L2-Slide-232-2010W

L2-Slide-232-2010W - Summary of Lecture Notes - ACTSC 232,...

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Summary of Lecture Notes - ACTSC 232, Winter 2010 Part 1 - Survival Distributions and Life Tables Lecture 2, January 7 1.2 The Force of Mortality Let Y > 0 be a positive and continuous r.v. with df F ( x ) = Pr { Y x } , pdf f ( x ) , and sf S ( x ) = Pr { Y > x } . Define function h ( x ) by h ( x ) = f ( x ) S ( x ) = - d dx S ( x ) S ( x ) = - d dx log S ( x ) . This function h ( x ) is called “ hazard rate function ” or “ failure rate 1
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function ” of Y in statistics and probability and “ force of mortality ” of Y in actuarial mathematics. Note that Y > 0 and S (0) = 1 . For any x 0 , Z x 0 h ( t ) dt = Z x 0 - d log S ( t ) = - (log S ( x ) - log S (0)) = - log S ( x ) . Therefore, for any x 0 , S ( x ) = e - R x 0 h ( t ) dt . For the positive and continuous random variable Y , the pdf f ( x ) , df F ( x ) , sf S ( x ) , and the force of mortality h ( x ) are equivalent in the sense that given any of the four functions, we can determine any other three 2
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L2-Slide-232-2010W - Summary of Lecture Notes - ACTSC 232,...

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