course4_1104 - Course 4 Fall 2004 Society of Actuaries...

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Course 4: Fall 2004 - 1 - GO ON TO NEXT PAGE **BEGINNING OF EXAMINATION** 1. You are given: (i) The annual number of claims for an insured has probability function: () ( ) 3 3 1 x x px q q x ⎛⎞ =− ⎜⎟ ⎝⎠ , 0,1, 2, 3 x = (ii) The prior density is q q 2 ) ( = π , 0 < q < 1. A randomly chosen insured has zero claims in Year 1. Using Bühlmann credibility, estimate the number of claims in Year 2 for the selected insured. (A) 0.33 (B) 0.50 (C) 1.00 (D) 1.33 (E) 1.50
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Course 4: Fall 2004 - 2 - GO ON TO NEXT PAGE 2. You are given the following random sample of 13 claim amounts: 99 133 175 216 250 277 651 698 735 745 791 906 947 Determine the smoothed empirical estimate of the 35 th percentile. (A) 219.4 (B) 231.3 (C) 234.7 (D) 246.6 (E) 256.8
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Course 4: Fall 2004 - 3 - GO ON TO NEXT PAGE 3. You are given: (i) Y is the annual number of discharges from a hospital. (ii) X is the number of beds in the hospital. (iii) Dummy D is 1 if the hospital is private and 0 if the hospital is public. (iv) The proposed model for the data is YX D = + + + β ε 12 3 . (v) To correct for heteroscedasticity, the model X DX X // / / = + ++ 3 is fitted to N = 393 observations, yielding 2 ˆ 3.1 = , 1 ˆ 2.8 = − and ± 3 28 = . (vi) For the fit in (v) above, the matrix of estimated variances and covariances of ± , 2 ± 1 and ± 3 is: 0.0035 0.1480 0.0357 0.1480 21.6520 16.9185 0.0357 16.9185 38.8423 ⎛⎞ ⎜⎟ −− ⎝⎠ Determine the upper limit of the symmetric 95% confidence interval for the difference between the mean annual number of discharges from private hospitals with 500 beds and the mean annual number of discharges from public hospitals with 500 beds. (A) 6 (B) 31 (C) 37 (D) 40 (E) 67
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Course 4: Fall 2004 - 4 - GO ON TO NEXT PAGE 4. For observation i of a survival study: d i is the left truncation point x i is the observed value if not right censored u i is the observed value if right censored You are given: Observation ( i ) i d i x i u 1 0 0.9 2 0 1.2 3 0 1.5 4 0 1.5 5 0 1.6 6 0 1.7 7 0 1.7 8 1.3 2.1 9 1.5 2.1 10 1.6 2.3 Determine the Kaplan-Meier Product-Limit estimate, 10 S (1.6). (A) Less than 0.55 (B) At least 0.55, but less than 0.60 (C) At least 0.60, but less than 0.65 (D) At least 0.65, but less than 0.70 (E) At least 0.70
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Course 4: Fall 2004 - 5 - GO ON TO NEXT PAGE 5. You are given: (i) Two classes of policyholders have the following severity distributions: Claim Amount Probability of Claim Amount for Class 1 Probability of Claim Amount for Class 2 250 0.5 0.7 2,500 0.3 0.2 60,000 0.2 0.1 (ii) Class 1 has twice as many claims as Class 2. A claim of 250 is observed. Determine the Bayesian estimate of the expected value of a second claim from the same policyholder. (A) Less than 10,200 (B) At least 10,200, but less than 10,400 (C) At least 10,400, but less than 10,600 (D) At least 10,600, but less than 10,800 (E) At least 10,800
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Course 4: Fall 2004 - 6 - GO ON TO NEXT PAGE 6. You are given the following three observations: 0.74 0.81 0.95 You fit a distribution with the following density function to the data: fx p x x p b g b g =+ < < 101 , , p > − 1 Determine the maximum likelihood estimate of p .
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course4_1104 - Course 4 Fall 2004 Society of Actuaries...

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