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ACTS 4304 FORMULA SUMMARY Lesson 1: Basic Probability Summary of Probability Concepts Probability Functions

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ACTS 4304 FORMULA SUMMARY Lesson 1: Basic Probability Summary of Probability Concepts Probability Functions F (x) = P r(X ≤ x) S(x) = 1 − F (x) dF (x) f(x) = dx H(x) = − ln S(x) dH(x) f(x) h(x) = = dx S(x) Functions of random variables Z 1 Expected Value E[g(x)] = g(x)f(x)dx −∞ 0 n n-th raw moment µn = E[X ] n n-th central moment µn = E[(X − µ) ] Variance σ2 = E[(X − µ)2] = E[X2] − µ2 µ µ0 − 3µ0 µ + 2µ3 Skewness γ = 3 = 3 2 1 σ3 σ3 µ µ0 − 4µ0 µ + 6µ0 µ2 − 3µ4 Kurtosis γ = 4 = 4 3 2 2 σ4 σ4 Moment generating function M(t) = E[etX ] Probability generating function P (z) = E[zX ] More concepts • Standard deviation (σ) is positive square root of variance • Coefficient of variation is CV = σ/µ • 100p-th percentile π is any point satisfying F (π−) ≤ p and F (π) ≥ p. If F is continuous, it is the unique point satisfying F (π) = p • Median is 50-th percentile; n-th quartile is 25n-th percentile • Mode is x which maximizes f(x) (n) n (n) • MX (0) = E[X ], where M is the n-th derivative (n) PX (0) • n! = P r(X = n) (n) • PX (1) is the n-th factorial moment of X. Bayes' Theorem P r(BjA)P r(A) P r(AjB) = P r(B) fY (yjx)fX (x) fX (xjy) = fY (y) Law of total probability 2 If Bi is a set of exhaustive (in other words, P r([iBi) = 1) and mutually exclusive (in other words P r(Bi \ Bj) = 0 for i 6= j) events, then for any event A, X X P r(A) = P r(A \ Bi) = P r(Bi)P r(AjBi) i i Correspondingly, for continuous distributions, Z P r(A) = P r(Ajx)f(x)dx Conditional Expectation Formula EX [X] = EY [EX [XjY ]] 3 Lesson 2: Parametric Distributions Forms of probability density functions for common distributions Distribution Probability density function f(x) Uniform c; x 2 [d; u] Beta cxa−1(θ − x)b−1; x 2 [0; θ] − x Exponential ce θ ; x ≥ 0 Weibull cxτ−1e−xτ /θτ ; x ≥ 0 α−1 − x Gamma cx e θ ; x ≥ 0 c Pareto (x+θ)α+1 ; x ≥ 0 c Single-parameter Pareto xα+1 ; x ≥ θ 2 2 ce−(ln x−µ) =2σ Lognormal x ; x > 0 Summary of Parametric Distribution Concepts • If X is a member of a scale family with scale parameter θ with value s, then cX is the same family and has the same parameter values as X except that the scale parameter θ has value cs. • All distributions in the tables are scale families with scale parameter θ except for lognormal and inverse Gaussian. • If X is lognormal with parameters µ and σ, then cX is lognormal with parameters µ + ln c and σ. • See the above table to learn the forms of commonly occurring distributions. Useful facts are d + u (u − d)2 Uniform on [d, u] E[X] = ; V arX = 2 12 u2 Uniform on [0, u] E[X2] = 3 Gamma V arX = αθ2 • If Y is single-parameter Pareto with parameters α and θ, then Y − θ is two-parameter Pareto with parameters. • X is in the linear exponential family if its probability density function can be expressed as p(x)er(θ)x f(x; θ) = q(θ) 4 Lesson 3: Variance For any random variables X and Y; E[aX + bY ] = aE[X] + bE[Y ] V ar(aX + bY ) = a2V ar(X) + 2abCov(X; Y ) + b2V ar(Y ) n ! n X X For independent random variables X1;X2; ··· ;Xn; V ar Xi = V ar(Xi) i=1 i=1 For independent identically distributed random variables (i.i.d.) X1;X2; ··· ;Xn; n ! X V ar Xi = nV ar(X) i=1 n 1 X The sample mean X¯ = X n i i=1 1 The variance of the sample mean V ar(X¯) = V ar(X) n Double expectation EX [X] = EY [EX [XjY ]] Conditional variance V arX [X] = EY [V arX [XjY ]] + V arY (EX [XjY ]) 5 Lesson 4: Mixtures and Splices Pn • If X is a mixture of n random variables with weights wi such that i=1 wi = 1, then the following can be expressed as a weighted average: n X Cumulative distribution function: FX (x) = wiFXi (x) i=1 n X Probability density function: fX (x) = wifXi (x) i=1 n k X k k-th raw moment: E[X ] = wiE[Xi ] i=1 • Conditional variance: V arX [X] = EI [V arX [XjI]] + V arI (EX [XjI]) • Splices: For a spliced distribution, the sum of the probabilities of being in each splice must add up to 1. 6 Lesson 5: Policy Limits All formulas assume P r(X < 0) = 0. Z 1 E[X] = S(x) dx 0 Z u Z u E[X ^ u] = xf(x) dx + u (1 − F (u)) = S(x) dx 0 0 Z 1 E[Xk] = kxk−1S(x) dx 0 Z u Z u E[(X ^ u)k] = xkf(x) dx + uk (1 − F (u)) = kxk−1S(x) dx 0 0 For inflation, if Y = (1 + r)X, then u E[Y ^ u] = (1 + r)E X ^ 1 + r 7 Lesson 6: Deductibles Payment per Loss: L FY L (x) = FX (x + d); if Y = (X − d)+ Z 1 E[(X − d)+] = (x − d)f(x) dx d Z 1 E[(X − d)+] = S(x) dx d E[X] = E[X ^ d] + E[(X − d)+] Payment per Payment: FX (x + d) − FX (d) P FY P (x) = ; if Y = (X − d)+jX > d 1 − FX (d) SX (x + d) P SY P (x) = ; if Y = (X − d)+jX > d SX (d) E[(X − d) ] E[X] − E[X ^ d] e (d) = + = − mean excess loss X S(d) S(d) R 1(x − d)f(x) dx e (d) = d X S(d) R 1 S(x) dx e (d) = d X S(d) E[X] = E[X ^ d] + e(d) (1 − F (d)) Mean excess loss for different distributions: eX (d) = θ for exponential θ − d e (d) = ; d < θ for uniform on [0; θ] X 2 θ − d e (d) = ; d < θ for beta with parameters a = 1; b; θ X 1 + b θ + d e (d) = for two-parameter Pareto X α − 1 ( d α−1 d ≥ θ eX (d) = α(θ−d)+d for single-parameter Pareto α−1 d ≤ θ If Y L;Y P are loss and payment random variables for franchise deductible of d, and XL;XP are loss and payment random variables for ordinary deductible of d, then E[Y L] = E[XL] + dS(d) E[Y P ] = E[XP ] + d 8 Lesson 7: Loss Elimination Ratio The Loss Elimination Ratio is defined as the proportion of the expected loss which the insurer doesn't pay as a result of an ordinary deductible d: E[X ^ d] E[(X − d) ] LER(d) = = 1 − + E[X] E[X] Loss Elimination Ratio for Certain Distributions: LER(d) = 1 − e−d/θ for an exponential θ α−1 LER(d) = 1 − for a Pareto with α > 1 d + θ (θ=d)α−1 LER(d) = 1 − for a single-parameter Pareto with α > 1; d ≥ θ α 9 Lesson 8: Risk Measures and Tail Weight −1 Value-at-Risk: V aRp(X) = πp = FX (p) Tail-Value-at-Risk: R 1 xf(x) dx V aRp(X) T V aRp(X) = E [XjX > V aRp(X)] = = 1 − F (V aRp(X)) R 1 V aRy(X) dy = p = V aR (X) + e (V aR (X)) = 1 − p p X p E[X] − E[X ^ V aR (X)] = V aR (X) + p p 1 − p Value-at-Risk and Tail-Value-at-Risk measures for some distributions: Distribution V aRp(X) T V aRp(X) Exponential −θ ln(1 − p) θ (1 − ln(1 − p)) − 1 − 1 Pareto θ (1 − p) α − 1 E[X] 1 + α (1 − p) α − 1 2 −zp=2 Normal µ + z σ µ + σ · ep p 1−p 2π µ+zpσ Φ(σ−zp) Lognormal e E[X] · 1−p 10 Lesson 9: Other Topics in Severity Coverage Modifications Policy limit - the maximum amount that the coverage will pay. In the presence of a deductible or other modifications, perform the other modifications first, then the policy limit. Maximum coverage loss is the stipulated amount considered in calculating the payment. Apply this limit first, and then the deductible. If u is the maximum coverage loss and d - the deductible, then Y L = X ^ u − X ^ d Coinsurance of α is the portion of each loss reimbursed by insurance. In the presence of the three modifications, E[Y L] = α (E[X ^ u] − E[X ^ d]) If r is the inflation factor, u d E[Y L] = α(1 + r) E X ^ − E X ^ 1 + r 1 + r 11 Lesson 10: Bonuses A typical bonus is a portion of the excess of r% of premiums over losses. If c is the portion of the excess, r is the loss ratio, P is earned premium, and X is losses, then B = max (0; c(rP − X)) = crP − c min(rP; X) = crP − c (X ^ rP ) For a two-parameter Pareto distribution with α = 2 and θ, θd E[X ^ d] = d + θ 12 Lesson 11: Discrete Distributions For a (a; b; 0) class distributions, pk b = a + ; pk = P r(X = k) pk−1 k Poisson Binomial Negative binomial Geometric n r n λ m n+r−1 β βn −λ· n! n m−n 1 pn e n q (1 − q) n 1+β 1+β (1+β)n+1 Mean λ mq rβ β Variance λ mq(1 − q) rβ(1 + β) β(1 + β) q β β a 0 − 1−q 1+β 1+β q β b λ (m + 1) 1−q (r − 1) 1+β 0 For a (a; b; 1) class distributions, p0 is arbitrary and p b k = a + for k = 2; 3; 4; ··· pk−1 k Zero-truncated distributions: T pn pn = ; n > 0 1 − p0 Zero-modified distributions: M M T pn = (1 − p0 )pn E[N] = cm V ar(N) = c(1 − c)m2 + cv; where M • c is 1 − p0 • m is the mean of the corresponding zero-truncated distribution • v is the variance of the corresponding zero-truncated distribution 13 Lesson 12: Poisson/Gamma Assume that in a portfolio of insureds, loss frequency follows a Poisson distribution with parameter λ, but λ is not fixed but varies by insured.
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  • CONTINUOUS DISTRIBUTIONS Laplace Transform

    CONTINUOUS DISTRIBUTIONS Laplace Transform

    J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 1 CONTINUOUS DISTRIBUTIONS Laplace transform (Laplace-Stieltjes transform) Definition The Laplace transform of a non-negative random variable X 0 with the probability density ≥ function f(x) is defined as st sX st f ∗(s)= ∞ e− f(t)dt = E[e− ] = ∞ e− dF (t) alsodenotedas (s) Z0 Z0 LX Mathematically it is the Laplace transform of the pdf function. • In dealing with continuous random variables the Laplace transform has the same role as • the generating function has in the case of discrete random variables. s – if X is a discrete integer-valued ( 0) r.v., then f ∗(s)= (e− ) ≥ G J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 2 Laplace transform of a sum Let X and Y be independent random variables with L-transforms fX∗ (s) and fY∗ (s). s(X+Y ) fX∗ +Y (s) = E[e− ] sX sY = E[e− e− ] sX sY = E[e− ]E[e− ] (independence) = fX∗ (s)fY∗ (s) fX∗ +Y (s)= fX∗ (s)fY∗ (s) J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 3 Calculating moments with the aid of Laplace transform By derivation one sees d sX sX f ∗0(s)= E[e− ]=E[ Xe− ] ds − Similarly, the nth derivative is n (n) d sX n sX f ∗ (s)= E[e− ] = E[( X) e− ] dsn − Evaluating these at s = 0 one gets E[X] = f ∗0(0) − 2 E[X ] =+f ∗00(0) . n n (n) E[X ] = ( 1) f ∗ (0) − J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 4 Laplace transform of a random sum Consider the random sum Y = X + + X 1 ··· N where the Xi are i.i.d.