summary - ACT370 Notes - Chapter 3 1. survival function:...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
ACT370 Notes - Chapter 3 1. time until death or age at death of a newborn - this is a survival function: \&X—!± & continuous, positive random variable; the of is distribution function \ probability that the newborn dies before (or on) attaining age ; J —B±&T<²\³B´ \ &B the of survival function \ probability that the newborn dies after attaining age , or =—B±&"µJ —B±&T<²\¶B ´B \ & equivalently, the probability that the newborn survives to at least age ; B if then , this is the probability that the D T<²B·\³D´&J —D±µJ —B±&=—B±µ=—D± \\ newborn dies between ages and ; BD it is always assumed that and (the newborn is alive with probability 1 ) and J— ±& ! =—!±&" \ 0 ¸ li m lim Bp _ B˜_ J —B±& " =—B±& ! J —B± and (the probability of the newborn surviving forever is 0); is a non-decreasing function and is a non-increasing function (as increases, there is an increasing =—B± B probability that the newborn dies by age ) the upper age limit on a survival distribution is denoted B ¹ = , which is usually a finite number (such as 100 or 110) for human survival - but for illustration purposes, some examples have = &_ 2. : the notation refers to an individual alive at age ; denotes time until death for —& B X—B& —B ±B the measuring the time until 's death from age ( continuous random variable —B ± B \&X—!± described above is a special case of this wit h , a newborn- also note that given that a newborn B&! survives to age , the future lifetime from age is - i.e., time until death from age is B B X—B±&\µB B age at death minus ); the distribution function of is which is the B X—B ±J —>±&T²X—B±³>´ X—B± ¸ probability tha t will die within the next years (by ag e ) - this probability is denoted ( ) B > Bº> >B ; ± J —>&±T²X—B&³>´ ¹ the complement of is denoted , which is the probability that ; > B : ±"µ; ±T²X—B&¶>´ —B ±> survives at least to time (and dies some time after ); > by the conventions of actuarial notation, we write and ; "" ; & ; : &: B B BB we assume tha t and (the probability of living forever is 0) _ B _B " : &! ¹ in the special case tha t we have and B& ! \&X—! ± : &=—B± ¸ B ! 3 .: Factorization of survival probability For and as a special case, so that 8³> ¸ :&: » : :&:»: > B 8B >µ 8 Bº 8 8 B 8B !! ¸¸ x+ : & =—Bº8± =—B± >l? B >B? Bº > ? B > B > B ?B ; &T²>·X—B±³>º?´&:» ;& ;µ;&: µ: & =—Bº>±µ=—Bº>º?± =—B± and again by notational convention, and note that >l " > l !l B B ; ; &; ¸ Example: for =—B ± !³B³"! & &" µ "! ! "!! ## , = \ for >> %% X—%± : J —>±&; & & µ ¸ & !³>³’ =—%º>± =—% ± ¼)% )>º > )>º> ) % )% —%º>± # "!! (since , if there are years remaining in 's lifetime distribution), == ! % & µ B —%±
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
so with this survival distribution, for someone at age , ' ' , %_ & ’ : &!± ’ % and for 5 % % # % $ % %% : & ! ;;;; & ± & ± &± * # ! $ $ %) ) % ) % ) % )% and for ³ % % 5% ; ; ; & " 5² ’± & &"± ³ )% ) % )% #l# % % % #% ;&;´; & && # ) µ’%´µ$’ ) % µ)% =—’¶´=—)¶ =—%¶
Background image of page 2
4. of - denotes the completed number of years until 's death, curtate future lifetime —B & O—B & —B& which is an integer of or more; corresponds to not surviving complete year (dying ! —B &" O—B&±!
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 62

summary - ACT370 Notes - Chapter 3 1. survival function:...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online