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# course4_1102 - Course 4 Fall 2002 Society of...

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Course 4: Fall 2002 -1- GO ON TO NEXT PAGE Fall 2002 Society of Actuaries **BEGINNING OF EXAMINATION** 1. For a stationary AR(2) process, you are given: r 1 05 = . r 2 02 = - Calculate f 2 . (A) - 08 (B) - 0 6 (C) - 0 2 (D) 0 6 (E) 0 8

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Course 4: Fall 2002 -2- GO ON TO NEXT PAGE 2. You are given the following claim data for automobile policies: 200 255 295 320 360 420 440 490 500 520 1020 Calculate the smoothed empirical estimate of the 45th percentile. (A) 358 (B) 371 (C) 384 (D) 390 (E) 396
Course 4: Fall 2002 -3- GO ON TO NEXT PAGE 3. You are given: (i) The number of claims made by an individual insured in a year has a Poisson distribution with mean λ . (ii) The prior distribution for λ is gamma with parameters a = 1 and q = 1.2 . Three claims are observed in Year 1, and no claims are observed in Year 2. Using Bühlmann credibility, estimate the number of claims in Year 3. (A) 1.35 (B) 1.36 (C) 1.40 (D) 1.41 (E) 1.43

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Course 4: Fall 2002 -4- GO ON TO NEXT PAGE 4. In a study of claim payment times, you are given: (i) The data were not truncated or censored. (ii) At most one claim was paid at any one time. (iii) The Nelson-Aalen estimate of the cumulative hazard function, H ( t ), immediately following the second paid claim, was 23/132. Determine the Nelson-Aalen estimate of the cumulative hazard function, H ( t ), immediately following the fourth paid claim. (A) 0.35 (B) 0.37 (C) 0.39 (D) 0.41 (E) 0.43
Course 4: Fall 2002 -5- GO ON TO NEXT PAGE 5. You fit the following model to eight observations: Y X = + + a b e You are given: \$ b = 2.065 X X i - = c h 2 42 Y Y i - = c h 2 182 Determine R 2 . (A) 0.48 (B) 0.62 (C) 0.83 (D) 0.91 (E) 0.98

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Course 4: Fall 2002 -6- GO ON TO NEXT PAGE 6. The number of claims follows a negative binomial distribution with parameters b and r , where b is unknown and r is known. You wish to estimate b based on n observations, where x is the mean of these observations. Determine the maximum likelihood estimate of b . (A) 2 x r (B) x r (C) x (D) rx (E) 2
Course 4: Fall 2002 -7- GO ON TO NEXT PAGE 7. You are given the following information about a credibility model: First Observation Unconditional Probability Bayesian Estimate of Second Observation 1 1/3 1.50 2 1/3 1.50 3 1/3 3.00 Determine the Bühlmann credibility estimate of the second observation, given that the first observation is 1. (A) 0.75 (B) 1.00 (C) 1.25 (D) 1.50 (E) 1.75

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Course 4: Fall 2002 -8- GO ON TO NEXT PAGE 8. For a survival study, you are given: (i) The Product-Limit estimator \$ S t 0 b g is used to construct confidence intervals for S t 0 b g . (ii) The 95% log-transformed confidence interval for S t 0 b g is 0.695 0.843 , b g . Determine \$ S t 0 b g . (A) 0.758 (B) 0.762 (C) 0.765 (D) 0.769 (E) 0.779
Course 4: Fall 2002 -9- GO ON TO NEXT PAGE 9. You are given the following information about an AR(1) model with mean 0: r r 2 3 0215 0100 0431 = = - = - . . . y T Calculate the forecasted value of y T + 1 . (A) –0.2 (B) –0.1 (C) 0.0 (D) 0.1 (E) 0.2

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Course 4: Fall 2002 -10- GO ON TO NEXT PAGE 10. A random sample of three claims from a dental insurance plan is given below: 225 525 950 Claims are assumed to follow a Pareto distribution with parameters q = 150 and a .
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course4_1102 - Course 4 Fall 2002 Society of...

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