Tutorial 4s.pdf - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801/2901 Probability and Statistics: Foundations of Actuarial Science Example Class 4 (Suggested Soultion) Problems 1. Let X be a continuous random variable with density function f ( x ) = | x | 10 1 ( - 2 x 4) . (a) Calculate the first and second moment of X . (b) Hence, find: i. Var( X ); ii. E (5 X - 3); iii. Var(5 X - 3) . Solution: (a) The first moment: E ( X ) = Z -∞ x · | x | 10 · 1 [ - 2 , 4] ( x ) dx = Z 4 - 2 x · | x | 10 dx = Z 0 - 2 - x 2 10 dx + Z 4 0 x 2 10 dx = - x 3 30 0 - 2 + x 3 30 4 0 = 28 15 . The second moment: E ( X 2 ) = Z -∞ x 2 · | x | 10 · 1 [ - 2 , 4] ( x ) dx = Z 4 - 2 x 2 · | x | 10 dx = Z 0 - 2 - x 3 10 dx + Z 4 0 x 3 10 dx = - x 4 40 0 - 2 + x 4 40 4 0 = 34 5 . (b) Var( X ) = E ( X 2 ) - [ E ( X )] 2 = 746 225 , E (5 X - 3) = 5 · E ( X ) - 3 = 19 3 , Var(5 X - 3) = 5 2 · Var( X ) = 746 9 . 2. An actuary determines that the claim size for a certain class of accidents is a random variable X with moment generating function M X ( t ) = 1 (1 - 2500 t ) 4 . 1
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Determine the standard deviation of the claim size for this class of accidents. Solution: Method 1: R X ( t ) = - 4 log(1 - 2500 t ) R 0 X ( t ) = 4 × 2500 1 - 2500 t R 00 X ( t ) = 4 × 2500 2 (1 - 2500 t ) 2 standard deviation = ( R 00 X (0)) 1 2 = 5 , 000 .
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