STA3007_0506_t02

# STA3007_0506_t02 - STA 3007 Applied Probability Tutorial 2...

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STA 3007 Applied Probability 2005 Tutorial 2 1. Introduction to Markov Chain (a) Exercise i. Consider the problem of sending a binary message, 0 or 1, through a signal channel consisting of several stages, where transmission through each stage is subject to a ﬁxed probability of error α . Suppose that X 0 = 0 is the signal that is sent and let X n be the signal that is received at the n th stage. Assume that { X n } is a Markov Chain with transition probabilities P 00 = P 11 = 1 - α and P 01 = P 10 = α , where 0 < α < 1 . (a) Determine Pr { X 0 = 0 , X 1 = 0 , X 2 = 0 } , the probability that no error occurs up to stage n=2. (b) Determine the probabilty that a correct signal is received at stage 2. 1

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2. Transition Probability Matrices of a Markov Chain (a) n - steps transition probabilities of a Markov Chain: We want to know P ( n ) ij = Pr { X m + n = j | X m = i } P ( n ) = PPPPP. ..P = P n Proof: P ( n ) ij = Pr { X n = j | X 0 = i } P ( n ) ij = X k =0 Pr { X n = j, X 1 = k | X 0 = i } ( Law of Total Probability
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## This note was uploaded on 05/21/2011 for the course STA 3007 taught by Professor Kb during the Spring '11 term at CUHK.

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STA3007_0506_t02 - STA 3007 Applied Probability Tutorial 2...

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