Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 18 November 9, 2010 1. There are n fish in a lake, some of which are green and the rest blue. Each day, Helen catches 1 fish. She is equally likely to catch any one of the n fish in the lake. She throws back all the fish, but paints each green fish blue before throwing it back in. Let Gi denote the event that there are i green fish left in the lake. (a) Show how to model this fishing exercise as a Markov chain, where {Gi} are the states. Explain why your model satisfies the Markov property. (b) Find the transition probabilities {pij }. (c) List the transient and the recurrent states. Textbook problem removed due to copyright restrictions. Drake, Fundamentals of Applied Probability Theory, Problem 5.02. 3. Consider the following Markov chain, with states labelled from s0, s1, . . . , s5: S 0 1/21/2 1/41 1 S S S SS 1 2 3 4 5 1/41/2 1/2 1/2 1/3 1/31/3 Given that the above process is in state s0 just before the first trial, determine by inspection the probability that: (a) The process enters s2 for the first time as the result of the kth trial. 2 Textbook problems are courtesy of Athena Scientific, and are used with permission.Page 1 ofMassachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) (b) The process never enters s4. (c) The process enters s2 and then leaves s2 on the next trial. (d) The process enters s1 for the first time on the third trial. (e) The process is in state s3 immediately after the nth trial. Page 2 of 2 Textbook problems are courtesy of Athena Scientific, and are used with permission.MIT OpenCourseWare http://ocw.mit.edu 6.041 / 6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit:
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