L3-Slide-232-2010W

L3-Slide-232-2010W - Lecture 3, January 12 1.3 Actuarial...

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Lecture 3, January 12 1.3 Actuarial Notation (a) The d.f. of T x is denoted by t q x = Pr { T x t } = F x ( t ) , t 0 . (b) The s.f. of T x is denoted by t p x = Pr { T x > t } = S x ( t ) = 1 - t q x , t 0 . (c) By convention, we write 1 q x = q x and 1 p x = p x . Hence, q x is the probability that ( x ) will die in one year or before age x + 1 and p x is the probability that ( x ) will be still survival after one year or beyond age x + 1 . (d) Note that q x = 1 , p x = 0 , t p 0 = S 0 ( t ) , 0 q x = 0 , and 0 p x = 1 . (e) Let u | t q x denote the probability that ( x ) survives u years and dies in the following t years or dies between ages x + u and x + u + t . Thus, u | t q x = Pr { u < T x < u + t } = u + t q x - u q x = u p x - u + t p x = u p x t q x + u . 1
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u | 1 q x = u | q x . Note that 0 | q x = q x . (f) Relationships: t + u p x = t p x u p x + t , t 0 , u 0 , f x ( t ) = t p x μ x + t , t 0 , t p x = e - R t 0 μ x + s ds = e - R x + t x μ s
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L3-Slide-232-2010W - Lecture 3, January 12 1.3 Actuarial...

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