ACTEX Learning Exam CS2 - Formula & Review Sheet

Sep 27, 2026

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Formula & Review Sheet for Exam CS2 from ACTEX Learning. 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.

LanguageEnglish

CategoryTechnology

FormatPDF

Pages15

Reading Time38 min

AccessFree to download

ESTIMATION AND GOODNESS OF FIT

REINSURANCE

RISK MODELS

COPULAS: MARGINAL AND JOINT DISTRIBUTIONS

Formula & Review Sheet for Exam CS2 Your quick-reference guide for the most important formulas and concepts you need. Want More Than Just the Formula Sheet? This 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. With the ACTEX Learning Bundle for Exam CS2, you get: Core Readings GOAL Practice Problems Instructional Videos Flashcards Formula & Review Sheets Learning Platform with a Study Planner and Topic Explorer Want an Extra Edge? Take your preparation even further with premium ACTEX resources designed to sharpen exam skills and boost confidence when it matters most. Popular Upgrades: One-On-One Tutoring Graded Mock Exams Learn More Click anywhere on this page or scan the QR code to learn more about offerings for this exam. ACTEX Learning Exam CS2 Formula & Review Sheet (updated 05/16/2024) ESTIMATION AND GOODNESS OF FIT Method of Moments: mj = 1n i1 xji j = 1 , 2 . . . r with xi = the ith value in the sample & n = the sample size Likelihood Function L (θ): Log-likelihood Function L (θ): Exponential Distribution: Gamma Distribution: Lognormal Distribution: Pareto Distribution: Burr Distribution: Three-Parameter Pareto Distribution: Weibull Distribution: n L (θ) = Q P (Xi = xi | θ) i=1 n L (θ) = Q f (xi | θ) i=1 n l (θ) = log L(θ) = P log P (Xi = xi | θ) i=1 n l (θ) = log L(θ) = P log f (xi | θ) i=1 F (x) = 1 − e −λx, x > 0 f (x) = Γλ(αα) xα−1e −λx, x > 0 log (X) ∼ N 􀀀µ, σ2 􀀁 then X ∼ log N 􀀀µ, σ2 􀀁 F (x) = 1 − 􀀒λ~~ ~~~~ ~~x 􀀓 α , x > 0 λα F (x) = 1 − (λ + xγ )α , x > 0 Γ(α + k)λα xk−1 fX (x) = Γ(α)Γ(k) (λ + x)α+k , x > 0 F (x) = 1 − e −cxγ , x > 0 (discrete) (continuous) (discrete) (continuous) f (x) = λe −λx, x > 0 α α E (X) = Var(X) = λ λ2 MLEs of µ and σ 2 are y¯ and s2y αλα f (x) = , x > 0 (λ + x)α+1 f (x) = cγxγ−1e −cxγ , x > 0 REINSURANCE 8 Excess of Loss Reinsurance: Payment of Insurer (Y ) = <> X if X ≤ M > M if X > M : E (Y ) =R0M xf (x)dx + MP(X > M ) MGF of Y: MY (t) = E 􀀀etY 􀀁 = R0M etxf (x)dx + etMP (X > M ) Payment of Reinsurer (Z) = X − Y E (Z) =R(x − M)f (x)dx 8 Payment of Insurer (Y ) = <> kX if kX ≤ M Effect of Inflation: > M if kX > M : E (Y ) =R0M /k kxf (x)dx + MP(X > M /k) n Likelihood Function: L (θ ) = Q f (xi ; θ ) × [(1 − F(M; θ ))]m 1 Reinsurer Claim: P (W < w) =RwM+M 1~~ ~~−fF(x()M)dx = F~~ ~~(w~~ ~~+1)(−MF)~~ ~~(M) g (w) = f1(MM)) Proportional Reinsurance: Insurer: Y = αX, 0 < α < 1 Reinsurer: Z = (1 − α)X Mean: E (Y ) = αE(X) E (Z) = (1 − α)E (X) Variance: Var (Y ) = α2 Var (X) Var (Z) = (1 − α)2 Var (X) 8 Policy Excess: Y = <> 0 if X ≤ L > X − L if X > L : [www.ACTEXLearning.com](www.ACTEXLearning.com) Need Help? Email [support@actexlearning.com](support@actexlearning.com) Copyright © ArchiMedia Advantage Inc. All Rights Reserved ACTEX Learning Exam CS 2 Formula & Review Sheet 2 RISK MODELS N Collective Risk Model: S = X1 + X2 + · · · + XN = P Xi i=1 ∞ P (S ≤ x) = P P (N = n)P (S ≤ x | N = n) n=0 E [S] = E[E(S | N)] = E[N]m 1 = E (N)E (X) Var[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 MS (t) = E[E[exp{tS} | N]] E [exp{tS} | N] = [MX (t)]N Compound Poisson: E [S] = λm 1 Var[S] = λm2 MS (t) = exp {λ (MX (t) − 1)} skew [S] = λm3 M (t) = 1Λ i1 λi R0∞ exp{tx}fi (x)dx = 1Λ i1 λiMi (t) MA (t) = exp{Λ(M(t) − 1)} Compound Binomial: E [S] = npm 1 Var[S] = npm2 − np2 m21 MS (t) = (pMX (t) + 1 − p)n skew [S] = npm3 − 3np2 m2 m 1 + 2np3 m31 Compound Negative Binomial: E [S] = kqpm 1 Var[S] = kqpm2 + kqp22 m21 MS (t) = pk / (1 − qMX (t))k skew [S] = 3kq2mp21~~ ~~m2 + 2kqp33m3~~1~~ + kqmp3 Individual Excess of Loss: SI = Y1 + Y2 + . . . + YN SR = Z1 + Z2 + . . . + ZN Aggregate Exce

FAQs

What topics are covered in the Exam CS2 Formula & Review Sheet?

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.

Which distribution formulas are included for quick reference?

The sheet includes exponential, gamma, lognormal, Pareto (including three-parameter), Burr, and Weibull distribution functions and related expressions.

How does the document treat copulas and joint distributions?

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.

Sep 27, 2026

Upload by nayy☆