Q#102: Logic behind the stationary distribution

Markov chains are stationary distributions

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Suppose you are given the following transition matrix:

P = np.array([[0.15, 0.75, 0.10],
[0.95, 0.0, 0.05],
[0.45, 0.20, 0.35]])

Explain what a stationary distribution is and how to calculate it. Then, use the given transition matrix above and find the stationary distribution.

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ANSWER

Understanding Stationary Distribution
A stationary distribution is a probability distribution that remains unchanged over successive iterations in a Markov Chain. In simpler terms, it represents the long-term behavior of the system, where the probabilities of being in each state stabilize. For a Markov Chain to have a stationary distribution, it must be both irreducible (all states can be reached from any state) and aperiodic (no fixed pattern in state transitions).

Calculating Stationary Distribution
To calculate the stationary distribution, we need to find a probability vector, π, such that:

π * P = π

Here, π is the stationary distribution vector, and P is the transition matrix. The equation implies that multiplying the stationary distribution vector by the transition matrix should result in the same distribution vector.

Using the Given Transition Matrix: Let’s consider the given transition matrix:

[0.15, 0.75, 0.10]
[0.95, 0.0, 0.05]
[0.45, 0.20, 0.35]

To find the stationary distribution vector, we can solve the equation:

π×P=π

Let π = [π₁, π₂, π₃], and substitute into the equation:

             [0.15, 0.75, 0.10]
[π1,​​π2,​​π3​​] X [0.95, 0.0, 0.05]   = [π1,​​π2,​​π3​​]
             [0.45, 0.20, 0.35]

Solving this system of equations will yield the values of π₁, π₂, and π₃, representing the stationary distribution of the given Markov Chain.

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