ( ) PD = PD = PC ( ) = PD ( ) = P( C D) = P( C) + P( D) P( C D) = = 0.67 ( ) ( ) P C D = 1 P C D = = Answer B Bayes Theorem -- Simplify with Trees: A blood test indicates the presence of a particular disease 95% of the time when the disease is actually present. The same test indicates the presence of the disease 0.5% of the time when the disease is not present. 1% of the population actually has the disease. Calculate the probability that a person has the disease given that the test indicates the presence of the disease. (A)0.34 (B) (C) (D) (E) 0.995

20 Solution: D = Person has the disease T = Test indicates the disease We need to find ( ) P D T = ( T) PT ( ) P D 39 Solution, cont.:.95 T.01 D.05 ~T.99 ~D.005 T.995 ~T = P( D T) ( ) = P ~ D T ( ) P D T = ( T) PT ( ) P D = = Answer B

21 Probability Rules: Law of Total Probability: Let E be an event. If A1, A, An partition the sample space, then P E = P A1 E + P A E P An E. ( ) ( ) ( ) ( ) 41 Theorem: Bayes Theorem: Let E be an event. If A1, A, An partition the sample space, then P( E A1) P( A1 E) = P( E) P( A1) P( A1 E) = P A P A E + + P A P A E ( 1) ( 1 ) ( n) ( n ) 4

22 Exercise: An insurance company issues life insurance policies in three separate categories: standard, preferred, and ultra-preferred. Of the company s policyholders, 50% are standard, 40% are preferred, and 10% are ultrapreferred. Each standard policyholder has probability of dying in the next year, each preferred policyholder has probability of dying in the next year, and each ultrapreferred policyholder has probability of 43 dying in the next year. Exercise, cont.: A policyholder dies in the next year. What is the probability that the deceased policyholder was ultra preferred? (A) (B) (C) (D) (E)

23 Solution.5.4 S P D.005 D.00 ( ) PU D U D.0001 = ( D) P( D) PU.0001 = = Answer D Exercise: The probability that a randomly chosen male has a circulation problem is 0.5. Males who have a circulation problem are twice as likely to be smokers as those who do not have a circulation problem. What is the conditional probability that a male has a circulation problem, given that he a smoker? (A) 1/4 (B) 1/3 (C) /5 (D) 1/ (E) /3 46

24 Solution: C = Circulatory problem S = Smoker We need to find P( C S). We do not know x = P( S ~ C). ( ) We do know that x = P S C since those who have a circulation problem are twice as likely to be smokers. 47 Solution, cont.:.5 C x S.5x.75 ~C x S.75x ( ) P C S = ( S) P( S) P C =.5 x x+.75x = 1.5 = Answer C 48

25 Expected Value: Definition: The expected value of X is defined by E( X) = xp( x) The expected value is also referred to as the mean of the random variable X and denoted by Greek letter μ. E( x ) = μ. A Property of Expected Value: E( ax+ b) = ae( X) + b 49 Variance: Definition: The variance of a random variable X is V( X) = E ( X μ) = ( x ) p( x μ ) Standard Deviation: σ = V( X). Notation: V( X ) = σ ( ) ( ) ( ) ( ) V X = E X E X = E X μ ( + ) = ( ) V ax b a V X 50

26 Exercise: A probability distribution of claim sizes is given in this table: Claim Size Probability Exercise, cont.: What percentage of the claims are within one standard deviation of the mean claim size? (A)45% (B) 55% (C) 68% (D) 85% (E) 100% 5

27 Solution: Claim Size Probability xp(x) x p(x) Total Solution, cont.: ( ) E( X ) ( ) xp( x) E X σ = = 55 = V X = = μ = = 475 σ = 475 = 1.8 A value is within one standard deviation of the mean if it is in the interval [ μ σ, μ + σ], that is, in the interval 33., [ ]

28 Solution, cont.: The values of x in this interval are 40, 50, 60, and Thus, the probability of being within one standard deviation of the mean is: p(40) + p(50) + p(60) + p(70) = = Z-Score: Definition: For any possible value x of a random variable, x - μ z = σ The z score measures the distance of x from E X = μ in standard deviation units. ( ) 56

29 Theorem: Chebychev s Theorem: For any random variable X, the probability that X is within k standard deviations of the 1 mean is at least 1. k P( μ - kσ X μ + kσ) 1-1 k 57 Additional Properties of V(X): V( X+ Y) = V( X) + V( Y) + cov ( X, Y) For X, Y independent V( X+ Y) = V( X) + V( Y) 58

30 Exercise: The profit for a new product is given by Z = 3X Y 5. X and Y are independent random variables with V(X) = 1 and V(Y) =. What is the variance of Z? (A) 1 (B) 5 (C) 7 (D) 11 (E) Solution: ( ) ( ) 3 ( ) 3 VZ ( ) = V 3X Y 5 = V 3X Y ( ) ( ) ( ) = V X+ Y = V X + V Y V( X) ( ) V( Y) () = independence = 91+ = 11 Answer D Note! Observe the wrong answer which you would obtain if you mistakenly wrote V(3X- Y) = V(3 X) - V( Y). This is choice C, and is a common careless mistake. 60

31 Exercise: A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 00 with a variance of 60. If a tax of 0% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 0% more expensive), what will be the variance of the annual cost of maintaining and repairing a car? (A) 08 (B) 60 (C) 70 (D) 31 (E) Solution:. Let X be the random variable for the present cost, and Y=1.X the random variable for the cost after 0% inflation. We are asked to find V(Y). VY ( ) = V(1. X) = 1. VX ( ) = = ( ) Answer E 6

32 Geometric Series Review: A geometric sequence is of the form 3 n a, ar, ar, ar,..., ar. The sum of the series for r 1 is given by: n+ 1 n 1 r a+ ar+ ar ar = a 1- r The number r is called the ratio. If r <1, we can sum the infinite geometric series: n 1 a+ ar+ ar ar +... = a 1- r 63 Geometric Distribution: ( ) k P X = k = q p, k = 0,1,,3, q p q p ( ) = V( X) = E X where X= the number of failures before the first success in a repeated series of independent success-failure trials with P Success = p. ( ) 64

33 Geometric Distribution Alternative: Here, you are looking at the number of trials needed to get to the first success. In this formulation, you are looking at Y = X+ 1. ( ) k 1 PY k q p k = =, = 1,,3, 1 = = p ( ) VY ( ) EY q p 65 Exercise: In modeling the number of claims filed by an individual under an automobile policy during a three-year period, an actuary makes the simplifying assumption that for all integers n 0, 1 pn+ 1 = pnwhere pn represents the probability 5 that the policyholder files n claims during the period. Under this assumption, what is the probability that a policyholder files more than one claim during the period? 66 (A) 0.04 (B) 0.16 (C) 0.0 (D) 0.80 (E) 0.96

34 Solution: We are not given p 0. Look at the first few terms: p0, p0, p0, p0, Solution, cont.: 3 n = p = p0 p0 1 = p0 = 5 68

35 Solution, cont.: PN ( > 1) = 1 PN ( 1) = 1 + = Answer A Note! The probability distribution has the form of a 1 geometric distribution with q =, so it must be 5 true that p 0 = p = Binomial Distribution: Given n independent, success-failure trials with PS () = p, P F = 1 p= q ( ) n k n-k PX ( = k) = p (1 p) k n p k n-k = () q, k = 0,1,, n k E( X) = np ( ) ( 1 ) V X = np p = npq 70

36 Notation Review: n = n( n ) ( ) ( n) r! Pn (,) r = Cn (,) r = r! n! = r!( n r)! nn ( 1) ( n r+ 1) = r! 10 10! = = =!8! 1 71 Example: Guessing on a 10 question multiple choice quiz with choices A, B, C, D, E. ( ) n = 10, P S =. = p, q = P( X = ) = (.) (.8).30 ( ) = 10 (.) = ( ) ( )( ) E X V X = = 1.6 7

37 Exercise: A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a oneyear period of time. Individual participants in the study drop out before the end of the study with probability 0. (independently of the other participants). What is the probability that at least 9 participants complete the study in one of the two groups, but not in both groups? 73 (A).096 (B).19 (C).35 (D).376 (E).469 Solution: Denote the random variables for the number of participants completing in each group by A and B. We need ( 9& < 9) ( 9& < 9) P A B or B A ( 9& 9) ( 9& 9) ( 9) ( 9) ( 9) ( 9) = P A B< + P B A< = P A P B< + P B P A< Ind The two groups are independent and have identical binomial probability distributions. 74

38 Solution, cont.: A is binomial with n=10 independent trials and probability of completion p=0.8. P( A 9) = P( A = 10) + P( A = 9) = (.) = PA ( < 9) = 1 P A 9 =.64 ( ) ( 9 ) =.376 PB ( < 9) =.64 ( 9) ( < 9) + ( 9) ( < 9).376(.64).376(.64).469 PB PA PB PB PA = + = 75 Answer E Harder Bayes Thrm./Binomial Exercise: A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company X s shipments, 10% of the vials are ineffective. For every other company, % of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective. 76

39 Bayes Thrm./Binomial Exercise, cont.: What is the probability that this shipment came from Company X? (A) 0.10 (B) 0.14 (C) 0.37 (D) 0.63 (E) Solution: X = Shipment came from company X I = Exactly 1 vial out of 30 tested is ineffective We are asked to find P( X I). If the shipment is from company X, the number of defectives in 30 components is a binomial random variable with n=30 and p=0.1. The probability of one defective in a batch of 30 from X is 30 9 PI ( X) = (.1)(.9 ) =

40 Solution, cont.: X = Shipment came from company X I = Exactly 1 vial out of 30 tested is ineffective We are asked to find P( X I). If the shipment isn t from company X, the number of defectives in 30 components is a binomial random variable with n=30 and p= PI ( ~ X) = (.0)(.98 ) = Solution, cont.:. X.141 I.08.8 ~X.334 I.67 ( ) P X I = ( I) PI () P X.08 = = Answer A 80

41 Poisson Distribution: X is Poisson with mean λ. PX ( k), k 1,,3, E( X) ( ) V X k e λ λ = = = k! = λ = λ 81 Example: Accidents occur at an average rate of λ = per month. Let X = the number of accidents in a month. 1 e P( X = 1 ) =.71 1! E X = = ( ) V( X) 8

42 Exercise: An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed? (A) 1/ 3 (B) 1 (C) (D) (E) 4 83 Solution: ( = ) = 3P( X = 4) P X λ λ 4 e λ e λ = 3! 4! 4 4λ = λ λ = Answer D 84

43 Hypergeometric Example: A company has 0 male and 30 female employees. 5 employees are chosen at random for drug testing. What is the probability that 3 males and females are chosen? Solution: Hypergeometric Probabilities: 1 A sample of size n is being taken from a finite population of size N. The population has a subgroup of size that is of interest. r n 3 The random variable of interest is X, the number of members of the subgroup in the sample taken. 86

44 Hypergeometric Probabilities, cont.: N- r r n- k k PX ( = k) =, N n k= 0,, n r E( X) = n N r r N- n V( X) = n 1 N N N Previous Example, cont.: X = number of males chosen in a sample of 5. N = 50 n = 5 r = k k PX ( = k) = EX ( ) = 5 = VX ( ) =

45 Negative Binomial Distribution: A series of independent trials has P(S) = p on each trial. Let X be the number of failures before success r. r+ k-1 k r PX ( = k) = q p, k= 0,1,,3, r -1 rq rq EX ( ) = VX ( ) = p p The special case with r = 1 is the geometric 89 random variable. Example: Play slot machine repeatedly with probability of P S = = p success on each independent play ( ).05. Find the probability of exactly 4 losses (failures) before the second win (success r=). 90

46 Example, cont.: Possible sequences: SFFFFS FSFFFS FFSFFS FFFSFS FFFFSS Single sequence probability : 5 Number of sequences: = 1 Solution: 5 ( ) (.05) (.95) Exercise: A company takes out an insurance policy to cover accidents at its manufacturing plant. The probability that one or more accidents will occur during any given month is 3/5. The number of accidents that occur in any given month is independent of the number of accidents that occur in all other months. 9

47 Exercise, cont.: Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs. (A) 0.01 (B) 0.1 (C) 0.3 (D) 0.9 (E) Solution: This is a negative binomial distribution problem. Success S = month with at least one accident Failure F = month with no accidents. Note that P(S) = p = 3/5. Let X be the number of months with no accidents before the fourth month with at least one accident i.e., the number of failures before the fourth success. X is negative binomial with r = 4 and p=3/5. 94

48 Solution, cont.: We are asked to find P( X 4) = 1 P( X 3) = 1 P X = 0 + P X = 1 + P X = + P X = P( X = 0) = = P 4 3 ( X = 1 ) = = 4 P ( X = 5 3 ) = = 4 3 P 6 3 ( X = 3 ) = = ( ) ( ) ( ) ( ) Solution, cont.: ( ) P X 3 = ( 4) = 1 P( X 3) P X =.7101 = =.8979 Answer D 96

49 Definition of Continuous Distribution: The probability density function of a random variable X is a real valued function satisfying: 1 3 fx ( ) 0 for all x. The total area bounded by the graph of y = f( x) and the x axis is 1. fxdx ( ) = 1 ( ) P a X b is given by the area under y = f( x) between x = a and x = b. b Pa ( X b) = fxdx ( ) a 97 Continuous Distribution Properties: Cumulative Distribution Function F(x) x Fx ( ) = PX ( x) = fudu ( ) Expected Value EX ( ) = xf( x) dx Expected value of a function of a continuous random variable E g( X) = g( x) f( x) dx Mean of Y = ax + b E ax+ b = ae X + b 98 ( ) ( )

50 Continuous Distribution Properties: Variance VX ( ) E[( X- μ) ] ( x- μ) fxdx ( ) = = Alternate Form of Variance Calculation VX ( ) = EX ( ) EX ( ) = EX ( ) μ [ ] Variance of Y = ax + b VaX b avx ( + ) = ( ) 99 Uniform Random Variable on [a, b]: 1, a x b = b a 0, otherwise fx ( ) - a + ( - ) = b = b a 1 EX ( ) VX ( ) 100

51 Exponential Distribution: Random variable T, parameter λ. T is often used to model waiting time, λ = rate. -λt -λt ft ( ) = λe, Ft ( ) = 1- e for t ET () = VT () = λ λ 101 Example: Waiting time for next accident. λ = accidents per month on average. ( ) P 0 T 1 = F(1) = 1 e = = 4 ( ) VT ( ) ET * Exponential waiting time Poisson number of events * 10

52 Useful Exponential Facts: lim x n -ax x e n x = lim = 0, for a > 0. ax x e -ax n! xe dx=, n+ 1 a for a > 0, and n a positive integer. n Exercise: The waiting time for the first claim from a good driver and the waiting time for the claim from a bad driver are independent and follow exponential distributions with 6 years and 3 years, respectively. What is the probability that the first claim from a good driver will be filed within 3 years and the first claim from a bad driver will be filed within years? 104

53 Exercise: (A) (B) (C) (D) (E) e e + e e e e + e e e + e e e + e Solution: Recall, the mean of the exponential is μ = 1/ λ. Thus if you are given the mean (as in this problem), you know that 1/ μ = λ. G: Waiting time for 1st accident for good driver B: Waiting time for 1st accident for bad driver 1 G: λg = FG( x) = 1 e 6 1 B: λb = FB( x) = 1 e 3 x 6 x 3 106

54 Solution, cont.: Find ( B ) P G 3 &. Note that G and B are independent. ( 3& ) = ( 3) ( ) = FG( 3) FB( ) P G B P G P B = 1 e 1 e = 1 e e + e Answer C 107 Exercise: The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company expects that 30% of high-risk drivers will be involved in an accident during the first 50 days of a calendar year. What portion of high-risk drivers are expected to be involved in an accident during the first 80 days of a calendar year? 108 (A) 0.15 (B) 0.34 (C) 0.43 (D) 0.57 (E) 0.66

55 Solution: T: time in days until the first accident for a high risk driver To find: PT ( 80) = F(80). We know Ft () = 1 e λt, but we don t know λ. Use the given probability for the first 50 days to find it. 109 Solution, cont.: PT ( 50) = F(50) = 1 = e λ ( ) ln 0.7 λ = 50 Now we have λ and can finish the problem. PT F e 80λ ( 80) = (80) = 1 =.4348 Answer B 110

56 Definitions: The mode of a continuous random variable is the value of x for which the density function f(x) is a maximum. The median m of a continuous random variable X is defined by Fm ( ) = PX ( m) = Let X be a continuous random variable and 0 p 1. The 100p th percentile of X is the number xp defined by Fx ( p) = p. Note that the 50th percentile is the median. 111 Exercise: An insurance policy reimburses dental expense, X, up to a maximum benefit of 50. The probability density function for X is: 0.004x ce for x 0 fx ( ) = 0 otherwise where c is constant. Calculate the median benefit for this policy. (A) 161 (B) 165 (C) 173 (D) 18 (E) 50 11

57 Solution: You can see by direct examination that X must be exponential with c =.004, since.004e is the density function for the exponential with λ =.004. (Some of our students integrated the density function and set the total area under the curve equal to 1, but that takes extra time.) x 113 Solution, cont.: Original expense X: cumulative distribution.004x Fx ( ) = 1 e. Thus the median m for X is obtained by solving the equation ( ) m m Fm e = = 0.50 = e ln (.50) m = = Actual benefit capped at 50. Since is less than 50, 50% of the benefits paid are still less than and 50% are greater. 114 Answer C

58 Normal Random Variable: μ = E( X) and σ = V( X) fx ( ) = 1 πσ ( x-μ ) e σ 115 Transformation to Standard Normal: Transform any normal random variable X with mean μ and standard deviation σ into a standard normal random variable Z with mean 0 and standard deviation 1. X μ 1 μ Z = = X σ σ σ Then probabilities can be calculated using the standard normal probability tables for Z. 116

59 Normal Distribution Table: 117 Standard Normal Example: X normal, μ = 500, σ = 100 P ( 600 X 750) = P Z ( 1 Z 1.5) = P = =

60 Central Limit Theorem: Let X1, X,, Xn be independent random variables all of which have the same probability distribution and thus the same mean μ and variance σ. If n is large, the sum S = X + X + + X 1 n will be approximately normal with mean and variance nσ. n μ 119 Exercise: An insurance company issues 150 vision care insurance policies. The number of claims filed by a policyholder under a vision care insurance policy during one year is a Poisson random variable with mean. Assume the numbers of claims filed by distinct policyholders are independent of one another. What is the approximate probability that there is a total of between 450 and 600 claims during a one-year period? 10 (A) 0.68 (B) 0.8 (C) 0.87 (D) 0.95 (E) 1.00

61 Solution: X i : number of claims on policy i, i=1,, 150. (Poisson) Xi : iid with mean μ = and variance σ =. The total number of claims is S = X1 + + X150 By the central limit theorem, S is approximately normal with ES ( ) = μ = 150() = 500 VS σ s ( ) = σ s = 150() = 500 S = 500 = Solution, cont.: Thus P ( 450 S 600) = P Z ( 1 Z ) = P = =.8191 Answer B 1

62 Normal Distribution Percentiles: The percentiles of the standard normal can be determined from the tables. For example, PZ ( 1.96) =.975 Thus the 97.5 percentile of the Z distribution is Commonly used percentiles of Z: Z P(Z<z) Example: X: normal random variable with mean μ and and standard deviation σ. Find x p the 100p th percentile of X using the 100p th percentile of Z. xp μ z p = xp = μ + zpσ σ For example, if X is a standard test score random variable with mean μ = 500 and standard deviation σ = 100 then the 99 th percentile of X is x.99 = μ + z.99σ = = 73.6 ( ) 14

63 Exercise: A charity receives 05 contributions. Contributions are assumed to be independent and identically distributed with mean 315 and standard deviation 50. Calculate the approximate 90th percentile for the distribution of the total contributions received. (A) 6,38,000 (B) 6,338,000 (C) 6,343,000 (D) 6,784,000 (E) 6,977, Solution: X i : number of contributions i, i=1,, 05. X i : iid with mean μ = 315 and variance σ = ( 50 ). The total contribution is S = X1+ + X05 By the central limit theorem, S is approximately normal with ES ( ) = μ = 315(05) = 6, 38,15 VS σ s ( ) = σ s = 50(05) = 16, 56, 500 S = 16, 56, 500 = 11, 50 16

64 Solution, cont.: Since z.90 = 1.8, the 90th percentile of S is ( ) s.90 = 6, 38, , 50 = 6, 34, Answer C 17 Theorem: If X1, X,, Xn are independent normal random variables with respective means μ 1, μ,, μn and respective variances σ 1, σ,, σ n, then X1+ X + + Xn is normal with mean μ 1+ μ + + μn and variance σ 1 + σ + + σ n. Note that this shows that you don t need large n (as required by the Central Limit Theorem) to have a normal sum. 18

65 Corollary: Let X1, X,, Xn be independent normal random variables all of which have the same probability distribution and thus the same mean μ and variance σ. For any n, the sum S = X1+ X + + Xn will be normal with mean n μ and variance nσ. 19 Corollary applied to sample mean X : Let X1, X,, Xn be iid normal random variables with mean μ and variance σ. The sample mean is defined to be S X Xn X = = n n For any n, the sample mean X will be normal with mean μ and variance σ n. 130

66 Exercise: Claims filed under auto insurance policies follow a normal distribution with mean 19,400 and standard deviation 5,000. What is the probability that the average of 5 randomly selected claims exceeds 0,000? (A) 0.01 (B) 0.15 (C) 0.7 (D) 0.33 (E) Solution: : claim amount on policy i, i=1,, 5. : iid with μ = 19, 400 and variance σ = X i Xi The average of 5 randomly selected claims is S X X5 X = = 5 5 EX ( ) = μ = 19, 400 σ 5000 VX ( ) = = = σ = = X

67 Solution, cont.: 0, , 400 P( 0, 000 < X) = P < Z 1, 000 (.6 Z) = P < =.743 Answer C 133 Exercise: A company manufactures a brand of light bulb with a lifetime in months that is normally distributed with mean 3 and variance 1. A consumer buys a number of these bulbs with the intention of replacing them successively as they burn out. The light bulbs have independent lifetimes. 134

68 Exercise: What is the smallest number of bulbs to be purchased so that the succession of light bulbs produces light for at least 40 months with probability at least 0.977? (A) 14 (B) 16 (C) 0 (D) 40 (E) Solution: X i : lifetime of light bulb i, i=1,, n. Xi : iid with μ = 3 and variance σ = 1. Total lifetime of the succession of n bulbs is S = X1+ X + + Xn ES () = μ = 3n S () VS = σ = n = n () S 1 σ S = n The succession of light bulbs produces light for at least 40 months with probability at least

69 Solution, cont.: S 3n 40 3n.977 = P( S 40) = P n n 40 3n = P Z n Z tables: P( Z ) = n = 3n n 40= 0 n Make the substitution x = n. 3x x 40= 0 x = n = 4 n = Answer B Definition: The pure premium for an insurance is the expected value of the amount paid on the insurance. The amount paid is usually a function of a random variable g(x), so to find pure premiums we use the theorem ( ) ( ) ( ) E g x = g x f x dx 138

70 Insurance with a cap or policy limit: An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder s loss, Y, follows a distribution with density function: y > 1 3 fy () = y 0 otherwise What is the expected value of the benefit paid under the insurance policy? (A) 1.0 (B) 1.3 (C) 1.8 (D) 1.9 (E) Solution: Let B=the random variable for the benefit paid. y, 1< y < 10 B = 10, 10 y 10 EB ( ) = y dy 10 dt y y 1 10 = y 10y = = 1.9 Answer D 140

71 Insurance with a deductible: The owner of an automobile insures it against damage by purchasing an insurance policy with a deductible of 50. In the event that the automobile is damaged, repair costs can be modeled by a uniform random variable on the interval (0, 1500). Determine the standard deviation of the insurance payment in the event that the automobile is damaged. (A) 361 (B) 403 (C) 433 (D) 464 (E) Solution: X: repair cost; Y: amount paid by insurance. Find the standard deviation σ = V( Y). Y Y 0, 0 < x 50 = x 50, 50 < x Density function of X is f(x) = 1/1500 on the interval (0, 1500). 14

72 Solution, cont.: ( ) EY = 0 dx ( x 50) dx ( x 50) = 3000 = Solution, cont.: ( ) EY = 0 dx ( x 50) dx = ( x 50) = 434,

73 Solution, cont.: ( ) = ( ) ( ) VY EY EY = 434, = 16, σ Y = ( ) V Y = 16, = Answer B 145 Exercise: A manufacturer s annual losses follow a distribution with density function.5( 0.6).5 for x > 0.6 fx ( ) = 3.5 x 0 otherwise To cover its losses, the manufacturer purchases an insurance policy with an annual deductible of. What is the mean of the manufacturer s annual losses not paid by the insurance policy? 146 (A) 0.84 (B) 0.88 (C) 0.93 (D) 0.95 (E) 1.00

74 Solution: X: actual cost; Y: part of loss not paid by policy. Find E(Y). Since there is a deductible of, x,.6< x< Y =, x 147 Solution, cont.: ( ) ( ) EY ( ) = x dx+ dx x x ( ) x ( ) x = = = Answer C 148

75 Exercise: An insurance policy is written to cover a loss, X, where X has a uniform distribution on [0, 1000]. At what level must a deductible be set in order for the expected payment to be 5% of what it would be with no deductible? (A) 50 (B) 375 (C) 500 (D) 65 (E) Solution: d: unknown deductible; Y: amount paid by insurance. 0, x< d Y = x d, x d 1 Density function for X: fx ( ) =, 0 x d E ( Y ) 0 = dx + ( x d) dx d ( x d) ( 1000 d) = = d 150

76 Solution, cont.: Find d such that EY ( ) = EX ( ).5. For the uniform X on [0, 1000], and.5e X = 15. ( ) EY ( ) =.5EX ( ) ( 1000 d) ( d) E( X ) = = = 50, 000 d = 500 Answer C 151 Finding the Density Function for Y=g(x): Example: Cost, X, is exponential with λ =.01. After inflation of 5%, the new cost is Y = 1.05 X. Find FY (). y.01x FX ( x) = e ( ) = ( ) = ( 1.05 ) Note that 1. F y P Y y P X Y Y Y Y = P X = FX = 1 y.01 e

77 Example, cont.: Y is exponential with.01 λ = Density function: Useful notation: Y ( ) = Y ( ) f y F y ( ) ( ) Sx ( ) = P X> x = 1 F x 153 Density Function When Inverse Exists: Case 1. g(x) is strictly increasing on the sample space for X. Let h(y) be the inverse function of g(x). The function h(y) will also be strictly increasing. In this case, we can find F Y (y) as follows: F y = PY ( y) = PgX ( ( ) y) Y ( ) ( ( )) ( h() y ) ( h() y ) = P h g X h() y = P X = F X 154

78 Density Function When Inverse Exists: Case. g(x) is strictly decreasing on the sample space for X. Let h(y) be the inverse function of g(x). The function h(y) will also be strictly decreasing. In this case, we can find F Y (y) as follows: F y = PY ( y) = PgX ( ( ) y) Y ( ) ( ( )) ( h() y ) X ( h() y ) = P h g X h() y = P X = S 155 Density Function When Inverse Exists: We can find the density function f Y (y) by differentiating F Y (y). The final result can be written in the same way for both cases: Y ( ) = X( ()) ( ) f y f h y h y 156

79 Exercise: The time, T, that a manufacturing system is out of operation has cumulative distribution function 1 for t > Ft () = t 0 otherwise The resulting cost to the company is Y = T. Determine the density function of Y, for y > 4. (A) 4 (B) 8 (C) 8 (D) 16 (E) 104 3/ 3 5 y y y y y 157 Solution: First find the cumulative distribution function for Y: Y ( ) = ( ) = ( ) = ( ) F y P Y y P T y P T y ( y) 4 = FT = 1 = 1 y y Then the density function for Y is: d d 4 4 fy() y = FY( y) = 1 dy dy = y y Answer A 158

80 Exercise: An investment account earns an annual interest rate R that follows a uniform distribution on the interval (0.04, 0.08). The value of a 10,000 initial investment in this account after one R year is given by V = 10, 000 e. Determine the cumulative distribution function, F(v), of V for values of v that satisfy 0<F(v)< Exercise, cont.: (A) (B) (C) (D) (E) 45 v /10,000 5e 0.04 v /10,000 10,000e 10, 408 v 10, ,833 10, v v 5 ln.04 10,

81 Solution: x a Uniform distribution fact to use here: Fx ( ) =. b a R is uniform on (0.04, 0.08) r.04 FR ( r) =, for 0.04 r Find the cumulative distribution function for V. 161 Solution, cont.: R ( ) = ( ) = ( 10, 000 ) F v P V v P e v v = P R ln 10, 000 v ln.04 v 10, 000 = FR ln = 10, v = 5 ln , 000 Answer E 16

82 Independent Random Variable Results: General results for the minimum or maximum of two independent random variables: Recall that the survival function of a random S () t = P( X > t) = 1 F t. variable X is X X( ) Recall that for X exponential we have F x e λx ( ) 1 x = and S( x) e λ =. 163 Independent Random Variable Results: X and Y independent random variables. Find survival function for Min=min(X, Y): SMin() t = P(min( X, Y) > t) = PX ( > t& Y> t) = PX ( > t) PY ( > t) X () Y() = S t S t independence 164

83 Independent Random Variable Results: X and Y independent random variables. Find cumulative distribution for Max=max(X, Y): FMax() t = P(max( X, Y) t) = PX ( t& Y t) = PX ( t) PY ( t) X () Y() = F t F t independence 165 Exponential Random Variable Results: Minimum of independent exponential random variables: X and Y with parameters β and λ. t t ( )) t S () t S t S t e β λ = = e = e β + λ () ( ) Min X Y Min=min(X, Y) is exponential with parameter β + λ. 166

84 Exercise: In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be independent, exponentially distributed random variables with respective means 1.0, 1.5, and.4. Determine the probability that the maximum of these losses exceeds 3. (A) 0.00 (B) (C) (D) 0.87 (E) Solution: X1, X, X3 : losses due to storm, fire, and theft, respectively. Find PMax [ > 3 ], where Max= max ( X1, X, X3) : F () t = F t F t F t () ( ) ( ) Max X X X 1 3 x x/1.5 x/.4 ( 1 e )( 1 e )( 1 e ) = PMax ( 3) = F Max (3) 3 3/1.5 3/.4 ( e )( e )( e ) = =.586 PMax> [ 3] = =.414 Answer E 168

85 Moments of a Random Variable: Definition: The n th n moment of X is E( X ). Moment Generating Function: Let X be a discrete random variable. The moment generating function MX ( t) is defined by M () t tx E( e ) tx e p( x) X = = 169 Finding the nth moment: Finding the nth moment using the moment generating function: ( n ) n n M (0) = x p( x) = E( X ) X 170

86 Discrete Random Variable Example: x p(x) X X X e tx e 0t =1 e 1t e t t t () = 1.5 ( ) + (.3) + (.) M t e e t t () 0 (.3) (.)( ) M t = + e + e ( 0) 0 1 (.3) 1 (.)( ) E( X) M = + + = 171 MGF Useful Properties: tb () X ( ) M t e M at ax+ b = If a random variable X has the moment generating function of a known distribution, then X has that distribution. For X and Y independent, M () t M () t M (). t X+ Y = X Y 17

87 Exercise: Let X1, X, X3 be a random sample from a discrete distribution with probability function 1 for x = 0 3 px ( ) = for x= otherwise Determine the moment generating function, Mt () of Y= X X X Exercise, cont.: (A) 19 8 t + e 7 7 (B) 1+ e t (C) (D) (E) 1 t + e e t + e t 174

88 Solution: Since each X i can be only 0 or 1, the product Y = X1, X, X3 can be only 0 or 1. In addition, Y is 1 if and only if all of the X i are 1. Thus 3 8 PY ( = 1) = = PY ( = 0) = 1 PY ( = 1) = 1 = MY () t = E( e ) = e + e = + e Yt t t t Answer A 175 Exercise: An actuary determines that the claim size for a certain class of accidents is a random variable, X, with moment generating function MX () t = 1 ( 1 500t ) 4 Determine the standard deviation of the claim size for this class of accidents. (A)1,340 (B) 5,000 (C) 8,660 (D) 10,000 (E) 11,

89 Solution: Use the derivatives of the moment generating function to find the first two moments and thus obtain V( X) = E( X ) E( X). M t = 1 500t X M t = t X = 10, t M t = t X () ( ) 4 5 () 4( ) ( 500) 5 ( ) 6 () 50, 000( ) ( 500) = 15, 000, 000( 1 500t ) Solution, cont.: M M X X 0 = 10, 000 ( ) 0 = 15, 000, 000 ( ) ( ) = ( ) ( ) V X E X E X = 15, 000, , 000 = 5, 000, 000 ( ) 5, 000, 000 5, 000 σ X = V X = = Answer B 178

90 Exercise: A company insures homes in three cities, J, K, and L. Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are: 3 M t = 1 t J K L () ( ) () = ( 1 ) () = ( 1 ) M t t M t t Exercise, cont.: Let X represent the combined losses from the three cities. Calculate E( X 3 ). (A)1,30 (B),08 (C) 5,760 (D) 8,000 (E) 10,

91 Solution: ( ) ( ) 3 Recall that E X = M X 0. First find MX Note that X = J+ K+ L where summands are independent. Thus X () = J+ K+ L( ) = MJ() t MK() t ML() t M t M t ( 1 t) ( 1 t) ( 1 t) ( 1 t ) = = 10 (). t 181 Solution, cont.: M t = t = t X X M t = t X () 10( 1 ) ( ) 0( 1 ) M t = t = t () 0( 1 ) ( ) 440( 1 ) 13 () 1( 440)( 1 ) ( ) 3 ( ) X ( ) ( t) = 10, E X = M 0 = 10, Answer E 18

92 Discrete Joint Probability Function: Definition: Let X and Y be discrete random variables. The joint probability function for X and Y is the function p( x, y) = P( X = x, Y = y). Note that: pxy (, ) = 1 x y Definition: The marginal probability functions of X and Y are defined by p ( x) = p( x, y) p ( y) = p( x, y) X y Y x 183 Exercise: A car dealership sells 0, 1, or luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold. 184

93 Exercise, cont.: PX ( = 0, Y= 0) = 1/6 PX ( = 1, Y= 0) = 1/1 PX ( = 1, Y= 1) = 1/6 PX ( =, Y= 0) = 1/1 PX ( =, Y= 1) = 1/3 PX ( =, Y= ) = 1/6 What is the variance of X? (A) 0.47 (B) 0.58 (C) 0.83 (D) 1.4 (E) Solution: First put the given information into a bivariate table and fill in the marginal probabilities for X. X Y 0 1 px ( x) 0 1/ /6=/1 1 1/1 1/6 0 3/1 1/1 1/3 1/6 7/1 186

94 Solution, cont.: ( ) E X ( ) E X = = = = VX ( ) = = Answer B 187 Definition: The joint probability density function for two continuous random variables X and Y is a continuous, real valued function f(x,y) satisfying: i) f(x,y) 0 for all x,y. 188

95 Definition: The joint probability density function for two continuous random variables X and Y is a continuous, real valued function f(x,y) satisfying: ii) The total volume bounded by the graph of z = f(x,y) and the x-y plane is 1. - fxydxdy (, ) = Definition: The joint probability density function for two continuous random variables X and Y is a continuous, real valued function f(x,y) satisfying: iii) P(a X b, c Y d) is given by the volume between the surface z = f(x,y) and the region in the x-y plane bounded by x = a, x = b, y = c and y = d. b d P( a X b, c Y d) = f( x, y) dy dx a c 190

96 Definition: Let f(x,y) be the joint density function for the continuous random variables X and Y. The marginal distribution functions of X and Y are defined by: ( ) f x = f( x, y) dy X ( ) f y = f( x, y) dx Y 191 Exercise: A device contains 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 0 < s <1 and 0 < t <1. What is the probability that the device fails during the first half hour of operation? 19

97 Exercise, cont.: (A) (B) (C) (D) (E) (, ) + (, ) (, ) + (, ) (, ) f s t dsdt (, ) f s t dsdt (, ) f s t dsdt f s t dsdt f s t ds dt f s t dsdt f s t ds dt 193 Solution: The device fails if either S < 1/ or T < 1/. S 1.5 A B.5 1 T 194

98 Solution, cont.: ( < 1/ or Y < 1/ ) f ( s, t) ds dt (, ) P S = + A (, ) (, ) = f s t ds dt B f s t ds dt f s t ds dt Answer E 195 Exercise: The future lifetimes (in months) of two components of a machine have the following joint density function: ( x y) 650, 0 < x< 50 y < 50 f( x, y) = 15, 000 0, otherwise What is the probability that both components are still functioning 0 months from now? 196

99 Exercise, cont.: 6 (A) 15, (B) 15, (C) 15, (D) 15, (E) 15, x x y x x y 0 0 ( 50 x ) ( 50 x ) ( 50 x ) ( 50 x ) y dy dx ( 50 x ) y dy dx y dy dx y dy dx y dy dx 197 Solution: Upper limits of integration in choices C and E are clearly incorrect. ( ) We need P X 0 & Y 0 from A, B or D. Density function is non-zero only in the first quadrant triangle bounded above by the line x+ y = 50 or y = 50 x. 198

100 Solution, cont.: In the diagram, below, we show the triangle and the region R where both components are still functioning after 0 months. y 50 0 R x 199 Solution, cont.: ( 0 & Y 0) f( x, y) dydx P X = R x = ( 50 x y) dy dx 15, Answer B 00

101 Exercise: 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 x+ y f( x, y) = for 0 < x< and 0 < y < 8 What is the probability that the device fails during its first hour of operation? (A).15 (B).141 (C).391 (D).65 (E) Solution: The device fails if either X < 1 or Y < 1. The set of pairs (x,y) for which this occurs is shown in the shaded region in the diagram below. y 1 A B 1 x 0

102 Solution, cont.: y B 1 A For the shaded region A, P X < 1 or Y < 1 ( ) (, ) = A f x y dxdy 1 x 03 Solution, cont.: y B 1 A 1 x Integrate over the unshaded rectangle B, to get the complementary probability. P X < 1 or Y < 1 ( ) 1 (, ) = B f x y dxdy x+ y = 1 dx dy

103 Solution, cont.: y B 1 A 1 x 1 1 x+ y dx dy 8 1 x = xy dy 1 1 = ( y) dy y = y + = 8 1 = Solution, cont.: y B 1 A 1 x ( Y ) P X < 1 or < 1 = =.65 Answer D 06

104 Exercise: A company is reviewing tornado damage claims under a farm insurance policy. Let X be the portion of a claim representing damage to the house and let Y be the portion of the same claim representing damage to the rest of the property. The joint density function of X and Y is 6 1- ( x+ y), x > 0, y > 0, and x+ y < 1 f( x, y) = 0, otherwise 07 Exercise, cont.: Determine the probability that the portion of a claim representing damage to the house is less than 0.. (A).360 (B).480 (C).488 (D).51 (E).50 08

105 Solution: Find (.) (, ) P X < = the region indicated in the diagram below. y 1 y = 1-x A A f x y dydx where A is. 1 x 09 Solution, cont.: y 1 y = 1-x A. 1 x A (, ) f x y dydx. 1 x 0 0 [ x ] = 6 1 y dy dx 10

106 Solution, cont.:. 1 x. ( 1 x) ( 1 x). 1 x y 6 [ 1 x y] dy dx = y xy dx 0. ( 1 x) = 6 ( 1 x) x( 1 x) dx 0 3. = 6 dx = 6 = Answer C Note: We can also work this using the marginal 11 for X. The calculations are basically the same. Definitions: Discrete Case. The conditional distribution of X given that Y=y is given by pxy (, ) PX ( = x Y= y) = px ( y) =. py () y Continuous Case. Let X and Y be continuous random variables with joint density function f(x,y). The conditional density for X given that Y=y is given by fxy (, ) fx ( Y= y) = fx ( y) =. f 1 Y () y

107 Conditional Expected Value: For discrete random variables, EY ( X x) ypy ( x) = = EX ( Y y) xpx ( y) When X and Y are continuous, the conditional expected values are given by y = = EY ( X= x) = yfy ( x) dy EX ( Y= y) = xfx ( y) dx x 13 Definitions: Two discrete random variables X and Y are independent if pxy (, ) = px( xp ) Y( y) for all pairs of outcomes (x,y). Two continuous random variables X and Y are independent if fxy (, ) = fx( xf ) Y( y) for all pairs (x,y). 14

108 Exercise: A diagnostic test for the presence of a disease has two possible outcomes: 1 for disease present and 0 for disease not present. Let X denote the disease state of a patient, and let Y denote the outcome of the diagnostic test. 15 Exercise, cont.: The joint probability function of X and Y is given by: P(X = 0, Y = 0) = P(X = 1, Y = 0) = P(X = 0, Y = 1) = 0.05 P(X = 1, Y = 1) = 0.15 Calculate Var( Y X =1). (A) 0.13 (B) 0.15 (C) 0.0 (D) 0.51 (E)

109 Solution: We can calculate this variance if we know the conditional distribution of Y given that X=1. px X Y 0 1 ( x) Solution, cont.: ( 0 X 1) PY = = = ( 1 X 1) PY = = = ( = 0& X= 1) P( X = 1) PY.05 = = ( = 1& X= 1) P( X = 1) PY.15 = =

110 Solution, cont.: ( ) ( ) Use V( X) = E X E X. EY ( X= 1 ) =.857 ( 0 ) ( 1 ) =.7143 ( ) ( ) ( ) EY X= 1 = =.7143 V( X ) =.7143 (.7143) =.04 Answer C 19 Exercise: Once a fire is reported to a fire insurance company, the company makes an initial estimate, X, of the amount it will pay to the claimant for the fire loss. When the claim is finally settled, the company pays an amount, Y, to the claimant. The company has determined that X and Y have the joint density function ( x 1) ( x 1) f( x, y) = y, x > 1, y > 1 x x 1 ( ) 0

111 Exercise, cont.: Given that the initial claim estimated by the company is, determine the probability that the final settlement amount is between 1 and 3. A) 1/9 B) /9 C) 1/3 D) /3 E) 8/9 1 Solution: To find P( 1< Y < 3 X = ) we need: 3 f(, y).5y f( y X = ) = = fx( ) fx( ) 3 y 1 fx ( ) = f(, y) dy =.5y dy = = y.5y 3 f( y X = ) = = = y f 1/4 X ( ) ( ) ( 1< < 3 = ) = 3 ( = ) P Y X f y X dy Answer E = y dy = y =