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stat2303_assignment02_ans

stat2303_assignment02_ans - THE UNIVERSITY OF HONG KONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT2303/2803 PROBABILITY MODELLING/STOCHASTIC MODELS ASSIGNMENT 2 SOLUTION 1. For 𝑃𝑃 1 , since there is only 1 class: {0,1,2}, all the states must be recurrent. Since 𝑝𝑝 𝑖𝑖𝑖𝑖 𝑛𝑛 > 0 for 𝑛𝑛 ≥ 2, 𝑖𝑖 = 0,1,2 . The period is 1 for all the states as periodicity is a class property. For 𝑃𝑃 2 , since there is only 1 class: {0,1,2,3}, all the states must be recurrent. For the periodicity, it can be shown that: n 1 2 3 4 5 6 7 8 9 𝑝𝑝 𝑖𝑖𝑖𝑖 𝑛𝑛 0 0 >0 0 0 >0 0 0 >0 Therefore, the period is 3 for all the states. For 𝑃𝑃 3 , the classes are {0,2}, {1} and {3,4}. 𝑓𝑓 0 = 1 2 + � 𝑝𝑝 02 𝑝𝑝 22 𝑛𝑛− 2 𝑝𝑝 20 𝑛𝑛 =2 = 1 𝑓𝑓 1 = 1 2 𝑓𝑓 3 = 1 2 + � 𝑝𝑝 02 𝑝𝑝 22 𝑛𝑛− 2 𝑝𝑝 20 𝑛𝑛 =2 = 1 Therefore, states 0, 2, 3 and 4 are recurrent while state 1 is transient. Since 𝑝𝑝 𝑖𝑖𝑖𝑖 𝑛𝑛 > 0 for all 𝑖𝑖 = 0,1,2,3,4; 𝑛𝑛 ≥ 0 , all states are aperiodic. (Note that it is not meaningful for state 1.) For 𝑃𝑃 4 , the classes are {0,1}, {2}, {3}, {4}. 𝑓𝑓 0 = 1 4 + � 𝑝𝑝 01 𝑝𝑝 11 𝑛𝑛− 2 𝑝𝑝 10 𝑛𝑛 =2 = 1 𝑓𝑓 2 = 1, 𝑓𝑓 3 = 2 3 , 𝑓𝑓 4 = 0 So, states 0, 1 and 2 are recurrent while other states are all transient.
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