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6.262.PS8

# 6.262.PS8 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 – Discrete Stochastic Processes Problem Set #8 Issued: April 10, 2010 Due: April 16, 2010 Reading : For this problem set, please study all of Sections 5.1 – 5.5. For the week of April 12 - 16, begin reading Chapter 6. 1) Exercise 5.1. (Assume the Markov chain is an arbitrary finite or countably infinite chain. Do the proof first for n = 1, then for n = 2, and then proceed by induction on n. Also does the limit, lim( ( )) ij n F n →∞ , necessarily always exist? Either show that it does, or else give a counterexample.) 2) a) Consider the Markov chain in Figure 5.1. Show that the solutions F ij ( ) given in Exercise 5.2 satisfy eq. (5.9) for this chain as well. For the special case p = q = 1/2, show that the chain is recurrent, (i.e., show that the solutions given in Exercise 5.2, F ij ( ) = 1, i,j , are correct) by showing that there is no smaller solution to eq. (5.9). (You may use, without proof, the symmetric behavior of the chain in this case, e.g., F 01 ( ) = F 10 ( ), F

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6.262.PS8 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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