Question

Can you help me with the following problem, explaining the solution step by step
Suppose that the probability that it will rain tomorrow if it is raining today is 0.6, and that the probability that the weather will be good tomorrow if the weather is good today is 0.3.
a. Define the states
b. Define the transition unit
c. Determine the corresponding Markov chain transition probability matrix.
d. Calculate the matrix T ^ 2 What are the probabilities that it will rain if there was good weather before?
and. Calculate the matrix T ^ 3 What is the probability that it will make good weather if there was good weather before?
d. Assuming that there is an equal probability of rain or good weather at first, what is the probability that it will rain on the third day?

Answer 1

a)Let denote the markov chain

{Xn: n=0,1,2,..)

The states of the chain are :

{weather is rainy, weather is good} = {1,2}, say

Let 1,2 denotes the states $weather\hspace{0.1cm} is \hspace{0.1cm}rainy, \hspace{0.1cm}weather \hspace{0.1cm}is \hspace{0.1cm}good$ respectively.

b)

Here the transition unit is per day.

c)

Transition matrix:

$T=\begin{bmatrix} 0.6& 0.4\\ 0.7& 0.3\end{bmatrix}$

d)

$T^{2}=\begin{bmatrix} 0.6& 0.4\\ 0.7& 0.3\end{bmatrix}\begin{bmatrix} 0.6& 0.4\\ 0.7& 0.3\end{bmatrix}=\begin{bmatrix} 0.64& 0.36\\ 0.63& 0.37\end{bmatrix}$

$P[X_{2}=1|X_{0}=2]=0.36$

$T^{3}=\begin{bmatrix} 0.6& 0.4\\ 0.7& 0.3\end{bmatrix}\begin{bmatrix} 0.64& 0.36\\ 0.63& 0.37\end{bmatrix}=\begin{bmatrix} 0.636& 0.364\\ 0.637& 0.363\end{bmatrix}$

$P[X_{3}=2|X_{0}=2]=0.363$

d)

Note,

$[\frac{1}{2},\frac{1}{2}]\begin{bmatrix} 0.6& 0.4\\ 0.7& 0.3\end{bmatrix}\begin{bmatrix} 0.6& 0.4\\ 0.7& 0.3\end{bmatrix}\begin{bmatrix} 0.6& 0.4\\ 0.7& 0.3\end{bmatrix}=[\frac{1}{2},\frac{1}{2}]\begin{bmatrix} 0.636& 0.364\\ 0.637& 0.363\end{bmatrix}=[0.6365,0.3635 ]$

Requred probability=$P[X_{3}=1]=0.6365$