n-step Transition Matrix Example

Example: n-step Transition Matrix

Following from Transition Matrix Example, let:

$\mathbf{P}=\Vert \begin{array}{cc}0.7& 0.3\\ 0.4& 0.6\end{array}\Vert$

Using the Chapman-Kolmogorov equation, we can compute the probability that if today is rains, then 4 days later it will be rains (${\mathbf{P}}_{00}^{(4)}$):

${\mathbf{P}}^{(4)}={\mathbf{P}}^4=\Vert \begin{array}{cc}0.5749& 0.4251\\ 0.5668& 0.4332\end{array}\Vert$

Thus, ${\mathbf{P}}_{00}^{(4)}=0.5749$

Interpretations:

• $P_{00}^{(4)}=0.5749$: If it rains today, probability it rains 4 days later is 57.49%
• $P_{01}^{(4)}=0.4251$: If it rains today, probability it’s sunny 4 days later is 42.51%
• $P_{10}^{(4)}=0.5668$: If it’s sunny today, probability it rains 4 days later is 56.68%
• $P_{11}^{(4)}=0.4332$: If it’s sunny today, probability it’s sunny 4 days later is 43.32%

NOTE

Each row sums to 1 (total probability from any starting state).

Related

• n-step Transition Matrix
• Transition Matrix Example