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Unformatted text preview: EE 376A/Stat 376A Handout #13 Information Theory Tuesday, Feburary 3, 2011 Prof. T. Cover Solutions to Homework Set #3 1. Random walk in a cube. A bird ies from room to room in a 3 3 3 cube (equally likely through each interior wall). What is the entropy rate? Solution: Random walk in a cube. The entropy rate of a random walk on a graph with equal weights is given by equation 4.41 in the text: H ( X ) = log(2 E ) H ( E 1 2 E ,..., E m 2 E ) There are 8 corners, 12 edges, 6 faces and 1 center. Corners have 3 edges, edges have 4 edges, faces have 5 edges and centers have 6 edges. Therefore, the total number of edges E = 54. So, H ( X ) = log(108) + 8 ( 3 108 log 3 108 ) + 12 ( 4 108 log 4 108 ) + 6 ( 5 108 log 5 108 ) + 1 ( 6 108 log 6 108 ) =2 . 03 bits. 2. Entropy of graphs. Consider a random walk on a (connected) graph with 3 edges. (a) Which graph has the lowest entropy rate? What is the rate? (b) Which has the highest entropy rate? Solution: Entropy of graphs . There are three choices for graphs with 3 edges (See Figure 1). The entropy rate is given by: H = i i j P ij log P ij = i W i W log( W i ) 1 GRAPH 1 GRAPH 2 GRAPH 3 Figure 1: Graphs with three edges (a) Graph 1 { W i } = { 2 , 2 , 2 } H = 3 ( 2 6 log(2) ) = 1 (b) Graph 2 { W i } = { 1 , 1 , 1 , 3 } H = 3 6 log(3) = 0 . 79 (c) Graph 3 { W i } = { 1 , 2 , 2 , 1 } H = 2 ( 2 6 log(2) ) = 0 . 667 Thus, Graph 1 has highest entropy while Graph 3 has the lowest. 3. Stationary processes. Let ...,X 1 ,X ,X 1 ,... be a stationary (not necessarily Markov) stochastic process. Which of the following statements are true? Prove or provide a counterexample. (a) H ( X n  X ) = H ( X n  X ) . (b) H ( X n  X ) H ( X n 1  X ) . (c) H ( X n  X n 1 1 ,X n +1 ) is nonincreasing in n . (d) H ( X n +1  X n 1 ,X 2 n +1 n +2 ) is nonincreasing in n . Solution: Stationary processes. (a) H ( X n  X ) = H ( X n  X ). This statement is true, since H ( X n  X ) = H ( X n ,X ) H ( X ) (1) H ( X n  X ) = H ( X n ,X ) H ( X ) (2) and H ( X n ,X ) = H ( X n ,X ) by stationarity. (Note that Pr( X n = a  X = b ) = Pr( X = a  X n = b ) in general.) 2 (b) H ( X n  X ) H ( X n 1  X ). This statement is not true in general, though it is true for first order Markov chains. A simple counterexample is a periodic process with period N . Let X ,X 1 ,X 2 ,...,X N 1 be i.i.d. Bern( 1 2 ) random variables and let X mN + k = X k for k = 0 ,...,N 1 and m = 1 , 1 , 2 , 2 ,... . Note that this is a stationary process. In this case, for n = mN , H ( X n  X ) = 0 and H ( X n 1  X...
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This note was uploaded on 04/05/2011 for the course EE 5368 taught by Professor Staff during the Spring '08 term at UT Arlington.
 Spring '08
 Staff

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