Tables for Exam C The reading material for Exam C includes a variety of textbooks. Each text has a set of probability distributions that are used in its readings. For those distributions used in more than one text, the choices of parameterization may not be the same in all of the books. This may be of educational value while you study, but could add a layer of uncertainty in the examination. For this latter reason, we have adopted one set of parameterizations to be used in examinations. This set will be based on Appendices A & B of Loss Models: From Data to Decisions by Klugman, Panjer and Willmot. A slightly revised version of these appendices is included in this note. A copy of this note will also be distributed to each candidate at the examination. Each text also has its own system of dedicated notation and terminology. Sometimes these may conflict. If alternative meanings could apply in an examination question, the symbols will be defined. For Exam C, in addition to the abridged table from Loss Models, sets of values from the standard normal and chi-square distributions will be available for use in examinations. These are also included in this note. When using the normal distribution, choose the nearest z-value to find the probability, or if the probability is given, choose the nearest z-value. No interpolation should be used. Example: If the given z-value is 0.759, and you need to find Pr(Z < 0.759) from the normal distribution table, then choose the probability for z-value = 0.76: Pr(Z < 0.76) = 0.7764. When using the normal approximation to a discrete distribution, use the continuity correction. 1 1 − x2 The density function for the standard normal distribution is φ()xe= 2 . 2π Excerpts from the Appendices to Loss Models: From Data to Decisions, 3rd edition August 27, 2009 Appendix A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by 1 x Γ(α; x)= tα 1e t dt, α > 0,x>0 Γ(α) − − Z0 ∞ α 1 t with Γ(α)= t − e− dt, α > 0. Z0 Also, define ∞ α 1 t G(α; x)= t − e− dt, x > 0. Zx At times we will need this integral for nonpositive values of α. Integration by parts produces the relationship xαe x 1 G(α; x)= − + G(α +1;x) − α α This can be repeated until the first argument of G is α + k, a positive number. Then it can be evaluated from G(α + k; x)=Γ(α + k)[1 Γ(α + k; x)]. − The incomplete beta function is given by x Γ(a + b) a 1 b 1 β(a, b; x)= t − (1 t) − dt, a > 0,b>0, 0 <x<1. Γ(a)Γ(b) − Z0 1 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 2 A.2 Transformed beta family A.2.2 Three-parameter distributions A.2.2.1 Generalized Pareto (beta of the second kind)–α, θ, τ Γ(α + τ) θαxτ 1 x f(x)= − F (x)=β(τ,α; u),u= Γ(α)Γ(τ) (x + θ)α+τ x + θ θkΓ(τ + k)Γ(α k) E[Xk]= − , τ<k<α Γ(α)Γ(τ) − θkτ(τ +1) (τ + k 1) E[Xk]= ··· − , if k is an integer (α 1) (α k) − ··· − θkΓ(τ + k)Γ(α k) E[(X x)k]= − β(τ + k, α k; u)+ xk[1 F (x)],k>τ ∧ Γ(α)Γ(τ) − − − τ 1 mode = θ − ,τ>1,else0 α +1 A.2.2.2 Burr (Burr Type XII, Singh-Maddala)–α, θ, γ αγ(x/θ)γ 1 f(x)= F (x)=1 uα,u= x[1 + (x/θ)γ ]α+1 − 1+(x/θ)γ θkΓ(1 + k/γ)Γ(α k/γ) E[Xk]= − , γ<k<αγ Γ(α) − 1/α 1/γ VaRp(X)=θ[(1 p)− 1] − − θkΓ(1 + k/γ)Γ(α k/γ) E[(X x)k]= − β(1 + k/γ, α k/γ;1 u)+ xkuα,k>γ ∧ Γ(α) − − − γ 1 1/γ mode = θ − ,γ>1, else 0 αγ +1 µ ¶ A.2.2.3 Inverse Burr (Dagum)–τ,θ,γ τγ(x/θ)γτ (x/θ)γ f(x)= F (x)=uτ ,u= x[1 + (x/θ)γ]τ+1 1+(x/θ)γ θkΓ(τ + k/γ)Γ(1 k/γ) E[Xk]= − , τγ<k<γ Γ(τ) − 1/τ 1/γ VaR p(X)=θ(p− 1)− − θkΓ(τ + k/γ)Γ(1 k/γ) E[(X x)k]= − β(τ + k/γ, 1 k/γ; u)+ xk[1 uτ ],k>τγ ∧ Γ(τ) − − − τγ 1 1/γ mode = θ − ,τγ>1, else 0 γ +1 µ ¶ APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 3 A.2.3 Two-parameter distributions A.2.3.1 Pareto (Pareto Type II, Lomax)–α, θ αθα θ α f(x)= F (x)=1 (x + θ)α+1 − x + θ µ ¶ θkΓ(k +1)Γ(α k) E[Xk]= − , 1 <k<α Γ(α) − θkk! E[Xk]= ,ifk is an integer (α 1) (α k) − ···1/α − VaRp(X)=θ[(1 p)− 1] − − 1/α θ(1 p)− TVaR (X)=VaR (X)+ − ,α>1 p p α 1 − α 1 θ θ − E[X x]= 1 ,α=1 ∧ α 1 − x + θ 6 − " µ ¶ # θ E[X x]= θ ln ,α=1 ∧ − x + θ µ ¶ θkΓ(k +1)Γ(α k) θ α E[(X x)k]= − β[k +1,α k; x/(x + θ)] + xk , all k ∧ Γ(α) − x + θ µ ¶ mode =0 A.2.3.2 Inverse Pareto–τ,θ τθxτ 1 x τ f(x)= − F (x)= (x + θ)τ+1 x + θ µ ¶ θkΓ(τ + k)Γ(1 k) E[Xk]= − , τ<k<1 Γ(τ) − θk( k)! E[Xk]= − ,ifk is a negative integer (τ 1) (τ + k) −1/τ ··· 1 VaR p(X)=θ[p− 1]− − x/(x+θ) τ k k τ+k 1 k k x E[(X x) ]=θ τ y − (1 y)− dy + x 1 ,k>τ ∧ − − x + θ − Z0 ∙ µ ¶ ¸ τ 1 mode = θ − ,τ>1, else 0 2 A.2.3.3 Loglogistic (Fisk)–γ,θ γ(x/θ)γ (x/θ)γ f(x)= F(x)=u, u = x[1 + (x/θ)γ]2 1+(x/θ)γ E[Xk]=θkΓ(1 + k/γ)Γ(1 k/γ), γ<k<γ 1 1/γ − − VaR p(X)=θ(p− 1)− − E[(X x)k]=θkΓ(1 + k/γ)Γ(1 k/γ)β(1 + k/γ, 1 k/γ; u)+ xk(1 u),k>γ ∧ − − − − γ 1 1/γ mode = θ − ,γ>1, else 0 γ +1 µ ¶ APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 4 A.2.3.4 Paralogistic–α, θ This is a Burr distribution with γ = α. α2(x/θ)α 1 f(x)= F (x)=1 uα,u= x[1 + (x/θ)α]α+1 − 1+(x/θ)α θkΓ(1 + k/α)Γ(α k/α) E[Xk]= − , α<k<α2 Γ(α) − 1/α 1/α VaRp(X)=θ[(1 p)− 1] − − θkΓ(1 + k/α)Γ(α k/α) E[(X x)k]= − β(1 + k/α, α k/α;1 u)+ xkuα,k>α ∧ Γ(α) − − − α 1 1/α mode = θ − ,α>1, else 0 α2 +1 µ ¶ A.2.3.5 Inverse paralogistic–τ,θ This is an inverse Burr distribution with γ = τ. 2 τ 2(x/θ)τ (x/θ)τ f(x)= F (x)=uτ ,u= x[1 + (x/θ)τ ]τ+1 1+(x/θ)τ θkΓ(τ + k/τ)Γ(1 k/τ) E[Xk]= − , τ 2 <k<τ Γ(τ) − 1/τ 1/τ VaR p(X)=θ(p− 1)− − θkΓ(τ + k/τ)Γ(1 k/τ) E[(X x)k]= − β(τ + k/τ, 1 k/τ; u)+ xk[1 uτ ],k>τ 2 ∧ Γ(τ) − − − mode = θ (τ 1)1/τ ,τ>1, else 0 − A.3 Transformed gamma family A.3.2 Two-parameter distributions A.3.2.1 Gamma–α, θ (x/θ)αe x/θ f(x)= − F (x)=Γ(α; x/θ) xΓ(α) k α k θ Γ(α + k) M(t)=(1 θt)− ,t<1/θ E[X ]= ,k>α − Γ(α) − E[Xk]=θk(α + k 1) α, if k is an integer − ··· θkΓ(α + k) E[(X x)k]= Γ(α + k; x/θ)+xk[1 Γ(α; x/θ)],k>α ∧ Γ(α) − − = α(α +1) (α + k 1)θkΓ(α + k; x/θ)+ xk[1 Γ(α; x/θ)],kan integer ··· − − mode = θ(α 1),α>1, else 0 − APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 5 A.3.2.2 Inverse gamma (Vinci)–α, θ (θ/x)αe θ/x f(x)= − F (x)=1 Γ(α; θ/x) xΓ(α) − θkΓ(α k) θk E[Xk]= − ,k<α E[Xk]= , if k is an integer Γ(α) (α 1) (α k) − ··· − θkΓ(α k) E[(X x)k]= − [1 Γ(α k; θ/x)] + xkΓ(α; θ/x) ∧ Γ(α) − − θkΓ(α k) = − G(α k; θ/x)+xkΓ(α; θ/x), all k Γ(α) − mode = θ/(α +1) A.3.2.3 Weibull–θ, τ τ (x/θ)τ τ(x/θ) e− (x/θ)τ f(x)= F (x)=1 e− x − E[Xk]=θkΓ(1 + k/τ),k>τ 1/τ − VaR p(X)=θ[ ln(1 p)] − − k k τ k (x/θ)τ E[(X x) ]=θ Γ(1 + k/τ)Γ[1 + k/τ;(x/θ) ]+x e− ,k>τ ∧ − τ 1 1/τ mode = θ − ,τ>1, else 0 τ µ ¶ A.3.2.4 Inverse Weibull (log Gompertz)–θ, τ τ (θ/x)τ τ(θ/x) e τ f(x)= − F (x)=e (θ/x) x − E[Xk]=θkΓ(1 k/τ),k<τ − 1/τ VaRp(X)=θ( ln p)− − k k τ k (θ/x)τ E[(X x) ]=θ Γ(1 k/τ) 1 Γ[1 k/τ;(θ/x) ] + x 1 e− , all k ∧ − { − − } − k τ k h(θ/x)τ i = θ Γ(1 k/τ)G[1 k/τ;(θ/x) ]+x 1 e− − − − τ 1/τ h i mode = θ τ +1 µ ¶ APPENDIX A.