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Notes-Part3 - Review Notes for Loss Models 1 ACTSC 431/831...

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Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2008 Part 3 – Frequency Models 1. Frequency models are used to model the number of events or claims. 2. A counting random variable N is a nonnegative integer-valued random variable with pf p k = P { N = k } , k = 0 , 1 , 2 , ... and the distribution of a counting random variable is called a counting distribution . (a) The pgf of a counting random variable is given by P N ( z ) = E ( z N ) = n =0 p n z n = p 0 + p 1 z + p 2 z 2 + · · · , in particular p 0 = P N (0), E ( N ) = P N (1) , E ( N ( N - 1)) = P N (1) , and E ( N 2 ) = P N (1) + P N (1) . 3. Three important counting distributions: (a) Poisson distribution with E ( N ) = V ar ( N ). (b) Binomial distribution with E ( N ) > V ar ( N ). (c) Negative binomial distribution with E ( N ) < V ar ( N ) . i. A negative binomial distribution NB (1 , β ) is called a geometric distribu- tion with pf Pr { N = k } = 1 1 + β β 1 + β k , k = 0 , 1 , 2 , ..., β > 0 . Alternately, a counting random variable N is said to be a geometric distrib- ution if the pf of N has the following form: Pr { N = k } = θ (1 - θ ) k , k = 0 , 1 , 2 , ..., 0 < θ < 1 with E ( N ) = 1 - θ θ and V ar ( N ) = 1 - θ θ 2 . ii. Theorem (The memoryless property of a geometric distribution): If N has a geometric distribution NB (1 , β ), then for any n = 0 , 1 , 2 , ..., N - n | N n has the same geometric distribution as that of N , namely Pr { N - n = k | N n } = Pr { N = k } for k = 0 , 1 , 2 , .... 1
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4. Relationships between P ( λ ) , NB ( r, β ) , and b ( n, p ): (a) A negative binomial distribution NB ( r, β ) is the mixture of a Poisson distribution P ( λ ) and a gamma distribution G ( r, β ), i.e. if X | Λ = λ has a Poisson distribution P ( λ ) and Λ has a gamma distribution G ( r, β ), then X has a negative binomial distribution NB ( r, β ). (b) A Poisson distribution P ( λ ) is the limit of a sequence of binomial distributions { b ( n, λ n ) } , namely if X n has a binomial distribution b ( n, λ n ), then lim n →∞ Pr { X n = k } = e - λ λ k k ! . 5. Sum of counting random variables: Let N 1 , ..., N n be independent counting ran- dom variables. Then the following table summarizes the distributions of some impor- tant sums of counting random variables.
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