some small changes
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@ -41,7 +41,7 @@
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\end{question}
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\end{question}
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We have
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We have
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\begin{IEEEeqnarray*}{rCl}
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\begin{IEEEeqnarray*}{rCl}
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\bP[X_1 = 0, X_2 = 0, X_3 = 1]
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&&\bP[X_1 = 0, X_2 = 0, X_3 = 1]\\
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&=& \bP[X_3 = 0 | X_2 = 0, X_1 = 0] \bP[X_2 = 0, X_1 = 0]\\
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&=& \bP[X_3 = 0 | X_2 = 0, X_1 = 0] \bP[X_2 = 0, X_1 = 0]\\
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&=& \bP[X_3 = 0 | X_2 = 0] \bP[X_2 = 0, X_1 = 0]\\
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&=& \bP[X_3 = 0 | X_2 = 0] \bP[X_2 = 0, X_1 = 0]\\
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&=& \bP[X_3 = 0 | X_2 = 0] \bP[X_2 = 0 | X_1 = 0] \bP[X_1 = 0]\\
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&=& \bP[X_3 = 0 | X_2 = 0] \bP[X_2 = 0 | X_1 = 0] \bP[X_1 = 0]\\
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@ -63,7 +63,8 @@
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More generally, consider a Matrix $P \in (0,1)^{n \times n}$
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More generally, consider a Matrix $P \in (0,1)^{n \times n}$
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whose rows sum up to $1$.
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whose rows sum up to $1$.
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Then we get a Markov Chain with $n$ states
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Then we get a Markov Chain with $n$ states
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by defining $\bP[X_{n+1} = i | X_{n} = j] = P_{i,j}$.
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by defining
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\[\bP[X_{n+1} = i | X_{n} = j] = P_{i,j}.\]
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\end{example}
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\end{example}
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\begin{definition}
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\begin{definition}
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@ -89,8 +90,10 @@
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\item $\bP[X_0 = i] = \alpha(i)$
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\item $\bP[X_0 = i] = \alpha(i)$
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for all $i \in E$,
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for all $i \in E$,
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\item $\bP[X_{n+1} = i_{n+1} | X_0 = i_0, X_1 = i_1, \ldots, X_{n} = i_{n}]
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\item \begin{IEEEeqnarray*}{rCl}
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= \bP[X_{n+1} = i_{n+1} | X_n = i_n]$
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&&\bP[X_{n+1} = i_{n+1} | X_0 = i_0, X_1 = i_1, \ldots, X_{n} = i_{n}]\\
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&=& \bP[X_{n+1} = i_{n+1} | X_n = i_n]
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\end{IEEEeqnarray*}
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for all $n = 0, \ldots$, $i_0,\ldots, i_{n+1} \in E$
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for all $n = 0, \ldots$, $i_0,\ldots, i_{n+1} \in E$
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(provided $\bP[X_0 = i_0, X_1 = i_1, \ldots, X_n = i_n] \neq 0$ ).
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(provided $\bP[X_0 = i_0, X_1 = i_1, \ldots, X_n = i_n] \neq 0$ ).
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\end{enumerate}
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\end{enumerate}
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