QuestionSet2

# QuestionSet2 - ACTSC 431/831 QUESTION SET 2 1 Suppose that...

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ACTSC 431/831 – QUESTION SET 2 1. Suppose that Λ GAM( r,β ). Given Λ = λ , let N POI( λ + μ ) where μ > 0 is a constant. (a) Show that N has pgf P ( z ) = X n =0 p n z n = e μ ( z - 1) [1 - β ( z - 1)] - r . (b) Find p 0 . (c) Show that P 0 ( z ) = [ μ + (1 + β - βz ) - 1 ] P ( z ), and hence (1 + β ) P 0 ( z ) = βzP 0 ( z ) + [ μ (1 + β ) + ] P ( z ) - μβzP ( z ) . (d) Use part (c) to show that p 1 = [ μ + rβ/ (1 + β )] p 0 and p n +1 = [( r + n ) β + μ (1 + β )] p n - μβp n - 1 ( n + 1)(1 + β ) for n = 1 , 2 , 3 ,... . (e) Assuming that μ = 0 . 5, r = 3, and β = 2 for this part only, use parts (b) and (d) to calculate p 0 , p 1 , p 2 , p 3 , and p 4 . (f) Show that P ( z ) = exp { θ ( n =1 k n z n - 1) } where θ = μ + r ln(1 + β ), k 1 = μ + rβ/ (1 + β ) θ and k n = r µ β 1 + β n for n = 2 , 3 , 4 ,... . (g) Assuming again that μ = 0 . 5, r = 3, and β = 2, use the compound Poisson recursive formula this time to calculate p 0 , p 1

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## This note was uploaded on 06/28/2009 for the course ACTSC 431 taught by Professor Laundriualt during the Spring '09 term at Waterloo.

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QuestionSet2 - ACTSC 431/831 QUESTION SET 2 1 Suppose that...

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