# hw9 - ECE320 Homework 9 Cornell University Spring 2006...

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ECE320 Homework 9 Spring 2006 Cornell University T.L.Fine Please hand in this assignment at the end of lecture on Tuesday, 11 April. Use only your assigned three-digit code and not your name. Throughout, give reasons for your answers. 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 .) (b) Identify the initial distribution π (1) for this Markov chain,. (c) Identify the one-step transition matrix P and see that you have a special case of all rows being identical. (d) Is this Markov chain homogeneous?

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