Midterm-SOL-431F08

Midterm-SOL-431F08 - SOLUTIONS TO MIDTERM TEST ACTSC...

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SOLUTIONS TO MIDTERM TEST – ACTSC 431/831, FALL 2008 1. (a) Let Y L be the per loss r.v. for the policy. Then Y L = 0 , X 20; X - 20 , 20 < X 100; 80 + 0 . 5( X - 100) , 100 < X 120; 90 , X > 120 . Thus, the mean of the loss for the policy is E ( Y L ) = Z 100 20 ( x - 20) 1 150 dx + Z 120 100 (80 + 0 . 5( x - 100)) 1 150 dx + Z 150 120 90 × 1 150 dx = 50 . 67 . (b) Let Y P be the per payment r.v. for the policy. Note that Y P (0 , 90] and Y P is a conditional random variable conditioning on X > 20 or Y P = Y L | X > 20. Thus, F P ( y ) = Pr { Y P y } = Pr { Y L y | X > 20 } = 0 , y 0; Pr { X - 20 y | X > 20 } , 0 < y 80; Pr { 80 + 0 . 5( X - 100) y | X > 20 } , 80 < y < 90; 1 , y 90; = 0 , y 0; Pr { 20 <X 20+ y } Pr { X> 20 } , 0 < y 80; Pr { 20
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Midterm-SOL-431F08 - SOLUTIONS TO MIDTERM TEST ACTSC...

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