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Includes severity and frequency distributions such as Uniform, Exponential, Weibull, Gamma, Beta, Pareto, Lognormal, and their extreme value counterparts (Fréchet, Gumbel, Weibull EV) plus frequency models like Poisson, Binomial, Geometric, and Negative Binomial. Provides formulas for scaling and transformations, mixture and splicing, moment/ probability generating functions, and tail and hazard rate limiting expressions for exam-style practice.",{"@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/capstone-exam-a4-short-term-actuarial-mathematics-formula-review-sheet/399190/",{"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/capstone-exam-a4-short-term-actuarial-mathematics-formula-review-sheet/399190.png","ImageObject",300,407,{"name":92,"@type":93},"\tJames","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-08","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},"What topics are covered in the Capstone Exam-A4 formula and review sheet?","Question",{"text":113,"@type":114},"It covers severity and frequency distributions, extreme value distributions, scaling and transformation, mixture and splicing, and tail and limiting distribution results used in actuarial mathematics practice.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Which distributions are listed under severity distributions?",{"text":118,"@type":114},"The sheet lists Uniform, Exponential, Weibull, Gamma, Beta, Pareto, Single P. Pareto, and Lognormal, along with key CDF and expectation/variance formulas where shown.",{"name":120,"@type":111,"acceptedAnswer":121},"What kind of formulas does the sheet provide for exams?",{"text":122,"@type":114},"It provides quick-reference formulas including distribution functions, moment generating functions, probability generating functions, and limiting expressions for survival, hazard rate, and mean excess loss.","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},399190,1790935444,{"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":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},2336474466412,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Formula & Review Sheet for A4 STAM  \nYour quick-reference guide for the most important formulas and concepts you need.  \nWant an Extra Edge?  \nTake your preparation even further with premium ACTEX resources designed to sharpen exam skills and boost confidence when it matters most.  \n Live Bootcamps  \nOne-On-One Tutoring  \nPopular Upgrades:  \nSheet?  \nWant More  \nFormula  \nThan Just the  \nFull Set of  \nManuals  \nGraded Mock Exam  \nPass Guarantee  \nExam-style Practice Questions  \nFormula & Review Sheets  \nThis free Formula & Review Sheet is just one part of the ACTEX Learning experience. Our complete study program is designed to help you understand concepts, practice effectively, and pass with confidence.  \nWith the ACTEX Learning Bundle for A4 STAM, you get:  \nLearn More  \nClick anywhere on this page or scan the QR code to learn more about offerings for this exam. ~~ ~~  \nACTEX Learning  \nCapstone Exam-A4 . Short-Term Actuarial Mathematics Formula & Review Sheet  \nLooking for comprehensive study materials for the Capstone Exam? Explore our:  \nAll-In-One Capstone Exam Review and Reference Materials  \nA4.7 Severity Models  \nSEVERITY DISTRIBUTIONS  \n\n|  |  |  |  |  |\n| --- | --- | --- | --- | --- |\n| Distribution | Probability density function | Formulas worth memorizing |  |  |\n| Uniform |  1 \u003Cbr>f (x) = b 􀀀 a ; a 􀀔 x 􀀔 b | F (x) = xb~~ ~~􀀀~~ ~~aa | E [X] = a~~ ~~+2~~ ~~b | Var (X) = (b~~ ~~a)2 |\n| Exponential | 1 􀀀 x\u003Cbr>f (x) = 􀀒 e 􀀒 ; x > 0 | F (x) = 1 􀀀 e 􀀀 ~~x~~􀀒 | E [X] = 􀀒 | Var (X) = 􀀒2 |\n| Weibull | f (x) = 􀀜x 􀀐 x􀀒 􀀑 􀀜 e􀀀􀀀 ~~x~~􀀒 􀀁 􀀜 ; x > 0 | F (x) = 1 􀀀 e􀀀􀀀 ~~x~~􀀒 􀀁 􀀜 |  |  |\n| Gamma | f (x) = 􀀀􀀀~~1􀀒~~(􀀁􀀋􀀋) x􀀋􀀀1e 􀀀 ~~x~~􀀒 ; x > 0 | F (x) = Pr (X􀀃 􀀕 􀀋)\u003Cbr>X 􀀃 is Poisson with 􀀕 = x􀀒 If 􀀋 is an integer. | E [X] = 􀀋􀀒 | Var (X) = 􀀋􀀒2 |\n| Beta | f (x) = 􀀀􀀀((aa)~~ ~~~~ ~~b(b)) xa􀀀1 (1 􀀀 x)b􀀀1 ;\u003Cbr>0 \u003C x \u003C 1 |  | E [X] = a~~ ~~~~ ~~b\u003Cbr>If a and b are integers. | E 􀀂X 2􀀃 =  a (a + 1) \u003Cbr>(a + b) (a + b + 1)\u003Cbr>If a and b are integers. |\n| Pareto | f (x) = (x~~ ~~􀀒􀀒􀀋)􀀋+1 ; x > 0 | F (x) = 1 􀀀 􀀒 x~~ ~~~~ ~~􀀒 􀀓 􀀋 | E [X] = 􀀋~~ ~~~~ ~~1\u003Cbr>If 􀀋 > 1 is an integer. |  |\n| Single P. Pareto |  􀀋􀀒􀀋  f (x) = x􀀋+1 ; x > 􀀒 | F (x) = 1 􀀀 􀀒 􀀒x􀀓 􀀋 | E [X] = 􀀋~~ ~~1 |  |\n| Lognormal | 1  (log x􀀀􀀖)2\u003Cbr>f (x) = ~~  ~~ e 􀀀 2􀀛 2 ; x > 0 x􀀛 p 2􀀙 | F (x) = N 􀀒 log~~ ~~x􀀛􀀀~~ ~~􀀖 􀀓 | E [X] = e􀀖+ ~~ 􀀛~~22 | E 􀀂X 2􀀃 = e2􀀖+2􀀛2 |\n|  |  |  |  |  |\n| Distribution | Moment Generating Function |  |  |  |\n| Exponential | MX (t) = 􀀕~~ ~~t ; t \u003C 􀀕 |  |  |  |\n| Weibull | MX (t) =R01 etx 􀀁 ~~􀀋~~􀀌 􀀐 ~~x~~􀀌 􀀑 􀀋􀀀1 e 􀀀(x/􀀌)􀀋 dx |  |  |  |\n| Gamma | MX (t) = 􀀐 ~~ ~~1􀀒t 􀀑 􀀋 ; t \u003C ~~1~~􀀒 |  |  |  |\n|  |  |  |  |  |\n\nEXTREME VALUE DISTRIBUTIONS  \nFréchet distribution F (x) = exp 􀀐􀀀􀀀1 + 􀀘 􀀀 x~~ ~~􀀖 􀀁􀀁 􀀀1/􀀘􀀑 ; 􀀘 > 0 and x > 􀀖 􀀀 ~~􀀒~~􀀘  \nGumbel Distribution: F (x) = exp 􀀀􀀀 exp 􀀀 􀀀 x~~ ~~􀀖 􀀁􀀁  \nWeibull EV distribution: F (x) = exp 􀀐􀀀􀀀1 + 􀀘 􀀀 x~~ ~~􀀖 􀀁􀀁 􀀀1/􀀘􀀑 ; 􀀘 \u003C 0 and x \u003C 􀀖 􀀀 ~~􀀒~~􀀘  \nGeneralized Pareto Distribution : G􀀘;􀀌 (x) = \u003C8:11 􀀀 e( 􀀘x􀀌/􀀌)􀀀1/􀀘 􀀘􀀘  00 where an􀀌d >00􀀔;xx􀀔􀀕0􀀀􀀌fo/􀀘r 􀀘for􀀕􀀘0\u003C, 0  \n[www.ACTEXLearning.com](www.ACTEXLearning.com)  \n Need Help? Email [support@actexlearning.com](support@actexlearning.com)  \nCopyright © ArchiMedia Advantage Inc. All Rights Reserved  \nACTEX Learning A4 . Short-Term Actuarial Mathematics Formula & Review Sheet 2  \nSCALING & TRANSFORMATION  \n\n| Scaling: | Y = cX ! FY (y) = FX 􀀀 yc􀀁 |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- |\n| Transformation: | Y = g (X) ! FY (y) = FX 􀀀g􀀀1(y)􀀁 If g is monotonically increasing.\u003Cbr>! FY (y) = SX 􀀀g􀀀1(y)􀀁 If g is monotonically decreasing. |  |  |  |  |  |\n| Constant Multiple: | Y = cX ! fY (y) = fX (k(y)) 􀀂 jk0 (y)j = fX 􀀀 yc 􀀁 􀀂 ~~1~~c . |  |  |  |  |  |\n| Power: | Y = X 1/􀀜 ! fY (y) = 􀀜 y􀀜 􀀀1fX (y􀀜 ) |  |  |  |  |  |\n| Exponential: | Y = g (X) ! fY (y) = ~~1~~y 􀀁 fX (log y)\u003Cbr>\u003Cbr>MIXING & SPLICING |  |  |  |  |  |\n| Mixture: | fX (x) = w1 f2 (x) + w2 f2 (x) + 􀀁 􀀁 􀀁 + wk fk (x) Where w 1 + w2 + 􀀁 􀀁 􀀁 + wk = 1 . |  |  |  |  |  |\n| Splices: | 8\u003Cbr>>>>>>\u003Cbr>fX (x) = \u003C>>>>>\u003Cbr>: | w 1 ","cbCaie5689u8phoH","https://ap.wps.com/l/cbCaie5689u8phoH","pdf",1370051,14,"English","# Severity Distributions\n# Extreme Value Distributions\n# Scaling & Transformation\n# Mixing & Splicing\n# Tails & Limiting Distributions\n# Frequency Distributions\n## Frequency Generating Functions\n# General Class of Distributions","[{\"question\":\"What topics are covered in the Capstone Exam-A4 formula and review sheet?\",\"answer\":\"It covers severity and frequency distributions, extreme value distributions, scaling and transformation, mixture and splicing, and tail and limiting distribution results used in actuarial mathematics practice.\"},{\"question\":\"Which distributions are listed under severity distributions?\",\"answer\":\"The sheet lists Uniform, Exponential, Weibull, Gamma, Beta, Pareto, Single P. Pareto, and Lognormal, along with key CDF and expectation/variance formulas where shown.\"},{\"question\":\"What kind of formulas does the sheet provide for exams?\",\"answer\":\"It provides quick-reference formulas including distribution functions, moment generating functions, probability generating functions, and limiting expressions for survival, hazard rate, and mean excess loss.\"}]","Capstone Exam-A4 - Short-Term Actuarial Mathematics Formula & Review Sheet | PDF",1790494373,35]