Markov Chains Calculator

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Example

Created on 2024-06-20Asked by Olivia Wilson (Solvelet student)

Consider a Markov chain with transition matrix

P = \begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix}.

Determine the steady-state distribution.

Solution

To find the steady-state distribution

\pi

of the Markov chain with transition matrix

P = \begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix},

we solve the system

\pi P = \pi

subject to

\sum \pi_i = 1

. 1. Write the steady-state equations:

\pi_1 = 0.7 \pi_1 + 0.4 \pi_2

\pi_2 = 0.3 \pi_1 + 0.6 \pi_2

2. Rewrite in matrix form:

\pi \begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix} = \pi

\begin{pmatrix} \pi_1 & \pi_2 \end{pmatrix} \begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix} = \begin{pmatrix} \pi_1 & \pi_2 \end{pmatrix}

3. Set up the system of linear equations:

\begin{cases} 0.7 \pi_1 + 0.4 \pi_2 = \pi_1 \\ 0.3 \pi_1 + 0.6 \pi_2 = \pi_2 \\ \pi_1 + \pi_2 = 1 \end{cases}

4. Simplify the first two equations:

0.7 \pi_1 + 0.4 \pi_2 = \pi_1 \implies -0.3 \pi_1 + 0.4 \pi_2 = 0 \implies 3 \pi_1 = 4 \pi_2

0.3 \pi_1 + 0.6 \pi_2 = \pi_2 \implies 0.3 \pi_1 - 0.4 \pi_2 = 0 \implies 3 \pi_1 = 4 \pi_2

5. Solve for

\pi_1

and

\pi_2

\pi_1 = \frac{4}{7} \pi_2 \quad \text{and} \quad \pi_1 + \pi_2 = 1

\frac{4}{7} \pi_2 + \pi_2 = 1

\frac{11}{7} \pi_2 = 1

\pi_2 = \frac{7}{11}

\pi_1 = 1 - \pi_2 = 1 - \frac{7}{11} = \frac{4}{11}

Therefore, the steady-state distribution is:

\pi = \left( \frac{4}{11}, \frac{7}{11} \right)

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Avery Nguyen on Solvelet

1. Construct the transition matrix for a simple Markov chain with states A, B, and C, where

P(A \to B) = 0.3

P(B \to C) = 0.5

, and

P(C \to A) = 0.2

.2. Find the steady-state distribution of a Markov chain with transition matrix

\begin{bmatrix} 0.6 & 0.4 \\ 0.2 & 0.8 \end{bmatrix}

DefinitionThey are mathematical systems that go from random state to random state on a state space. They are Markov processes, a type of stochastic process, where the transition probability from one state to the next depends only on the current state and not on the entire history that precedes it. If the rain tomorrow is based only on whether it is raining today and not on any previous days (as in a simple weather model), then it can be represented by a Markov chain.

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