09_28_11 - 1. An insurance policy reimburses a loss up to a...

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1. An insurance policy reimburses a loss up to a benefit limit of C and has a deductible of d . The policyholder’s loss, X, follows a distribution with density function f ( x ). Find the expected value of the benefit paid under the insurance policy? Benefit Paid = + + < - < d x d x d d x d x C C C 0 E ( Benefit Paid ) = ( ) ( ) ( ) ( ) + - + + + d C d d d C C dx x f dx x f d x dx x f 0 X X 0 X . For example, if X has an Exponential distribution with mean θ , E ( Benefit Paid ) = ( ) ( ) ( ) + - + + d C d d C C dx x f dx x f d x X X = ( ) + - + - + - d x C d d x C C dx dx d x e e 1 1 θ θ θ θ = ( ) d x C d d x x C C e e e d x + - + - - - + - - - θ θ θ θ = ( ) θ θ θ θ d d C e e + - - - . For example, if d = 2, C = 10, X has an Exponential distribution with mean θ = 5, E ( Benefit Paid ) = ( ) ( ) ( ) + - 12 X 12 2 X 10 2 dx x f dx x f x = ( ) + - - - 12 5 12 2 5 5 1 10 5 1 2 dx dx x x x e e = ( ) 12 10 2 12 5 2 5 5 5 - + - - - - - - x x x e e e x = 5 12 5 2 5 5 - - - e e = 5 e – 0.4 – 5 e – 2.4 2.898 .
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Suppose X has an Gamma distribution with parameters α (integer) and θ . E ( Benefit Paid ) = ( ) ( ) ( ) + - + + d C d d C C dx x f dx x f d x X X = ( ) ( ) ( ) + - + - - + - - Γ Γ d x C d d x C C dx x dx x
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This note was uploaded on 10/13/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.

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09_28_11 - 1. An insurance policy reimburses a loss up to a...

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