AMS410_HWSOL3 - AMS 410 Actuarial Mathematics Fall 2009...

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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009 Homework 3 solution: Continuous distribution 10/20/2009 1. The number of claims that an insurance company gets is a random variable, and the prob- ability of getting k claims is ( 18 k ) . 15 k . 85 (18- k ) , k = 0 , 1 , ··· , 18 . What's the mode of the distribution? f (0) = 0 . 0536 , f (1) = 0 . 1704 , f (2) = 0 . 2556 , f (3) = 0 . 2406 , ... So f ( x ) is maximized at x = 2 . So the mode of the distribution is 2. 2. The moment generating function of a random variable X is M X ( t ) = λ λ- t r , λ > , r > , t < λ . What's V ar ( X ) ? M X ( t ) = r · λ r 1 λ- t r +1 , thus E ( X ) = M X (0) = r λ . M 00 X ( t ) = r ( r + 1) · λ r 1 λ- t r +2 , thus E ( X 2 ) = M 00 X (0) = r ( r +1) λ 2 . So V ar ( X ) = E ( X 2 )- [ E ( X )] 2 = r λ 2 . Note: If you can recognize that this is the mgf of a Negative Binomial distribution with parameters r and λ , you can write down the variance without the calcuation. 3. An insurer's annual weather related-loss, X , is a random variable with density function f ( x ) = ( 2 . 5(200) 2 . 5 x 3 . 5 for x > 200 otherwise . Calculate the di erence between the 30th percentile and...
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This note was uploaded on 03/13/2011 for the course AMS 410 taught by Professor Yang,y during the Spring '08 term at SUNY Stony Brook.

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AMS410_HWSOL3 - AMS 410 Actuarial Mathematics Fall 2009...

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