STA3007_0506_t02_ssol

# STA3007_0506_t02_ssol - STA 3007 Applied Probability...

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STA 3007 Applied Probability 2005 Tutorial 2 1. Introduction to Markov Chain i. (a) The Markov Matrix: P = " 1 - α α α 1 - α # Pr { X 0 = 0 , X 1 = 0 , X 2 = 0 } = Pr { X 2 = 0 , X 1 = 0 | X 0 = 0 } Pr { X 0 = 0 } ( Conditional Probability ) Pr { X 0 = 0 , X 1 = 0 , X 2 = 0 } = Pr { X 2 = 0 | X 1 = 0 , X 0 = 0 } Pr { X 1 = 0 | X 0 = 0 } Pr { X 0 = 0 } ( Conditional Probability ) Pr { X 0 = 0 , X 1 = 0 , X 2 = 0 } = Pr { X 2 = 0 | X 1 = 0 } Pr { X 1 = 0 | X 0 = 0 } Pr { X 0 = 0 } ( MarkovProperty ) Pr { X 0 = 0 , X 1 = 0 , X 2 = 0 } = P 00 P 00 Pr { X 0 = 0 } Pr { X 0 = 0 , X 1 = 0 , X 2 = 0 } = (1 - α )(1 - α ) (b) Pr { X 2 = 0 | X 0 = 0 } = 1 X k =0 Pr { X 2 = 0 | X 1 = k, X 0 = 0 } Pr { X 1 = k | X 0 = 0 } Pr { X 2 = 0 | X 0 = 0 } = Pr { X 2 = 0 | X 1 = 1 } Pr { X 1 = 1 | X 0 = 0 } + Pr { X 2 = 0 | X 1 = 0 } Pr { X 1 = 0 | X 0 = 0 } Pr { X 2 = 0 | X 0 = 0 } = P 10 P 01 + P 00 P 00 = α 2 + (1 - α ) 2 1

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2. Transition Probability Matrices of a Markov Chain i. (a) P 2 = 0 . 1 0 . 2 0 . 7 0 . 2 0 . 2 0 . 6 0 . 6 0 . 1 0
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STA3007_0506_t02_ssol - STA 3007 Applied Probability...

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