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09hw13

09hw13 - (a P X 1 = 1 | X = 0,X 2 = 1(b P X = X 1(c P X 1 6...

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EE 464, Mitra Spring 2009 Homework 12 - Last one! Due FRIDAY, May 1, 2009, 10am This problem set investigates the use of the central limit theorem and properties of discrete time Markov chains. 1. L-G 7.27. 2. Consider an integer valued, i.i.d. sequence X n , define ¯ M n = max 1 i n X i M n = min 1 i n X i Determine if either of these sequences form a Markov Chain. 3. Consider a discrete-time Markov Chain with two states labeled 0 and 1 , known probability transi- tion matrix and known initial state probability mass function: P [ X 0 = j ] . Determine the following probabilities in terms of the one-step transition probabilities and initial state probabilities:
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Unformatted text preview: (a) P [ X 1 = 1 | X = 0 ,X 2 = 1] (b) P [ X = X 1 ] (c) P [ X 1 6 = X 2 ] 4. Consider a sequence of independent tosses of a coin with probability of heads, p . Let X n be the total number of tosses which yielded a heads given n tosses. (a) Verify that X n is a Markov Chain. Explicitly note the state space. (b) Determine the transition probability matrix. Is the Markov Chain homogeneous or non-homogeneous. (c) Repeat (a) and (b) if X n is deﬁned as the total number of tosses which yielded heads minus the total number of tosses which yielded tails given n tosses....
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