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Unformatted text preview: ECE 496 HOMEWORK ASSIGNMENT III Spring 2007 1. Consider the threestate Markov chain in Diagram 1. (a) Find the transition matrix P and the stationary distribution π * for this Markov chain. (b) Find labels for the arrows in Diagram 2 so that the resulting Markov chain has the same stationary distribution as the Markov chain in (a). 2. Consider the rather trivial GA that operates on binary “strings” of length L = 1 and populations of size n = 2 with fitness function f (1) = 5 and f (0) = 1, and with p m = . 01 (note that crossover is somewhat meaningless here). Assume fitnessproportional selection. (a) List all the possible populations and give each one a label. (b) Draw the transition diagram for the Markov chain on population space corre sponding to this GA. (c) Find the transition matrix P of the Markov chain. (d) Find (at least approximately) the stationary distribution of the Markov chain. (e) Based on your answer to (c), find the longterm average fitness of the population, that is, find ¯ f ∞ = lim T →∞ 1 T T X t =1 ¯ f ( t ) , where ¯ f ( t ) is the average fitness of the population at time t . Note that ¯ f ∞ is independent of the initial population. 3. Mitchell asserts in Chapter 4 that the number of possible populations of size n for a standard GA operating on binary strings of length L is given by 2 L + n 1 2 L 1 ! . This problem steps you through a proof of that fact. (a) Show that there are 2 L populations in which only one string is represented and ` 2 L n ´ populations in which exactly n strings are represented. (b) Show that there are ` 2 L 2 ´` n 1 1 ´ populations in which exactly two strings are rep resented. (Suggestion: first you need to pick which two strings are represented; then you have to pick a “dividing line” in the population so that one of the strings...
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 Spring '07
 DELCHAMPS
 Algorithms, Markov chain, Empty set, representative, binary strings

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