A4-S-431F08 - SOLUTIONS TO ASSIGNMENT 4 ACTSC 431/831, FALL...

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SOLUTIONS TO ASSIGNMENT 4 – ACTSC 431/831, FALL 2008 1. The conditional distribution of X , given Θ = θ , is a Poisson distribution with mean λθ . The pgf of X is given by P X ( z ) = M Θ ( λ ( z - 1)) , where M Θ ( t ) is the mgf of Θ. Since Θ is infinitely divisible, for any n = 1 , 2 ,..., there exists a mgf M n ( t ) so that M Θ ( t ) = [ M n ( t )] n . Thus, P X ( z ) = [ M n ( λ ( z - 1))] n . Since M n ( λ ( z - 1)) is still a pgf that is the pgf of a mixed Poisson distribution, in which the mixing distribution has the mgf M n ( t ). Hence, X is infinitely divisible. 2. (a) Let v = Pr { X j 50 } and let I j = 1 if X j 50 and 0 if X j > 50. Note that N P * = I 1 + ... + I N L . If N L is infinitely divisible, then for any n = 1 , 2 ,..., there exists a pgf P n ( z ) so that P N L ( z ) = [ P n ( z )] n . Thus, the pgf of N P * is P N P * ( z ) = P N L (1 + v ( z - 1)) = [ P n (1 + v ( z - 1))] n , which means that N P * is also infinitely divisible since P n (1+ v ( z - 1)) is still a pgf that is the pgf of a compound frequency model, in which P n ( z ) is the pgf of the primary distribution and the secondary distribution is the Bernoulli distribution
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This note was uploaded on 02/02/2010 for the course ACTSC 331 taught by Professor David during the Fall '09 term at Waterloo.

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A4-S-431F08 - SOLUTIONS TO ASSIGNMENT 4 ACTSC 431/831, FALL...

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