# zz-20 - Utah State University ECE 6010 Stochastic Processes...

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Utah State University ECE 6010 Stochastic Processes Homework #11 Due Friday Dec 10, 2004 These problems come from the Leon-Garcia text. 1. Let M n denote the sequence of sample means from an i.i.d. random process X n : M n = X 1 + x 2 + · · · + X n n (a) Is M n a Markov process? (b) If so, find the state transition p.m.f. f M n ( x | M n - 1 = y ) 2. An urn initially contains five black balls and five white balls. The following experiment is repeated indefinitely. A ball is drawn from the urn; if the ball is white it is put back in the urn, otherwise it is left out. Let X n be the number of black balls remaining in the urn after n draws from the urn. (a) Is X n a Markov process? If so, find the appropriate transition probabilities. (b) Do the transition probabilities depend on n ? 3. Let X n be the Bernoulli i.i.d. process and let Y n be Y n = X n + X n - 1 . (a) Show that Y n is not a Markov process. (b) Now consider the vector process Z n = ( X n , X n - 1 ). Show that Z n is a Markov process.

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