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The state transition matrix can be viewed as the dynamics of the Markov chain because the probability distribution of the state at any time t+1, Pr(Xt), can be uniquely determined by the probability distribution of the state at the previous time t and the state transition matrix, satisfying the relationship: Pr(Xt+1=j)=∑i=1Npij⋅Pr(Xt=i), where i,j∈X are arbitrary states in X, and N=#(X), the total number of states in X.
The [[Transitional Probability Matrix]] can be viewed as the [[Dynamics]] of the [[Markov Chain]] because the probability distribution of the state at any time [math]t+1[/math], [math]Pr(X_t)[/math], can be uniquely determined by the state probability distribution at the previous moment, i.e., [math]Pr(X_t)[/math] and [[Transitional Probability Matrix]], satisfying the relation:
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Pr(X_{t+1}=j)=\sum_{i=1}^N p_{ij}\cdot Pr(X_t=i),
Pr(X_{t+1}=j)=\sum_{i=1}^N p_{ij}\cdot Pr(X_t=i),
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Here [math]i,j\in \mathcal{X}[/math] are all arbitrary states in [math]\mathcal{X}[/math], and [math]N=\#(\mathcal{X})[/math] is the total number of states in [math]\mathcal{X}[/math].
The following table presents three different transition probability matrices and their EI values:
The following table presents three different transition probability matrices and their EI values: