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ACTSC 232 Review - ACTSC 232 Review Basic Probability Sx t...

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ACTSC 232 Review Basic Probability : S x t : S x 0 = 1; S x = 0; t S x t 0; S x t diff; lim t t S x t = 0 and lim t t 2 S x t = 0 S x t = S 0 x t S 0 x ; F x t = 1 S x t ; f x t = d dt F x t x t = x t : force of mortality : x = S 0 ' x S 0 x = d dx ln S 0 x ; x t = S x ' t S x t S 0 x = e 0 x s ds ; S x t = e x x t s ds = e 0 t x s ds ; Actuarial Notation : T x : future life time random variable; T 0 : age at death random variable K x = floor T x : curtate life time random variable t p x = S x t ; t q x = F x t ; t u q x = F x t u F x t = S x t S x t u t p x t q x = 1; t u p x = t p x u p x t ; t u q x = t u q x t q x = t p x t u p x = t p x u q x t t p x = l x t l x ; t q x = t d x l x ; f x t = t p x x t ; F Kx k = k 1 q x Life Expectancy : e o x = E T x : complete life expectancy at age x; e x = E K x : curtate life expectance at age x H x = T x ^n : temporary life time random variable; G x = K x ^n : curtate temporary life time random variable e o x: n = E H x : temporary life expectancy at age x; e x: n = E G x : curtate temporary life expectance at age x e o x = E T x = 0 t p x dt; E T x 2 = 0 2 t t p x dt; e o x: n = 0 n t p x dt e x = E K x = k = 1 k p x
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