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Unformatted text preview: 1 Week 3 Demography Outline: & Survival models & Hazard Rates and Survival Models & Actuarial Notation & Life table Reading Sherris, Ch 3 (req) 2 Survival Models & Probability model to calculate the probability a life will survive expected payments for insurance and annuity contracts & Survival Function: Continuous random variable ageatdeath X Distribution function F X ( x ) Probability density function F X ( x ) = f X ( x ) Survival Function (for a Newborn): s ( x ) = Pr ( X > x ) = 1 F X ( x ) x For a life aged 0: F X (0) = 0 so that s (0) = 1 3 Survival Function for Life Aged x & Note: Conditional probability Consider firstly joint probability Pr ( A and B ) = Pr ( A ) Pr ( B j A ) = Pr ( B ) Pr ( A j B ) where Pr ( A j B ) is the probability that A occurs given that B has already occurred. Hence the conditional probability Pr ( B j A ) = Pr ( A and B ) Pr ( A ) & Life aged x the probability that the life will survive to age z ( z > x ) ! Conditional probability Pr ( X > z j X > x ) = Pr ( X > z and X > x ) Pr ( X > x ) = s ( z ) s ( x ) 4 Discussion & If s ( x ) = 1 x 100 for x < 100 (a) Determine the probability that a life aged 20 will survive to age 65 (b) (*) Find the probability that a life aged 20 will die between 40 and 60. 5 Future Lifetime & A life aged x is denoted by the symbol ( x ) & Future lifetime random variable is T ( x ) = X x & International Actuarial Notation q x = Pr [( x ) will die within a year ] = Pr [ T ( x ) 1] p x = Pr [( x ) will survive at least a year ] = Pr [ T ( x ) > 1] t q x = Pr [( x ) will die within a t years ] = Pr [ T ( x ) t ] t t p x = 1 t q x = Pr [ T ( x ) > t ] 6 Survival Functions and Actuarial Notation...
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