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Thread: discrete markov stochastic process

  1. #1
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    discrete markov stochastic process

    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
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  2. #2
    Moo
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    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 $\displaystyle A^n$
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