# discrete markov stochastic process

#### omer.jack

hi guys

I have a simple problem driving me crazy the topic is stochastic process and markov process, this is the problem:

assuming every day's weather has only two options: rainy or sunny.
the probability that in certain day the weather is the same as the day before is $$\displaystyle p$$. the probability of change in weather is $$\displaystyle q=1-p$$.

now i will define $$\displaystyle (x_{[n]})_{n>=0}$$ as a discrete time stochastic process as: $$\displaystyle x_{[n]}=1$$ if the n'th day was rainy and $$\displaystyle x_{[n]}=0$$ if the n'th day was sunny.

the question is: given that today, n=0, is a rainy day, find the probability to rain in the n'th day. clue: you can organize the data in a matrix.
i've tried few approaches but nothing comes out, i know its simple, will be happy for assistance. thanks #### Moo

MHF Hall of Honor
Hello,

Let A be the transition matrix. It's a 2x2 matrix because there are only two states. Since the probability of staying in the same state the next day is p, you will have p in the diagonal. And q otherwise.

$$\displaystyle A=\begin{pmatrix} p&q \\ q&p\end{pmatrix}$$

it can be diagonalized : its eigenvalues are 1 (eigenvector (1,1)) and p-q (eigenvector (1,-1)).
So $$\displaystyle A=PDP^{-1}$$, where $$\displaystyle P=\begin{pmatrix}1&1\\1&-1\end{pmatrix}$$ (matrix of the eigenvectors) and $$\displaystyle D=\begin{pmatrix} 1&0 \\ 0&p-q\end{pmatrix}$$
Also note that $$\displaystyle P^{-1}=\frac 12\begin{pmatrix} 1&1 \\ 1&-1\end{pmatrix}$$

Hence $$\displaystyle A^n=P\begin{pmatrix} 1^n & 0 \\ 0&(p-q)^n\end{pmatrix}$$
But the element $$\displaystyle (i,j)$$ in $$\displaystyle A^n$$ denotes $$\displaystyle P(X_n=j\mid X_0=i)$$ (more or less)
And you're looking for $$\displaystyle P(X_n=0 \mid X_0=0)$$, so you'll be looking at the (1,1) element of the matrix $$\displaystyle A^n$$