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Test2-S08-Solutions

Test2-S08-Solutions - ACTSC 431 Loss Models 1 Spring 2008...

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ACTSC 431 - Loss Models 1 Spring 2008 TEST #2 1. (22 marks) Suppose that the ground-up loss X is a discrete random variable (r.v.) with probability mass function (p.m.f.) p X ( x ) = 8 < : 0 : 5 ; x = 2 0 : 3 ; x = 4 0 : 2 ; x = 5 . and that the number of losses N L has a zero-modi°ed NB( r = 5 ; ° = 0 : 4 ) distribution with a probability mass at 0 , p M 0 , of 0 : 1 . We consider a policy with an ordinary deductible of 2 . (a) (7 marks) Prove that the distribution of the number of (non-zero) payments N p is a zero-modi°ed NB ( r = 5 ; ° = 0 : 2) with probability mass at 0 equal to 0 : 3387 . Solution: Here, the probability that a loss results in a zero-payment is the probability that the ground-up loss is 2 since when X = 2 , Y L = max ( X ° 2 ; 0) = 0 . It follows that Pr ( Y L = 0) = Pr ( X = 2) = 0 : 5 . We also know that P N p ( t ) = P N L (0 : 5 + 0 : 5 t ) , where P N L ( t ) = 0 : 1 + 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 ² 1 1 ° 0 : 4 ( t ° 1) ³ 5 ° ² 1 1 + 0 : 4 ³ 5 ! . One arrives at P N p ( t ) = 0 : 1 + 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 ² 1 1 ° 0 : 4 (0 : 5 + 0 : 5 t ° 1) ³ 5 ° ² 1 1 + 0 : 4 ³ 5 ! = 0 : 1 + 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 ² 1 1 ° 0 : 2 ( t ° 1) ³ 5 ° ² 1 1 + 0 : 4 ³ 5 ! . (1) To prove that N p is a zero-modi°ed NB ( r = 5 ; ° = 0 : 2) with probability mass at 0 equal to 0 : 3387 , one shall prove that P N p ( t ) = 0 : 3387 + 1 ° 0 : 3387 1 ° ´ 1 1 : 2 µ 5 ² 1 1 ° 0 : 2 ( t ° 1) ³ 5 ° ² 1 1 : 2 ³ 5 ! . 1

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Simple modi°cations of (1) lead to P N p ( t ) = 0 : 1 + 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 ² 1 1 ° 0 : 2 ( t ° 1) ³ 5 ° ² 1 1 + 0 : 2 ³ 5 + ² 1 1 + 0 : 2 ³ 5 ° ² 1 1 + 0 : 4 ³ 5 ! = 0 : 1 + 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 ² 1 1 + 0 : 2 ³ 5 ° ² 1 1 + 0 : 4 ³ 5 ! + 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 ² 1 1 ° 0 : 2 ( t ° 1) ³ 5 ° ² 1 1 + 0 : 2 ³ 5 = 0 : 1 + 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 1 ° ² 1 1 + 0 : 4 ³ 5 ° 1 ° ² 1 1 + 0 : 2 ³ 5 !! + 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 ² 1 1 ° 0 : 2 ( t ° 1) ³ 5 ° = 1 ° 1 ° 0 : 1 1 ° ° 1 1+0 : 4 ± 5 1 ° ² 1 1 + 0 : 2 ³ 5 !
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Test2-S08-Solutions - ACTSC 431 Loss Models 1 Spring 2008...

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