Test2-version2-Solutions

# Test2-version2-Solutions - ACTSC 431 Loss Models 1 FALL...

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Unformatted text preview: ACTSC 431 - Loss Models 1 FALL 2007 TEST #2 Name : ID Number : 1. (8 marks) Suppose that the per payment r.v. Y p is PAR ( & = 5 ;¡ = 200) distributed. In addition, the number of payments N p has a zero-modi&ed BIN( n = 4 ;q = 0 : 4 ) distribution with a probability mass at , p M , of : 1 . The actuary responsible of reviewing the previous actuary¡s work decides to use a slightly di/erent model for the next period based on the claim experience : & the per payment r.v. Y p has the same distribution & the distribution of N p (see above) will be used instead for the number of losses N L We assume that a loss results in a (non-zero) payment with probability : 9 (independently of N L ). (a) (4 marks) Prove that the new distribution for N p is a zero-modi&ed BIN ( n = 4 ;q = 0 : 36) with proba- bility mass at equal to : 1395 . Solution: We know that P N p ( t ) = P N L (0 : 1 + 0 : 9 t ) , where P N L ( t ) = 0 : 1 + 1 ¡ : 1 1 ¡ (0 : 6) 4 & (0 : 6 + 0 : 4 t ) 4 ¡ (0 : 6) 4 ¡ . It follows that P N p ( t ) = : 1 + 1 ¡ : 1 1 ¡ (0 : 6) 4 & (0 : 6 + 0 : 4 (0 : 1 + 0 : 9 t )) 4 ¡ (0 : 6) 4 ¡ = : 1 + 1 ¡ : 1 1 ¡ (0 : 6) 4 & (0 : 64 + 0 : 36 t ) 4 ¡ (0 : 6) 4 ¡ . (1) To prove that N p is a zero-modi&ed BIN ( n = 4 ;q = 0 : 36) with probability mass at equal to : 1395 , one shall prove that P N p ( t ) = 0 : 1395 + 1 ¡ : 1395 1 ¡ (0 : 64) 4 & (0 : 64 + 0 : 36 t ) 4 ¡ (0 : 64) 4 ¡ . Simple modi&cations of (1) lead to P N p ( t ) = : 1 + 1 ¡ : 1 1 ¡ (0 : 6) 4 & (0 : 64 + 0 : 36 t ) 4 ¡ (0 : 64) 4 + (0 : 64) 4 ¡ (0 : 6) 4 ¡ = : 1 + 1 ¡ : 1 1 ¡ (0 : 6) 4 & (0 : 64) 4 ¡ (0 : 6) 4 ¡ + 1 ¡ : 1 1 ¡ (0 : 6) 4 & (0 : 64 + 0 : 36 t ) 4 ¡ (0 : 64) 4 ¡ = : 1 + 1 ¡ : 1 1 ¡ (0 : 6) 4 & 1 ¡ (0 : 6) 4 ¡ & 1 ¡ (0 : 64) 4 ¡¡ + 1 ¡ : 1 1 ¡ (0 : 6) 4 & (0 : 64 + 0 : 36 t ) 4 ¡ (0 :...
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## This note was uploaded on 11/23/2010 for the course ACTSC act431 taught by Professor Na during the Spring '09 term at Waterloo.

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Test2-version2-Solutions - ACTSC 431 Loss Models 1 FALL...

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