# summary - ACT370 Notes Chapter 3 1 survival function\X time...

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

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 —%±

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

View Full Document
so with this survival distribution, for someone at age , ' ' , %_ & ’ : &!± ’ % and for 5 % % # % \$ % %% : & ! ;;;; & ± & ± &± * # ! \$ \$ %) ) % ) % ) % )% and for ³ % % 5% ; ; ; & " 5² ’± & &"± ³ )% ) % )% #l# % % % #% ;&;´; & && # ) µ’%´µ\$’ ) % µ)% =—’¶´=—)¶ =—%¶
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&±!

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 62

summary - ACT370 Notes Chapter 3 1 survival function\X time...

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

View Full Document
Ask a homework question - tutors are online