Editing Examples of Markov chains (section)

==== Predicting the weather ====

The weather on day 0 (today) is known to be sunny.  This is represented by an initial state vector in which the "sunny" entry is 100%, and the "rainy" entry is 0%:

: <math>
    \mathbf{x}^{(0)} = \begin{bmatrix}
        1 & 0
    \end{bmatrix}
</math>

The weather on day 1 (tomorrow) can be predicted by multiplying the state vector from day 0 by the transition matrix:

: <math>
    \mathbf{x}^{(1)} = \mathbf{x}^{(0)} P  = 
    \begin{bmatrix}
        1 & 0
    \end{bmatrix}
    \begin{bmatrix}
        0.9 & 0.1 \\
        0.5 & 0.5
    \end{bmatrix}
    
    = \begin{bmatrix}
        0.9 & 0.1
    \end{bmatrix} 
</math>

Thus, there is a 90% chance that day 1 will also be sunny.

The weather on day 2 (the day after tomorrow) can be predicted in the same way, from the state vector we computed for day 1:

: <math>
    \mathbf{x}^{(2)} =\mathbf{x}^{(1)} P  = \mathbf{x}^{(0)} P^2 
    = \begin{bmatrix}
        1 & 0
    \end{bmatrix}
    \begin{bmatrix}
        0.9 & 0.1 \\
        0.5 & 0.5
    \end{bmatrix}^2
    
    = \begin{bmatrix}
        0.86 & 0.14
    \end{bmatrix} 
</math>
or
: <math>
    \mathbf{x}^{(2)} =\mathbf{x}^{(1)} P 
    = \begin{bmatrix}
        0.9 & 0.1
    \end{bmatrix}
    \begin{bmatrix}
        0.9 & 0.1 \\
        0.5 & 0.5
    \end{bmatrix}
    
    = \begin{bmatrix}
        0.86 & 0.14
    \end{bmatrix} 
</math>

General rules for day ''n'' are:

: <math>
    \mathbf{x}^{(n)} = \mathbf{x}^{(n-1)} P 
</math>

: <math>
    \mathbf{x}^{(n)} = \mathbf{x}^{(0)} P^n 
</math>