Consider three r.v.'s X1, X2, X3 with finite sample space X; E {1,..., K}. The sequence X;, i = 1,2, 3, is called a Markov chain if Pr (X3 = i| X2 = j, X1 = k) = Pr (X3 = i | X2 = j) = Pr (X2 = i| X1 = j) . That is, Pr (X3 = i | X2 = j) = Pi does not depend on X1, and it is the same as Pr (X2 = i | X = j). We call Pji the transition probabilities. Let P = [Pi] denote the (K x K) matrix of transition probabilties. %3D Let T = (T1,... , TK)' be a probability vector (i.e., Tk > 0 and T = 1) with Tk Pri = T;Pjk
Consider three r.v.'s X1, X2, X3 with finite sample space X; E {1,..., K}. The sequence X;, i = 1,2, 3, is called a Markov chain if Pr (X3 = i| X2 = j, X1 = k) = Pr (X3 = i | X2 = j) = Pr (X2 = i| X1 = j) . That is, Pr (X3 = i | X2 = j) = Pi does not depend on X1, and it is the same as Pr (X2 = i | X = j). We call Pji the transition probabilities. Let P = [Pi] denote the (K x K) matrix of transition probabilties. %3D Let T = (T1,... , TK)' be a probability vector (i.e., Tk > 0 and T = 1) with Tk Pri = T;Pjk
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Consider three r.v.'s X1, X2, X3 with finite sample space X; E {1,..., K}. The sequence X¡, i = 1, 2, 3,
is called a Markov chain if
Pr (X3 = i| X2 = j, X1 = k) = Pr (X3 = i | X2 = j)
= Pr (X2 = i| X1 = j) .
That is, Pr (X3 = i| X2 = j) = Pi does not depend on X1, and it is the same as Pr (X2 = i| X1 = j).
We call Pji the transition probabilities.
Let P = [Pi] denote the (K × K) matrix of transition probabilties.
Let T = (T1,..., TK)' be a probability vector (i.e., T; > 0 and Tk = 1) with
Tk Pkj = T;Pjk
%3D
for any pair of states j and k. If q1 = T, show that q2 = q3 = r as well (T is called an equilibrium
distribution").
By the law of total probability φΣ, Τ P- Σ, ΤP πι Σ, P= πι
By Bayes' theorem q2i =
92i = Ti
%3D
q1 is a probability vector and P is a stochastic matrix = q2 = q3 = 91.
By definition of conditional probability Pj
Pr(X1=k,X2=j)
Pr(X1=k)
= Tj.
%3D
%3D
none of these](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98ee32e6-25e5-42b1-a959-4ad347765da5%2Fb1dcc110-87d4-4aca-994f-345ddb2e8080%2F9ym8xg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider three r.v.'s X1, X2, X3 with finite sample space X; E {1,..., K}. The sequence X¡, i = 1, 2, 3,
is called a Markov chain if
Pr (X3 = i| X2 = j, X1 = k) = Pr (X3 = i | X2 = j)
= Pr (X2 = i| X1 = j) .
That is, Pr (X3 = i| X2 = j) = Pi does not depend on X1, and it is the same as Pr (X2 = i| X1 = j).
We call Pji the transition probabilities.
Let P = [Pi] denote the (K × K) matrix of transition probabilties.
Let T = (T1,..., TK)' be a probability vector (i.e., T; > 0 and Tk = 1) with
Tk Pkj = T;Pjk
%3D
for any pair of states j and k. If q1 = T, show that q2 = q3 = r as well (T is called an equilibrium
distribution").
By the law of total probability φΣ, Τ P- Σ, ΤP πι Σ, P= πι
By Bayes' theorem q2i =
92i = Ti
%3D
q1 is a probability vector and P is a stochastic matrix = q2 = q3 = 91.
By definition of conditional probability Pj
Pr(X1=k,X2=j)
Pr(X1=k)
= Tj.
%3D
%3D
none of these
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