Section2

# Section2 - Deﬁne X n = max Z 1,Z 2,Z n Show that X n,n...

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1. A total of m white and m black balls are distributed among two urns, with each urn containing m balls. At each stage, a ball is randomly selected from each urn and the two selected balls are interchanged. Let X n denote the number of black balls in urn 1 after the nth interchange. Give the transition probabilities of the Markov Chain { X n ,n 1 } 2. Suppose, Z n ,n 1 are i.i.d. representing outcomes of successive throws of a die.
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Unformatted text preview: Deﬁne, X n = max { Z 1 ,Z 2 ,...,Z n } . Show that, { X n ,n ≥ 1 } is a Markov Chain and give its transition matrix P. 3. We are given the Markovian stochastic process ( X n ) ,n ≥ 0 with one-step transition probabilities matrix P = . 2 0 . 5 0 . 3 . 3 0 . 1 0 . 6 . 1 0 . 1 0 . 8 . Calculate the following probabilities: P ( X 3 = 2 | X 2 = 1 ,X 1 = 3) P ( X 3 = 2 | X 1 = 3) 1...
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