Vo 3, No Iraioa Joura of Busiss ad Maagm Saisica Aaysis ad Emirica Sudy for Lif Isurac Xiag Jiag, Jia Ni Guagi Mdica ad Pharmacuica Cog, Naig 532, Chia T: 86-77-2853-644 E-mai: gj@26com Wii Wu Guagi Idusria Vocaioa Tchica Cog, Naig 53, Chia T: 86-77-385-662 E-mai: gwuwii@63com Absrac Accordig o h ucraiy of isurac accid, i his aric, w iroduc h imorac of isurac ad h aicaio of robabiiy saisics i h isurac, brify dscrib svra aramr mods abou if disribuio (i if mods), discuss h sabishm ad us of h if ab, ad saisicay aayz h rofi ad oss ad isurd amou i if isurac Th rsu coud ossss crai dircioa maigs ad rfrc fucio for racica wors Kywords: Lif isurac, Probabiiy saisics, Saisica aaysis, Lif mod, Emirica Sudy Iroducio Th robabiiy saisics is aid abroad o isurac, icudig accrua ad discou, surviva mod ad if ab, saisica aaysis of surviva ad dah isurac ad saisica aaysis of rmium ad comsaio Isurac is a bargai which rad o righ ad obigaio bw isurr ad isurac aica This bargai rguas i gra ha h isurac aica mus ay h isurr crai amou rmium, ad h isurr romis o ay h isurd amou wh h covioa isurac accid occurs Th isurac cagoris ar ui abudac, bcaus of h diffr sadards, isurac ca divids io rory isurac, iabiiy isurac ad if isurac Th firs ad scod isurac ar cad o-if isurac, h hird is cad if isurac Th cacuaio of if isurac rmium is a comicad usio irsy, h comosiios of rmium ar drmid by may facors Th isurac amou usd o ay for h rmium simy is cad aura rmium Bu h isurr ados from isurac aica is o oy h aura rmium, bu aso various fs ha ach oraio roducs ad various ossib riss Morovr, as h mai body of coomic, h vur isurac ag shoud cosidr crai rofis A hos comoss gross rmium Th gross rmium ca b cacuad basd o aura rmium Tha is, cacuas h aura rmium accordig o h uivac uaio firsy, ad h adds h cssary riss, fs, rofis ad ohr facors Ad h cacuaio of gross rmium ca aso b cacuad by usig h uivac uaio basd o h cosrvaiv fs irs ra ad wighd coffici of dah ra Th disic characrisic of isurac is h ucraiy haig of h isurac accid, such as o s if, raffic accid ad corms This characrisic aso drmis h imora ro of robabiiy saisics i isurac cagoris sig, isurac oraio adig ad ohr ascs So, if you wa o b a viab acuary, you mus b o o of saisics 2 Surviva mod Amog h cagoris of isurac, if isurac is a radiioa ad aso imora isurac Lif isurac ras o o s if characrisic, hrfor, i is cssary o rsarch o s characrisic ad surviva mod 2 Rmaiig Lifim Suosig T () is h isurr s rmaiig ifim, aary, i is a o-gaiv coiuous radom variab Marig h f () is h disribuio dsiy fucio of T (), ad is disribuio fucio is ( ) P{ T ( ) } f ( ) d By a aaracs, wh, T () dos h rmaiig ifim of oaa ifa, ad ( ) dos h disribuio fucio of oaa ifa s rmaiig ifim 22 Commo mars A rs, iraioa acuaria scic uss h radiioa mar (Zhag, 23, P39-83), whr dos h 76
Iraioa Joura of Busiss ad Maagm Ocobr, 28 robabiiy of yas od if s dah ra i yars, dos h robabiiy of yas od if s surviva ui yars ar, s dos h robabiiy of h yas od if s surviva i + s yars ad dah ra i yars afr ha, + dos h yas od if s codiio robabiiy of h dah ra o h codiio of is + s ag s Ad h, hr ar () (), ( ), (2) s P{sT s + } s+ s, (3) - ( s + ) P{T > s + T ( ) > s}, + s ( s) (4) ( s + ) ( s) P{T s + T ( ) > s}, + s ( s) If, aways is omid, shorig as, ad, c s 23 Ecaio of Rmaiig ifim If suosig is h caio of isura s rmaiig ifim wh h is yas od, h i ca b asiy rovd, f ( ) d ( ( ) d d 24 Dah isiy Dfi h yas od if s dah isiy (Zhag, 23, P39-83) as f ( ) d ( ()) W ca + ( ) d s ha, h dah robabiiy of yas od if i h scio of (, + d ) ca b rssd as foows, P{T + d} i + d + d Thrfor, h caio of yas od if s rmaiig ifim ca aso cacuas Th dah isiy ca aso b dfid as d +, imm h igra of his formua, w ca g d ha { } + sds I is obvious ha h dah isiy of yas od if s rmaiig im ad h disribuio fucio of i ar dfid cusivy by ach ohr 3 Svra aramr mods abou if disribuio 3 Lif mod of D Moivr (724) Lif mod of D Moivr cosidrs ha hr is a maima agω, ad h rmaiig ifim T () of yas od if i h scio of (, ω ) obys h uaiy disribuio, i f ( ), < < ω, so w ca g h disribuio ω fucio ( ), < < ω, ad h dah isiy is +, < < ω ω ω 32 Lif mod of Gomrz (825) (Zou, 25) This mod cosidrs ha h dah isiy icrasig is oia icras, ha is, + + BC > B > C > By comarig wih Lif mod of D Moivr, if mod of Gomrz rfcs h cours of if rfraby, ad hrows off h maimum ag assumio Th cacuaio bcoms asy Th disribuio fucio of if mod of Gomrz is B ( ) ( C ) I 86, Maham dd h mod of C + Gomrz H suosd ha h dah isiy was + A + BC > A > B >, C >, ad h h disribuio fucio was B ( ) A ( C ) If c, h dah isiy of Gomrz s ad C Maham s if mod ar cosa Th h disribuio fucio of T () is oia disribuio fucio Ahough i is sim wih mah, i ca o rfc h if-sa of o 33 Lif mod of Wibu (Zou, 25) I 939, Wibu u forward ha h dah isiy of o was o oia icras, bu icras wih owr, 77
Vo 3, No Iraioa Joura of Busiss ad Maagm ha is + + ) K > > Ad h disribuio fucio of i is K + ( ) [( + ) ] Th + abov-miod if mod is vry hfu 34 Aicaio isac of if mod Eam : Suosd ha h dah of o aggrga obys h mod of Maham, w g h daa, 3, 2 4, 3 6,h cacua o g 4 Souio, Accordig o h mod of dah ad h ow daa, sabish h foowig sysm of uaios, 2 A + BC 3 A + BC 4 A + BC 2 3 4 3, 4, 6, Sovig h abov sysm of uaios, w ca g A 2, B 25, C 2, ad h h dah isiy + is + 2 + 25 2,h 25 ( ) 2 (2 ),wha foows 2 is 25 --2 - (2 ) 268 2 35 Igra rmaiig ifim Dfiiio K ( ) [ T ( )] is igra yars which yas od o wi iv hrough, somims cad rmaiig ifim Th robabiiy disribuio of igra radom variab is { ( ) } P{ T ( ) + } +, (,,2, Λ, ) Th caio of ) is cad h yas od o s c igra rmaiig ifim, sig as Th P{ ) }, or + P{ ) } Th advaag of usig h igra rmaiig ifim of yas od o is ha i rdigss h rmaiig ifim of yas od o, ad i is covi o h cacuaio of caio of rmaiig ifim Maig S( ) T( ) ), S() is h fracio ar of h yars which yas od o has ivd I is asy o s ha S() is a coiuous radom variab o h igra of (,) Assumig ha ) ad S() ar idd, h h codiioa disribuio of S() do o dd o ) wh h vau of ) is giv, ad u + P{ S() u ) } + Prsuoss abov formua ua o H (u), H (u) is a crai fucio which has ohig o do wih Thr wih, u + H( ), +,2, Λ Suosig H ( u) u, ha is, S() is a homogous disribuio i h igra of (,), h + V ar ( T ( )) Var( )) + 2 2 Bcaus ha h disribuio of ) a dds o, hrfor, ca srucur a ids of ab by mas of Th commo if ab of isurac is ab of 4 Th comosiio ad aicaio of if ab Through h rsarch o h disribuio fucio of if, ha is a radom variab, h robabiiy dsiy fucio ad caio, h robabiiy of crai o s dah i crai im or i isaaous ca b rssd as survivorshi fucio ad som scia symbos Tha is o say, w ca sima h if sa of isura i oioa ag I racic, w of us h if o sad for his Th survivorshi fucio ad if ab ar cocd wih ach ohr, ad h survivorshi fucio ca b cacuad hrough if ab Lif ab ar summary ab which b sabishd accordig o h saisic daa of ach ag s dah i crai im 4 Aufbau rici of if ab O h basis of rici of argr umbr, h surviva robabiiy of ach ag o ca b cacuad by obsrvd daa (simaig frucy by frucy) Th commo symbos icud, w-bor if umbr is, is h ag, ad is h umos ag 78
Iraioa Joura of Busiss ad Maagm Ocobr, 28 42 Characrisics of if ab Th Aufbau rici of if ab is sim, ad is o drmid by h oa disribuio (o-aramr mhod) Th surviva caio umbr of w-bor if wi iv o ag X is s(), h dah caio umbr of w-bor if wi di bw h ag of ad + is d s Esciay, wh, mar as d - +,d - + Th sum of yars which w-bor if iv hrough o h igra of ad + is L dy, ad h y oa umbr of h idividua s rmaiig if, which ca iv o ag, is T ydy, h ca g T 43 Aicaio isac of if ab Eam 2: Th ow vau is ),as cacua h foowig vaus, () d 3, 2 3,3 3, 3, (2) h dah robabiiy of 2 yars od o i h igra of 5 o 55, (3) h avrag if of h o Souio: () d 3 3-3, 2 3 5 5/7, 3 3 3 6 3/7 3 4 4 /7, 3 (2) 3 5 2 5 55 /6, (3) T 5 ( 5 ) d 3 3 44 Aufbau rici of scio-uima if ab (as s i Tab ) Th hah of w moys which g hrough h frsh hysica amiaio cs h hah of od moys which go hrough h hysica amiaio a a og im ago Th forc of scio wi disaar wih im 45 Th aicaio of scio-uima if ab Th aicaio of if ab ad h rsu of cacuaio rsss as h ab 2 46 Th assumio abou fracio ag 46 Surviva siuaio i if ab Usig bacgroud, if ab rovid h disribuio of igra ag, bu somim w d o aaysis h surviva siuaio, so w sc h disribuio of crai ag basd o h wo daa bordr uo, ad sima h surviva siuaio of fracio ag 462 Basic ricis ad commo mhod (iroaio mhod) Basic ricis ad commo mhod (iroaio mhod) icud () Assumio of h uaiy disribuio (iar iroaio), s(+) (-)s()+s(+), <<, (2) Assumio of h dah forc (gomrica iroaio), s(+) s() (-) s(+), <<, (3) Assumio of Baducc (harmoious iroaio), +, << s( + ) s( ) s( + ) Eam 3: Th ow vau is ), as cacua h foowig vaus udr hr diffr fracio ag, 5 3, 525 5, 35 Souio, udr h hr assumios, () Bcaus 3 3 3, so, 69 3 7, 7 3 5 3 UDD 5 3, 4 5 3C - 3 69, 7 79
Vo 3, No Iraioa Joura of Busiss ad Maagm 5 3 Baducci 5 3 3 + 53 39 (2) 525 5 5 3 + 5 5 25 55, Bcaus 5 5, 5 5 9, 55, 45 so, 525 5 UDD +9 256 5, 45 525 5C +9 ( ) 5422, 525 5 i 8 44 45 25 Baducc +9 25 5847 44 + 25 (3) 35 UDD 3, 5 695 3 35C -( 3 )- 69, 7 35 Baducci 3 3 + 53 69 5 5 Th isurac rofi ad oss aaysis basd o cra imi horm 5 Th cra imi horm ad is maig Th cra imi horm ois ou ha if o radom variab is iducd by umrous radom facors, ad vry succssiv chag has fw fucios, so i coud dduc ha h radom variab which dscribs h radom homa obys h orma schoo So o ruir h sum of radom variabs i crai ara, w shoud oy sadardiz i ad aroimay comus i by h orma schoo Th cra imi horm has dircioa maig for h isurac idusry, ad hrough i, w ca sima ad rdic h rofi ad oss of o isurac comay, ad h aw of argr umbrs is h basd o sabish h modr isurac idusry Th foowig am aias o h imora fucio ad cocr aicaio of h aw of argr umbrs ad h cra imi horm i h isurac idusry 52 Eam aaysis ad souio Eam 4: Suos ha hr ar o a ar i h isurac i o isurac comay, ad vry o ays rmium of 2 Yua o h comay vry yar, ad vryo s dah robabiiy i o yar is 6, ad his famiy umbrs coud draw Yua from h isurac comay wh h dis, h aswr h usios, () how big h oss robabiiy of h isurac comay is?, ad (2) how big h robabiiy ha h rofi of h isurac comay i o yar is o ss ha 4 Yua? 4 Souio, Suosig ha h umbr of h dah i o i o yar is X, so X ~ B(, ),, 6,, ad h yar icom of h isurac comay is 2 X () If h isurac comay oss moy i busiss, so 2 X <, ad X > 2 rom h cra imi horm, X 2 2 6 { 2} P > P > Φ( ) Φ(77693) 6 994 So h isurac comay woud o oss moy i busiss (2) If h yar rofi of h isurac comay is o ss ha 4 Yua, so 2 X 4 ad X 8 X 8 8 6 P{ 8} P Φ( ) Φ(259) 995 6 994 So h robabiiy ha h yar rofi of h isurac comay is o ss ha 4 Yua is 995 53 Rsu aaysis ad discussio Through aaysis ad souio of h am, w coud cary udrsad why so may isurac comais com io isc, bcaus h oss robabiiy of isurac comay amos is zro, ad why isurac y is br for h isurac comay i so may isurac ys ushd by h isurac comais a rs, ad w coud f ha h robabiiy saisics is surroudig us, ad is aicaio is vry siv
Iraioa Joura of Busiss ad Maagm Ocobr, 28 6 Saisica aaysis of isurd amou i if isurac Th dah isurac is h isurac y aig huma dah as h isurac sadard, ad if h isura dis i h im imi of corac, h isurr shoud ay isurd amou o h isura, ad if h isura iv afr h im imi of corac, h isurr ds o ay h isurd amou Thrfor, h cofirmaio of isurd amou dds o h im imi of corac ad h robabiiy characr of isura s rmidig if, ad i aso shoud cosidr may facors such as h rofi of isurac comay, h gross icom ad ayou Accordig o diffr aym ims of isurd amou, h aym mhod of isurd amou ca b dividd io wo sors, i ayig i h a of h yar wh h isura dis, ad ayig wh h isura dis irs, w iroduc h firs sor Suosig ha h corac rguas ha afr h isura isurs his if wh h is yars od, if h dis i h fuur yars, h isurr shoud ay h isurd amou of o moy ui o h isura i h yar h dis, ad if h isura coud iv afr + yars, h isurr woud ay ohig Suos b () dos h isurd amou aid i h h yar, v () dos h discou facor i h h yar So,, b() < K ( ),, ad h rs vau of h isurd amou is od as Z ( (, ) >, )), K, so ( ) ( ) v, < ), Z, ( )) b( ) v, ) > Th caio ad h variac of h rs isurd amou vau rscivy ar A ( ) E[ Z ad, ω ( ))] v P{ ) < + } v + 2 2 2( + ) 2, ( )] EZ, ( ) A ( ) + v A ( ) Var [ Z No, If i h abov formua is ua o h imi ag, h isurac is h dah isurac for if, ad h rs ω caio vau of h isurd amou is ( ω ) EZ, ω ( )) v + A X + I h dah isurac for if, h aym of h isurd amou is h a of h dah yar, so udr h hyohsis ha h isurd amou is o moy ui ad h irs is h cosa, h foowig raioshi coud com io isc A ( ) v + ν A ( ω) ω + 7 Cocusios Emirica sudy idicad ha hrough amos 2 yars dvom, h acuaria mahmaics had graduay dvod o b a rofssioa subjc from a sor of scia comuaio mhod, ad is aicaio rag was graduay adig, ad i had crai dircioa maigs ad br rfrcd fucio for h racica wor of isurac Bcaus of h gh, i h aric, w oy iroduc fw mos basic ad sim cos abou isurac, ad h sudy ad aicaio abou h igraio of isurac, robabiiy saisics ad acuaria mahmaics shoud b furhr discussd by us Rfrcs HUGrbr (999) Th Mahmaics of Comoud Irs Shaghai, Shaghai Word Boo Pubishig Comay Oc, 999 P62-89 Li, Wira (26) Th Paramr Esimaio of Dah orc ucio Basd o Scio-Uima Lif Tab Joura of Hifi Uivrsiy of Tchoogy (Naura Scic) No6 P972-976 Mi, Xiaoi (24) Saisics i Lif Isurac Thory Bijig Saisics No Wagya (28) Acuaria Scic of Lif Isurac Bijig, Chia Rmi Uivrsiy Prss May, 28 P2-7 Wi, Zogshu (24) Tuoria of Probabiiy Thory ad Mahmaica Saisics Bijig, Highr Educaio Prss Aug, 24 Zhag, Miyu (23) Sudy of Rsidua Lif ucio i Riabiiy Thory Joura of Lazhou Uivrsiy of Tchoogy No5 P39-83 Zhou, Jiagiog (26) Th Cosrucig Thory of Lif Tab (2d Ediio) Tiaji, Naai Uivrsiy Prss P9-3 Zou, Gogmig (25) Surviva Mods ad Thir Esimaio Shaghai, Shaghai Uivrsiy of iac & 8
Vo 3, No Iraioa Joura of Busiss ad Maagm Ecoomics Prss Aug, 25 Tab Scio-uima if ab Ag ara Dah roorio Surviva Numbr a h bgiig of h rm Dah umbr i h rm Surviva yars i h ag ara Sum of rmaiig if Avrag rmaiig if a h bgiig of h rm -+ r ds T - 463 463 273 7287758 7388-7 246 99537 745 635 7387485 7422 7-28 39 99292 38 578 738585 7438-26 26 96973 7387758 7388-2 93 9874 92 98694 7288785 7382 2-3 65 98648 64 9867 799 7289 Tab 2 Us ad comuaio of scio-uima if ab [] [+] [+2] [+3] [+4] [+5] +5 7 75 249 33 388 474 545 75 7 9 272 342 424 58 596 76 72 29 297 374 463 566 652 77 73 228 324 49 57 62 74 78 74 249 354 447 554 678 78 79 75 273 387 489 67 742 855 8 76 298 424 535 664 82 936 8 77 326 464 586 727 889 24 82 Tab 3 ucio comariso aaysis of if ab udr hr hyohss fucio v disribuio cosa dah forc Bucci -- ( ) - - ( ) y+ y -- y ( y ) + ( ) fr() --u 2 [ ( ) ] 82