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Prove Chapman-Kolmogorov equations step by step with more explanations. Especially show how to obtain 2.4 (Solve Exercise 4 in the second image). Again, I need explanations for every step and solve Exercise 4 to verify 2.4. The Definition is in the third image

Prove Chapman-Kolmogorov equations step by step with more explanations. Especially show how to obtain 2.4 (Solve Exercise 4 in the second image). Again, I need explanations for every step and solve Exercise 4 to verify 2.4. The Definition is in the third image

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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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Prove Chapman-Kolmogorov equations step by step with more explanations. Especially show how to obtain 2.4 (Solve Exercise 4 in the second image). Again, I need explanations for every step and solve Exercise 4 to verify 2.4. The Definition is in the third image.

(n) DEFINΙTION 2.5 probability of transferring from state i to state j in n time steps, The n-step transition probability, denoted p', is the p = Prob{X, = j\Xo = i}. The n-step transition matrix is denoted as P(n) (P). For the cases n = () аnd n 3D 1, pi (1) = Pji and (0) Pji 1, j = i, 0, j + i, = where 8ji represents the Kronecker delta symbol. Then P(1) where I is the identity matrix. Р аnd P(0) — І, Relationships exist between the n-step transition probabilities and s-step and (n – s)-step transition probabilities. These relationships are known as the Chapman-Kolmogorov equations: (n) Pji (п-s) (s) jk Pki 0 < s < n. k=1
Transcribed Image Text:(n) DEFINΙTION 2.5 probability of transferring from state i to state j in n time steps, The n-step transition probability, denoted p', is the p = Prob{X, = j\Xo = i}. The n-step transition matrix is denoted as P(n) (P). For the cases n = () аnd n 3D 1, pi (1) = Pji and (0) Pji 1, j = i, 0, j + i, = where 8ji represents the Kronecker delta symbol. Then P(1) where I is the identity matrix. Р аnd P(0) — І, Relationships exist between the n-step transition probabilities and s-step and (n – s)-step transition probabilities. These relationships are known as the Chapman-Kolmogorov equations: (n) Pji (п-s) (s) jk Pki 0 < s < n. k=1
Verification of the Chapman-Kolmogorov equations can be shown as follows (Stewart, 1994): P = Prob{X, = j|Xo = i}, Pji Prob{Xn = j, X, = k|Xo = i}, (2.3) How Prob{X, = j|X, = k, Xo = i}Prob{X, = k|Xo = i}, (2.4) k=1 Prob{X, = j|X, = k}Prob{X, = k|Xo = i}, (2.5) k=1 (п-s) (s) jk (2.6) k=1 where equations (2.3)-(2.6) hold for 0 <s < n. The relationship (2.4) follows from conditional probabilities (Exercise 4). The relationship (2.5) follows
Transcribed Image Text:Verification of the Chapman-Kolmogorov equations can be shown as follows (Stewart, 1994): P = Prob{X, = j|Xo = i}, Pji Prob{Xn = j, X, = k|Xo = i}, (2.3) How Prob{X, = j|X, = k, Xo = i}Prob{X, = k|Xo = i}, (2.4) k=1 Prob{X, = j|X, = k}Prob{X, = k|Xo = i}, (2.5) k=1 (п-s) (s) jk (2.6) k=1 where equations (2.3)-(2.6) hold for 0 <s < n. The relationship (2.4) follows from conditional probabilities (Exercise 4). The relationship (2.5) follows
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