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Unformatted text preview: Credibility I ActSCi 432/832 1 / 16 Distributions Summary X is Binomial with parameters n (positive integer) and p ∈ (0 , 1) p . f . f ( x ) = n x p x (1 p ) n x for x = 0 , 1 , 2 , 3 ,...... n E ( X ) = np , Var ( X ) = np (1 p ) X is Bernoulli with parameters p ∈ (0 , 1) → X is Binomial with n = 1 X is Poisson with parameter λ > p . f . f ( x ) = λ x e λ x ! for x = 0 , 1 , 2 , 3 ,...... E ( X ) = λ, Var ( X ) = λ 2 / 16 Distributions Summary X is Gamma with parameters α > and β > p . f . f ( x ) = x α 1 e x /β Γ( α ) β α for x > E ( X k ) = β k Γ( α + k ) Γ( α ) , if k > α Note that Γ( α + 1) = α Γ( α ) for α > and Γ( α + 1) = α ! for α = 0 , 1 , 2 ,.... X is Exponential with parameters β > → X is Gamma with α = 1 X is Pareto with parameters α > 0 and β > p . f . f ( x ) = αβ α ( x + β ) α +1 for x > E ( X ) = β α 1 for α > 1 , Var ( X ) = αβ 2 ( α 1) 2 ( α 2) for α > 2 3 / 16 Introduction Suppose a new policyholder has a risk – say a fleet of cars – to insure. Insurer will look up standard rates and charge, say, M per unit of risk. This the the manual premium as it comes from the handbook or manual. After a few years, the experience of the individual risk is available, which is ¯ X . Now the true premium is E ( X ) which is to be estimated from M or ¯ X . Which one is a better indicator of E ( X )? 4 / 16 The credibility theory is to take into account both information in a weighted average form: P = Z ¯ X + (1 Z ) M P is the credibility estimate of E ( X ) Z , 0 ≤ Z ≤ 1, is the credibility factor and measures the...
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 Fall '09
 davidlandriault
 Variance, Probability theory, yp

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