Part3-Notes-431-2011-F

# Part3-Notes-431-2011-F - Review Notes for Loss Models 1...

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Unformatted text preview: Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2011 Part 3 – Frequency Models 1. Frequency models are used to model the number of events or claims/losses. 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 ) = ∞ X n =0 p n z n = p + p 1 z + p 2 z 2 + ··· , in particular p = P N (0), E ( N ) = P N (1) , E ( N ( N- 1)) = P 00 N (1) , and E ( N 2 ) = P 00 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 > . Alternately, a counting random variable N is said to be a geometric distribu- tion if the pf of N has the following form: Pr { N = k } = θ (1- θ ) k , k = 0 , 1 , 2 ,..., < 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 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...
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Part3-Notes-431-2011-F - Review Notes for Loss Models 1...

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