Transcription of 8'(x) -I - UCLA
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8'(x) -I - UCLA
Chapter 3: Survival Distributions and Life Tables Distribution function of X: Force of mortality flea:): Fx(:r) = Pr(X S; :1;). /1(:1:). Survival function B(.1:): 8' (x). sex). Relations between survival functions and Probability of death between age :r and force of mortality: age y: Pr(.r < X S; z) ( (z) - Fx (:1:). - B(Z). exp ( -I "(Y)d ll ). Probability of death between age age y given survival to age :r:: and nPx exp (. - ! x+n p.(y)ely ). Pr(:1; < X S; zlX > Derivatives: d dt t(jx d Notations: dt tPx tPx . It (:r: + t). d tlJx PriT(.r) tl -T. dt '" prob. (3:) dies within t years d distribution function of T(a:) -L. dt x tPx Pr[T(:c) > t] d -1. attains age ;1; +t Mean and variance of T and ](: E[T(:r)] complete expectation of life Pr[t < Tel') t + 'Ill t+ul]x - t])a' t+u])x t(jx =./. o tP:B elt tPx' u(]x-t-t curtate expectation of life ex). Relations with survival functions: 00. :T(:r:) ] . o ,J. t . tPx u,t , 2. - ex 00. Curtate future lifetime (K(:r) greatest Vo:r[K(.))
Chapter 8: Benefit Reserves . Notations: br . death benefit payable at the end of year of death for the j-th policy year . 71J~l: benefit premium paid at the beginning of the j-th policy year
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