soln9a - ECE320 Solution Notes 9 Cornell University Spring...

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ECE320 Solution Notes 9 Spring 2006 Cornell University T.L.Fine 1. Recall the Bernoulli process of Section 3.6 in which the outcome space X = { 0 , 1 } and the probability of a sequence of binary-valued random variables is given by P ( X 1 = x 1 , . . . , X n = x n ) = p P n 1 x i (1 - p ) n - P n 1 x i for some 0 < p < 1 . (a) Show that the Bernoulli process is also a Markov chain by evaluating the conditional probability P ( X n = x n | x 1 = x 1 , . . . , X n - 1 = x n - 1 and showing that it satisfies the Markov condition in that it does not depend upon x 1 , . . . , x n - 2 . (Unusually, it will also turn out not to depend upon x n - 1 .) P ( X n = x n | X 1 = x 1 , . . . , X n - 1 = x n - 1 ) = P ( X 1 = x 1 , . . . , X n = x n ) P ( X 1 = x 1 , . . . , X n - 1 = x n - 1 ) = p x n + P n - 1 1 x i (1 - p ) n - x n - P n - 1 1 x i p P n - 1 1 x i (1 - p ) n - 1 - P n - 1 1 x i = p x n (1 - p ) 1 - x n . The condition for being a Markov chain is satisfied as the conditional probability does not depend upon x 1 , . . . , x n - 2 . This Markov chain has only the two states X = { 0 , 1 } . (b) Identify the initial distribution π (1) for this Markov chain. π (1) = [ P ( X 1 = 0) , P ( X 1 = 1)] = [1 - p, p ] , and sums to one, as it should. (c) Identify the one-step transition matrix P and see that you have a special case of all rows being identical. P = [ p i,j ] , p i,j = P ( X 2 = ξ j | X 1 = ξ i ) , P = 1 - p p 1 - p p . The rows sum to one, as they should. They are also identical, which means that P ( X 2 = ξ j | X 1 = ξ i ) = P ( X 2 = ξ j ) .
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