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The random variables , $1, 2,... are independent and identically distributed with distribution P( = 0) = 1/4 and P(§ = j) = c/j for j = 1,2,3. Let Xo = 0 and Xn = max($₁,..., En) for n = 1, 2,.... (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix.

The random variables , $1, 2,... are independent and identically distributed with distribution P( = 0) = 1/4 and P(§ = j) = c/j for j = 1,2,3. Let Xo = 0 and Xn = max($₁,..., En) for n = 1, 2,.... (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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The random variables §, 1, $2,... are independent and identically distributed with distribution P(§ = 0) = 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁,..., En) for n = 1, 2, .... (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X4 = 3, X₂ = 1|X₁ = 3).
Transcribed Image Text:The random variables §, 1, $2,... are independent and identically distributed with distribution P(§ = 0) = 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁,..., En) for n = 1, 2, .... (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X4 = 3, X₂ = 1|X₁ = 3).
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