Markov chain problem: probability of raining at any given day

Question

I have a very annoying problem. I am just beginning to learn the Markov chains, and I cant figure out how should I solve this problem.

So I have a transition table showing the state transitions of rain and sunny for yesterday and today.

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----------------------| today sunny | today raining|

yesterday sunny---| 0.9-----------| 0.1----------|

yesterday raining-| 0.5------------| 0.5----------|

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I do not know anything about the actual state and I should find the probability of raining at any given day (so kind of probability of training after an infinite number of days).

How should I calculate this? (My problem that is not a finite problem and I am not aware of the actual status).

Thank you for your help in advance!

Answer 1

Let $x=(x_1,x_2)$ be a steady state, where $x_1$ and $x_2$ are the probabilities of a sunny and a rainy day, respectively. Then: $$x_1=x_0M;\\ x_2=x_1M;\\ \vdots\\ x_{n+1}=x_nM \Rightarrow \\ x=xM \Rightarrow \\ \begin{pmatrix}x_1&x_2\end{pmatrix}=\begin{pmatrix}x_1&x_2\end{pmatrix}\begin{pmatrix}0.9&0.1\\ 0.5&0.5\end{pmatrix} \Rightarrow \\ \begin{cases}0.1x_1-0.5x_2=0 \\ x_1+x_2=1\end{cases} \Rightarrow \\ (x_1,x_2)=(\frac56,\frac16)$$ Hence, the probability of a rainy day is $\frac16$.

Answer 2

Welcome to MSE! Hint: You have to consider a discrete-time Markov chain with transition matrix $$P =\left(\begin{array}{cc} 0.9 & 0.1\\0.5 & 0.5\end{array}\right).$$ Given an initial distribution, say $x_0 = (0.5,0.5)$, the state after $n\geq 0$ time steps is $x_n = x_0 P^n$.