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stat2303_tutorial07

stat2303_tutorial07 - THE UNIVERSITY OF HONG KONG...

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Page 1 of 6 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT2303 Probability Modelling (2010-2011) STAT2803 Stochastic Models (2010-2011) Tutorial 7 1. (Cont’d Tutorial 6 Question 1) A Markov chain 𝑋𝑋 0 , 𝑋𝑋 1 has the transition probability matrix as follows: 𝑃𝑃 = 0.7 0.2 0.1 0 0.6 0.4 0.5 0 0.5 Draw the state transition diagram.
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STAT2303/2803 Probability Modelling/Stochastic Models Semester 1, 2010-11 Page 2 of 6 2. A two-state Markov chain has the transition probability matrix 𝑷𝑷 = 0 1 0 1 1 − 𝛼𝛼 𝛼𝛼 𝛽𝛽 1 − 𝛽𝛽 a) Denote 𝑇𝑇 𝑖𝑖 the minimum value of 𝑛𝑛 such that 𝑋𝑋 𝑛𝑛 = 𝑖𝑖 , i.e. 𝑇𝑇 𝑖𝑖 = 𝑚𝑚𝑖𝑖𝑛𝑛 { 𝑛𝑛 : 𝑋𝑋 𝑛𝑛 = 𝑖𝑖 , 𝑛𝑛 ≥ 1} . Determine the first return probability, 𝑃𝑃 ( 𝑇𝑇 0 = 𝑛𝑛 | 𝑋𝑋 0 = 0) . For 𝑛𝑛 = 1 , 𝑃𝑃 ( 𝑇𝑇 0 = 1| 𝑋𝑋 0 = 0) = 𝑃𝑃 ( 𝑋𝑋 1 = 0| 𝑋𝑋 0 = 0) = 𝑝𝑝 00 = 1 − 𝛼𝛼 For 𝑛𝑛 = 2 , 𝑃𝑃 ( 𝑇𝑇 0 = 2| 𝑋𝑋 0 = 0) = 𝑃𝑃 ( 𝑋𝑋 2 = 0, 𝑋𝑋 1 = 1| 𝑋𝑋 0 = 0) = 𝑃𝑃 ( 𝑋𝑋 2 = 0| 𝑋𝑋 1 = 1) 𝑃𝑃 ( 𝑋𝑋 1 = 1| 𝑋𝑋 0 = 0) = 𝑝𝑝 01 𝑝𝑝 10 = 𝛼𝛼𝛽𝛽 For 𝑛𝑛 > 2 , 𝑃𝑃 ( 𝑇𝑇 0 = 𝑛𝑛 | 𝑋𝑋 0 = 0) = 𝑝𝑝 01 ( 𝑝𝑝 11 ) 𝑛𝑛− 2 𝑝𝑝 10
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