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The document provides multiple exercises and worked solutions on survival and mortality models, including converting survival functions to c.d.f. and p.d.f., deriving future-lifetime survival functions, and computing multi-year death or survival probabilities. It also includes calculations under Gompertz’s law, constant force of mortality, constant force of interest, and UDD assumptions, along with use of life table quantities and select-and-ultimate mortality excerpts.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/midterm-exam-i-actuarial-mathematical-models-i-exercise-solutions/399166/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/midterm-exam-i-actuarial-mathematical-models-i-exercise-solutions/399166.png","ImageObject",300,407,{"name":92,"@type":93},"Hazel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-27",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How is the c.d.f. derived from the given survival function in Exercise 1?","Question",{"text":113,"@type":114},"Use F0(t)=1−S0(t) for t≥0, where S0(t) is provided. Then differentiate to obtain the p.d.f. as the derivative of F0(t).","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What assumptions are used to compute probabilities over an interval in Exercise 3?",{"text":118,"@type":114},"The document applies Constant Force of Mortality (CFM) for parts that interpolate between integral ages, and it applies the UDD (uniform distribution of deaths) assumption for the probability of dying within two years when forces of mortality are given.",{"name":120,"@type":111,"acceptedAnswer":121},"How does Exercise 2 handle a mortality table to find the probability of death within two years?",{"text":122,"@type":114},"It uses life table relationships between survival and death probabilities (e.g., expressing two-year death probability as 2q21=1−2p21) and computes required intermediate quantities from the provided table values.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},399166,1791267160,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":29,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},137441390410,"https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984","1  \nKing Saud University Academic Year (G) 2020{2021  \nCollege of Sciences Academic Year (H) 1442  \nMathematics Department Bachelor AFM: M. Eddahbi  \nOctober 4, 2020 from 9 to 11 AM, ACTU-362-372 (two pages + formula sheet)  \nMidterm Exam I, Actuarial Mathematical Models I  \nExercise 1  \nThe survival function S0 (t) of the age-at death T0 is given by  \n 1   \nS0 (t) = for t 􀀕 0  \n(1 + t)2 ;  \n1. Find F0 (t) the c.d.f of T0  \n2. Find f0 (t) the p.d.f of T0  \n3. Find Sx (t) the survival function of the future life time Tx of an age x:  \n4. Calculate 2p 10 .  \n5. Calculate 10j5q20 .  \nSolution:  \n1. The c.d.f. F0 (t) of T0 is given by F0 (t) = 1 􀀀 S0 (t) = 1 􀀀 (1t)2 for all t 􀀕 0 :  \n2. The p.d.f. f0 (t) of T0 is given by f0 (t) = F00(t) = 􀀐 1 􀀀 (1t)2 􀀑 0 = (1t)3 for all t 􀀕 0 :  \n3. The survival function Sx (t) of future life time Tx is given in terms of S0 (t) by: (1t)3 Sx (t) = S0x0(+x)~~ ~~t) = ~~  ~~(1+xt))2 = (1(+x~~ ~~)2t)2 :  \n4. We know that tpx = Sx (t), then 2p 10 = S10 (2) = (1(+110)22)2 = ((~~ ~~1113))22 = 0:71598:  \n5. We know that mjnqx = m+nqx 􀀀 m qx = mpx 􀀀 m+npx , then  \n10j5q20 = 10p20 􀀀 15p20 = S20 (10) 􀀀 S20 (15)  \n (1 + 20)2   (1 + 20)2   \n= 􀀀 = 0:11862:  \n(1 + 20 + 10)2 (1 + 20 + 15)2  \n2  \nExercise 2  \n1. You are given the following mortality table:  \n\n| x | 20 | 21 | 22 | 23 | 24 |\n| --- | --- | --- | --- | --- | --- |\n| qx |  |  |  | 0:0700 |  |\n| `x | 30 ; 000 |  |  |  | 23 ; 900 |\n| dx | 1 ; 200 |  |  |  |  |\n\nDetermine the probability that a life age 21 will die within two years.  \n2. CANCELED You are given kjqx = 0:01 + 1~~k~~00 for k = 0 ; 1 ; : : : ; 9. Calculate 4px+6  \n3. Find tpx when the Mortality follows Gompertz's law (􀀖x = Bcx ) . You are given that 5p60 = 0:95 and 10p60 = 0:87 . Determine 30p60 .  \nSolution:  \n1. The required probability is 2 q21 = 1 􀀀 ~~`~~`~~23~~21 . We need `21 and `23 which will be backed up as follows  \n`21 = `20 􀀀 d20 = 30000 􀀀 1200 = 28800  \nnow we shall use qx = `~~ x~~`x~~x+1~~ or equivalently `x = `1~~+1~~q~~ ~~x , thus `23 = 1~~24~~q23 = 1~~2~~~~9~~0~~0~~:~~0~~07 = 25698:92: So we get  \n25698:92  \n2 q21 = 1 􀀀 = 0:1076764:  \n28800  \n2. CANCELED By de􀀌nition of tpx we can write  \n4px+6 = ``xx and kj qx = `x+k~~ ~~`xx+1+k () `x+k 􀀀 `x+1+k = kj qx `x  \nhence  \n9 9 `x 􀀀 `x+10 = X (`x+k 􀀀 `x+1+k) = `x X  \nk=0 k=0  \n5 5 `x 􀀀 `x+6 = X (`x+k 􀀀 `x+1+k) = `x X  \nk=0 k=0  \nkj qx  \nkj qx  \nthus  \n9  \n`x+10 = `x 1 􀀀X k=0  \ntherefore  \nkj qx! and `x+6 = `x  \n51 􀀀X  \nk=0  \nkj qx!  \n4px+6 = ``xx = 11~~ ~~􀀀~~ ~~PP9~~k~~5k~~0~~0~~ ~~kkq~~ ~~xq~~ ~~x = 11~~ ~~􀀀~~ ~~PP9~~k~~5k~~0~~0((00::0101~~ ~~~~ ~~00::01k01k)) = 00::5579 = 0:6962:  \n3  \n3. By de􀀌nition of tpx we can write  \ntpx = e􀀀Rxx+t 􀀖 sds = e􀀀B Rxx+t csds = e􀀀B Rt0 cx+sds = e 􀀀 lBn~~c~~(xc) (ct􀀀1)  \nTake the log for x = 60, t = 5 and t = 10 in the previous formula we get,  \nBc60 (c5 􀀀 1) Bc60 (c10 􀀀 1)  \n~~ ~~ = 􀀀 ln(0:95) and ~~ ~~ = 􀀀 ln(0:87)  \nln(c) ln(c)  \nDividing the 􀀌rst into the second,  \nc5 + 1 = ~~ ~~lnln((00::8795)) = 2:715015 c = p5 1:715015 = 1:11392  \nc30 􀀀 1 = 24:4453  \nBc60ln(c30(c)~~ ~~􀀀~~ ~~1) = 􀀒 Bc60ln(c(5c)􀀀~~ ~~1)􀀓 􀀒 c30c5~~ ~~􀀀􀀀~~ ~~11􀀓= 􀀀 ln(5p60 ) 􀀒 c30c5~~ ~~􀀀􀀀~~ ~~11􀀓  \n= (􀀀 ln 0:95) 􀀒 c30c5~~ ~~􀀀􀀀~~ ~~11􀀓 = 1:753641 30p60 = e􀀀1:753641 = 0:1731  \nExercise 3  \n1. Assume that the life table function is given by `x = 10000e􀀀0:05x , x 􀀕 0. 􀀌nd 3j10q25  \n2. Assume that the life table function at some ages is given as follows: (i) `40 = 94 ; 000, (ii)`41 = 93 ; 000, (ii) `42 = 92 ; 000  \nAssuming Constant Force of Mortality between integral ages, 􀀌nd 1:3q40:5 .  \n3. The force of mortality at some ages of a certain mortality table are given: (i) 􀀖60:5 = 0:02, (ii)  \n􀀖61:5 = 0:04 and (iii) 􀀖62:5 = 0:06 . Under UDD assumption, calculate the probability that a person age 60:5 will die within two years.  \nSolution:  \n1. We have  \n3j10q25 = 13 q25 􀀀 3 q25 = 3p25 􀀀 13p25 =  \n`28  \n`25  \n􀀀  \n`38  \n`25  \n=  `28 􀀀 `38 = e􀀀0:05􀀂28 􀀀 e􀀀0:05􀀂38  \n`25 e􀀀0:05􀀂25  \n= e􀀀0:05􀀂3 􀀀 e􀀀0:05􀀂13 = 0:3386  \n2. We can write under","cbCairsGrhEt6iZR","https://ap.wps.com/l/cbCairsGrhEt6iZR","pdf",178375,"English","# Midterm Exam I, Actuarial Mathematical Models I\n## Exercise 1\n## Exercise 2\n## Exercise 3\n## Exercise 4","[{\"question\":\"How is the c.d.f. derived from the given survival function in Exercise 1?\",\"answer\":\"Use F0(t)=1−S0(t) for t≥0, where S0(t) is provided. Then differentiate to obtain the p.d.f. as the derivative of F0(t).\"},{\"question\":\"What assumptions are used to compute probabilities over an interval in Exercise 3?\",\"answer\":\"The document applies Constant Force of Mortality (CFM) for parts that interpolate between integral ages, and it applies the UDD (uniform distribution of deaths) assumption for the probability of dying within two years when forces of mortality are given.\"},{\"question\":\"How does Exercise 2 handle a mortality table to find the probability of death within two years?\",\"answer\":\"It uses life table relationships between survival and death probabilities (e.g., expressing two-year death probability as 2q21=1−2p21) and computes required intermediate quantities from the provided table values.\"}]","Midterm Exam I, Actuarial Mathematical Models I - Exercise Solutions | PDF",1790494106,15]