Let X₁, be the Markov chain with state space Z and transition probability P2,2+1 = P₁ P2,2-1 = 1- P₁ where p > 1/2. Assume X₁ = 0. (a) Let Y= min{Xo, X₁,...}. What is the distribution of Y? (b) For positive integer k, let T = why e(k) = ke(1). min{n: X₂ = k} and let e(k)= E[T]. Explain (c) Find e(1). Hint: part (b) might be helpful. (d) Use (c) to give another proof that e(1) = ∞o if p = 1/2.
Let X₁, be the Markov chain with state space Z and transition probability P2,2+1 = P₁ P2,2-1 = 1- P₁ where p > 1/2. Assume X₁ = 0. (a) Let Y= min{Xo, X₁,...}. What is the distribution of Y? (b) For positive integer k, let T = why e(k) = ke(1). min{n: X₂ = k} and let e(k)= E[T]. Explain (c) Find e(1). Hint: part (b) might be helpful. (d) Use (c) to give another proof that e(1) = ∞o if p = 1/2.
A First Course in Probability (10th Edition)
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Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![Let X, be the Markov chain with state space Z and transition probability
Pa,z+1 = P, Pa,2-1 = 1- p,
where p > 1/2. Assume X, = 0.
(a) Let Y = min{Xo, X1,...}. What is the distribution of Y?
(b) For positive integer k, let T = min{n : X, = k} and let e(k) = E[TR]. Explain
why e(k) = ke(1).
(c) Find e(1). Hint: part (b) might be helpful.
(d) Use (c) to give another proof that e(1) = o if p = 1/2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a4d3cf4-4f2b-4627-a22e-664fd6c2491a%2F55b0e50e-1423-408a-b076-bc96310c8bca%2Fvyws7us_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let X, be the Markov chain with state space Z and transition probability
Pa,z+1 = P, Pa,2-1 = 1- p,
where p > 1/2. Assume X, = 0.
(a) Let Y = min{Xo, X1,...}. What is the distribution of Y?
(b) For positive integer k, let T = min{n : X, = k} and let e(k) = E[TR]. Explain
why e(k) = ke(1).
(c) Find e(1). Hint: part (b) might be helpful.
(d) Use (c) to give another proof that e(1) = o if p = 1/2.
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