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A1-431F08

A1-431F08 - 120 if X> 160 the policy pays 70 Thus the per...

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ASSIGNMENT 1 – ACTSC 431/831, FALL 2008 Due at the beginning of the tutorial on Wednesday, September 24 . 1. Let Y be the total loss in an insurance portfolio and N be the number of claims in the portfolio. If N = 0, then Y = 0. If 1 N 5, then Y has an exponential distribution with mean 100. If N > 5, then Y has an exponential distribution with mean 200. It is given that Pr { N = 0 } = 0 . 3 , Pr { 1 N 5 } = 0 . 5 , and Pr { N > 5 } = 0 . 2 . (a) Let F Y ( y ) be the distribution function of Y . Calculate F Y ( y ) for all y ( -∞ , ). [5 marks] (b) Calculate the probability that the total loss in the portfolio will be greater than 500. [5 marks] (c) Calculate the probability of Pr { 0 < Y < 300 } . [5 marks] (d) Calculate the mean of the total loss. [5 marks] (e) Calculate the 20th percentile of the total loss. [5 marks] (f) Calculate the 80th percentile of the total loss. [5 marks] 2. The ground-up loss X for an insurance policy has a uniform distribution U (0 , 200). If X 80, the policy pays nothing; if 80 < X 120, the policy pays the amount of the loss above 80; if 120 < X 160, the policy pays 40 plus 75% of the loss above
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Unformatted text preview: 120; if X > 160, the policy pays 70. Thus, the per loss random variable for the policy, denoted by Y L , is Y L = , X ≤ 80; X-80 , 80 < X ≤ 120; 40 + 0 . 75( X-120) , 120 < X ≤ 160; 70 , X > 160 . Let Y P be the per payment random variable for the policy. Then Y P = Y L | X > 80. (a) Determine F Y L ( y ), the cdf of Y L , for all y ∈ (-∞ , ∞ ). [10 marks] (b) Determine F Y P ( y ), the cdf of Y P , for all y ∈ (-∞ , ∞ ). [10 marks] (c) Calculate the probability of Pr { 40 ≤ Y L ≤ 50 } . [5 marks] (d) Calculate the probability of Pr { 30 ≤ Y P < 70 } . [5 marks] (e) Calculate the mean of the per loss of the policy. [5 marks] (f) Determine the median of the per payment of the policy. [5 marks] (g) Determine the 90th percentile of the per payment of the policy. [5 marks] [Total: 75 marks] 1...
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