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Next: Transition Probabilities Up: Discrete Time Markov Chains Previous: Discrete Time Markov Chains

State Probabilities

The state probability, denoted as $\pi_{j}(n)$, is the probability that the process is in state $j$ at time $n$.

\begin{displaymath}
\pi_{j}(n) = Pr \{ X_{n}=j \}
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The state probability vector is denoted as $\Pi(n)$, and consists of all of the state probabilities for a given time $n$.

\begin{displaymath}
\Pi(n) = \left[ \begin{array}{cccc} \pi_{0}(n) & \pi_{1}(n) & \pi_{2}(n) & \cdots \end{array} \right]
\end{displaymath}

Note that the sum over the elements in $\Pi(n)$ is equal to 1.

\begin{displaymath}
\sum_{j} \pi_{j}(n) = 1
\end{displaymath}