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Includes a quick-reference set of key estimation and goodness-of-fit expressions, likelihood and log-likelihood formulations, and common distribution functions such as exponential, gamma, lognormal, Pareto (including three-parameter), Burr, and Weibull. Provides risk-model fundamentals including collective/compound models, Poisson-type constructions, and excess-of-loss and proportional reinsurance claims. Also summarizes copula components, Sklar’s theorem, and named copulas with marginal and joint distribution notation.",{"@graph":69,"@context":122},[70,84,105],{"@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":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/actex-learning-exam-cs2-formula-review-sheet/398999/",{"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/actex-learning-exam-cs2-formula-review-sheet/398999.png","ImageObject",300,407,{"name":92,"@type":93},"nayy☆","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-16","2026-09-27",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What topics are covered in the Exam CS2 Formula & Review Sheet?","Question",{"text":112,"@type":113},"It covers estimation and goodness-of-fit formulas, likelihood/log-likelihood expressions, common distribution functions, reinsurance payment expectations, risk models, and copulas including Sklar’s theorem.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Which distribution formulas are included for quick reference?",{"text":117,"@type":113},"The sheet includes exponential, gamma, lognormal, Pareto (including three-parameter), Burr, and Weibull distribution functions and related expressions.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the document treat copulas and joint distributions?",{"text":121,"@type":113},"It defines copulas in terms of marginal and joint distribution functions and states Sklar’s theorem, then lists specific copula forms such as product, co-monotonic, counter-monotonic, Gumbel, Frank, and Clayton, plus Gaussian and Student’s t copulas.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},398999,1791534689,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"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},962090893153,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Formula & Review Sheet for Exam CS2  \nYour quick-reference guide for the most important formulas and concepts you need.  \nWant More Than Just the Formula Sheet?  \nThis free Formula & Review Sheet is just one part of the ACTEX Learning  \nexperience. Our complete study program is designed to help you understand concepts, practice effectively, and pass with confidence.  \nWith the ACTEX Learning Bundle for Exam CS2, you get:  \n Core Readings  \n GOAL Practice Problems  \n Instructional Videos  \n Flashcards  \n Formula & Review Sheets  \n Learning Platform with a Study Planner and Topic Explorer  \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.  \nPopular Upgrades:  \n One-On-One Tutoring  \n Graded Mock Exams  \nLearn More  \nClick anywhere on this page or scan the QR code to learn more about offerings for this exam.  \nACTEX Learning  \nExam CS2 Formula & Review Sheet  \n(updated 05/16/2024)  \nESTIMATION AND GOODNESS OF FIT  \nMethod of Moments: mj = 1n i1 xji j = 1 , 2 . . . r  \nwith xi = the ith value in the sample & n = the sample size  \nLikelihood Function L (θ):  \nLog-likelihood Function L (θ):  \nExponential Distribution:  \nGamma Distribution:  \nLognormal Distribution:  \nPareto Distribution:  \nBurr Distribution:  \nThree-Parameter Pareto Distribution:  \nWeibull Distribution:  \nn  \nL (θ) = Q P (Xi = xi | θ)  \ni=1  \nn  \nL (θ) = Q f (xi | θ)  \ni=1  \nn  \nl (θ) = log L(θ) = P log P (Xi = xi | θ)  \ni=1  \nn  \nl (θ) = log L(θ) = P log f (xi | θ) i=1  \nF (x) = 1 − e −λx, x > 0  \nf (x) = Γλ(αα) xα−1e −λx, x > 0  \nlog (X) ∼ N 􀀀µ, σ2 􀀁 then X ∼ log N 􀀀µ, σ2 􀀁 F (x) = 1 − 􀀒λ~~ ~~~~ ~~x 􀀓 α , x > 0  \n λα  F (x) = 1 − (λ + xγ )α , x > 0  \nΓ(α + k)λα xk−1  fX (x) = Γ(α)Γ(k) (λ + x)α+k , x > 0 F (x) = 1 − e −cxγ , x > 0  \n(discrete)  \n(continuous)  \n(discrete)  \n(continuous)  \nf (x) = λe −λx, x > 0 α  α  \nE (X) = Var(X) =  \nλ λ2  \nMLEs of µ and σ 2 are y¯ and s2y  αλα   \nf (x) = , x > 0  \n(λ + x)α+1  \nf (x) = cγxγ−1e −cxγ , x > 0  \nREINSURANCE  \n8  \nExcess of Loss Reinsurance: Payment of Insurer (Y ) = \u003C> X if X ≤ M  \n> M if X > M  \n:  \nE (Y ) =R0M xf (x)dx + MP(X > M )  \nMGF of Y: MY (t) = E 􀀀etY 􀀁 = R0M etxf (x)dx + etMP (X > M )  \nPayment of Reinsurer (Z) = X − Y  \nE (Z) =R(x − M)f (x)dx  \n8  \nPayment of Insurer (Y ) = \u003C> kX if kX ≤ M  \nEffect of Inflation:  \n> M if kX > M  \n:  \nE (Y ) =R0M /k kxf (x)dx + MP(X > M /k)  \nn  \nLikelihood Function: L (θ  ) = Q f (xi ; θ  ) × [(1 − F(M; θ  ))]m  \n1  \nReinsurer Claim: P (W \u003C w) =RwM+M 1~~ ~~−fF(x()M)dx = F~~ ~~(w~~ ~~+1)(−MF)~~ ~~(M) g (w) = f1(MM)) Proportional Reinsurance: Insurer: Y = αX, 0 \u003C α \u003C 1 Reinsurer: Z = (1 − α)X  \nMean: E (Y ) = αE(X) E (Z) = (1 − α)E (X)  \nVariance: Var (Y ) = α2 Var (X) Var (Z) = (1 − α)2 Var (X)  \n8  \nPolicy Excess: Y = \u003C> 0 if X ≤ L  \n> X − L if X > L  \n:  \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 Exam CS 2 Formula & Review Sheet 2  \nRISK MODELS  \nN  \nCollective Risk Model: S = X1 + X2 + · · · + XN = P Xi  \ni=1  \n∞  \nP (S ≤ x) = P P (N = n)P (S ≤ x | N = n)  \nn=0  \nE [S] = E[E(S | N)] = E[N]m 1 = E (N)E (X)  \nVar[S] = E[Var[S | N]] + Var[E[S | N]] = E[N] 􀀀m2 − m21􀀁 + Var[N]m21 = E (N)V (X) + V(N)[E(X)]2  \nMS (t) = E[E[exp{tS} | N]] E [exp{tS} | N] = [MX (t)]N  \nCompound Poisson: E [S] = λm 1 Var[S] = λm2  \nMS (t) = exp {λ (MX (t) − 1)} skew [S] = λm3  \nM (t) = 1Λ i1 λi R0∞ exp{tx}fi (x)dx = 1Λ i1 λiMi (t)  \nMA (t) = exp{Λ(M(t) − 1)}  \nCompound Binomial: E [S] = npm 1 Var[S] = npm2 − np2 m21  \nMS (t) = (pMX (t) + 1 − p)n skew [S] = npm3 − 3np2 m2 m 1 + 2np3 m31  \nCompound Negative Binomial:  \nE [S] = kqpm 1 Var[S] = kqpm2 + kqp22 m21  \nMS (t) = pk / (1 − qMX (t))k skew [S] = 3kq2mp21~~ ~~m2 + 2kqp33m3~~1~~ + kqmp3  \nIndividual Excess of Loss: SI = Y1 + Y2 + . . . + YN SR = Z1 + Z2 + . . . + ZN  \nAggregate Exce","cbCain8mSlmmFCNd","https://ap.wps.com/l/cbCain8mSlmmFCNd","pdf",1036090,15,"English","# ESTIMATION AND GOODNESS OF FIT\n# REINSURANCE\n# RISK MODELS\n# COPULAS: MARGINAL AND JOINT DISTRIBUTIONS","[{\"question\":\"What topics are covered in the Exam CS2 Formula \\u0026 Review Sheet?\",\"answer\":\"It covers estimation and goodness-of-fit formulas, likelihood/log-likelihood expressions, common distribution functions, reinsurance payment expectations, risk models, and copulas including Sklar’s theorem.\"},{\"question\":\"Which distribution formulas are included for quick reference?\",\"answer\":\"The sheet includes exponential, gamma, lognormal, Pareto (including three-parameter), Burr, and Weibull distribution functions and related expressions.\"},{\"question\":\"How does the document treat copulas and joint distributions?\",\"answer\":\"It defines copulas in terms of marginal and joint distribution functions and states Sklar’s theorem, then lists specific copula forms such as product, co-monotonic, counter-monotonic, Gumbel, Frank, and Clayton, plus Gaussian and Student’s t copulas.\"}]","ACTEX Learning Exam CS2 - Formula & Review Sheet | PDF",1790491005,38]