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# rec18 - (3 both From a sequential sample space and by using...

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Massachusetts Institute of Technology 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 18 November 9, 2010 1. There are n fsh in a lake, some oF which are green and the rest blue. Each day, Helen catches 1 fsh. She is equally likely to catch any one oF the n fsh in the lake. She throws back all the fsh, but paints each green fsh blue beFore throwing it back in. Let G i denote the event that there are i green fsh leFt in the lake. (a) Show how to model this fshing exercise as a Markov chain, where { G i } are the states. Explain why your model satisfes the Markov property. (b) ±ind the transition probabilities { p ij } . (c) List the transient and the recurrent states. 2. Problem 5.02, From Fundamentals of Applied Probability (Drake). Consider the Following three-state discrete-transition Markov chain: 0.3 0.3 0.2 0.1 0.1 0.3 S S 1 2 3 S 0.4 0.6 0.7 Determine the three-step transition probabilities r 11 (3), r 12 (3), and r 13

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Unformatted text preview: (3) both From a sequential sample space and by using the equation r ij ( n + 1) = ∑ k r ik ( n ) p kj in an e²ective manner. 3. Consider the Following Markov chain, with states labelled From s , s 1 , . . . , s 5 : S 1/2 1/2 1/4 1 1 S S S S S 1 2 3 4 5 1/4 1/2 1/2 1/2 1/3 1/3 1/3 Given that the above process is in state s just beFore the frst trial, determine by inspection the probability that: (a) The process enters s 2 For the frst time as the result oF the k th trial. Page 1 oF 2 Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) (b) The process never enters s 4 . (c) The process enters s 2 and then leaves s 2 on the next trial. (d) The process enters s 1 for the Frst time on the third trial. (e) The process is in state s 3 immediately after the n th trial. Page 2 of 2...
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rec18 - (3 both From a sequential sample space and by using...

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