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Solutions13

# Solutions13 - EE464 Homework#12 Solutions Problem#1 Note...

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EE464 Homework #12 Solutions Problem #1 Note that for a Poisson random variable with parameter λ we have M X ( t ) = exp( λ ( e t - 1)). Since the X’s are iid we have M S ( t ) = exp(80 λ ( e t - 1)) and so we see that S is Poisson with parameter 80 × . 25 = 20. We now plot this theoretical pdf vs the approximation given in eqn 7.30. where the mean and variance are both 20. Note how closely these plots agree. Problem #2 (a) We first note that ¯ M n = max( ¯ M n - 1 , X n ) for n > 1 and that ¯ M 1 = X 1 . Since this system has memory 1 we conclude that it is indeed a Markov chain. To show this rigorously we write P ( ¯ M n = m n | ¯ M n - 1 = m n - 1 , ..., ¯ M 1 = m 1 ) = P (max( ¯ M n - 1 , X n ) = m n | ¯ M n - 1 = m n - 1 , ..., ¯ M 1 = m 1 ) = P (max( m n - 1 , X n ) = m n | ¯ M n - 1 = m n - 1 , ..., ¯ M 1 = m 1 ) = P (max( m n - 1 , X n ) = m n ) = P ( ¯ M n = m n | ¯ M n - 1 = m n - 1 ), where we have used the fact that the X n ’s are inde- pendent of the past ¯ M n ’s, since the previous ¯ M n ’s are functions of only the 1

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previous X n ’s and the sequence is iid.
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Solutions13 - EE464 Homework#12 Solutions Problem#1 Note...

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