Find The Vector Of Stable Probabilities Calculator

This calculator finds the steady-state or vector of stable probabilities for a given regular Markov chain transition matrix (2×2 or 3×3).

Calculator

Enter the probabilities for the 2×2 transition matrix P:

Enter the probabilities for the 3×3 transition matrix P (rows must sum to 1):

Transition Matrix P and Stable Vector s

P	State 1	State 2	State 3	Stable Prob (s)
From 1	0.7	0.3	0.1	–
From 2	0.2	0.8	0.1	–
From 3	0.2	0.2	0.6	–

What is the Vector of Stable Probabilities?

The vector of stable probabilities, also known as the steady-state vector or stationary distribution, represents the long-run probabilities of a Markov chain being in each of its states. If you run a regular Markov chain for a very large number of steps, the probability of being in any particular state will converge to a fixed value, regardless of the initial state. This collection of fixed probabilities for all states forms the vector of stable probabilities.

A Markov chain is regular if some power of its transition matrix has all positive entries. For such chains, a unique vector of stable probabilities exists. This vector s satisfies the equation s * P = s, where P is the transition matrix, and the sum of the elements in s is 1. Our vector of stable probabilities calculator helps you find this vector s.

Anyone working with systems that can be modeled as Markov chains, such as economists, biologists, engineers, and computer scientists, might use a vector of stable probabilities calculator to understand long-term system behavior.

A common misconception is that the system reaches the stable state and stops changing. In reality, the system continues to transition between states, but the *probability* of being in each state remains constant over time once the steady state is reached.

Vector of Stable Probabilities Formula and Mathematical Explanation

Let P be the transition matrix of a regular Markov chain with states {1, 2, …, n}. The vector of stable probabilities s = [s1, s2, ..., sn] is a row vector that satisfies:

1. s * P = s
2. s1 + s2 + ... + sn = 1

The equation s * P = s can be rewritten as s * (P - I) = 0, where I is the identity matrix, or more explicitly as a system of linear equations. For a 2×2 matrix:

s1 = p11*s1 + p21*s2
s2 = p12*s1 + p22*s2
s1 + s2 = 1

From s1 + s2 = 1, we get s2 = 1 - s1. Substituting into the first equation:

s1 = p11*s1 + p21*(1 - s1)
s1 = p11*s1 + p21 - p21*s1
s1 * (1 - p11 + p21) = p21
s1 = p21 / (1 - p11 + p21)
s1 = p21 / (p12 + p21) (since p12 = 1 – p11)
And s2 = p12 / (p12 + p21)

For a 3×3 matrix, we solve the system:

(p11-1)*s1 + p21*s2 + p31*s3 = 0
p12*s1 + (p22-1)*s2 + p32*s3 = 0
s1 + s2 + s3 = 1
This system can be solved using methods like Gaussian elimination or Cramer's rule. Our vector of stable probabilities calculator handles this for you.

Variables Table

Variable	Meaning	Unit	Typical Range
P	Transition Probability Matrix	–	Matrix of probabilities
pij	Probability of transitioning from state i to state j	Probability	0 to 1
s	Vector of stable probabilities	–	Vector of probabilities
si	Stable probability of being in state i	Probability	0 to 1

Practical Examples (Real-World Use Cases)

Example 1: Brand Switching

Suppose customers choose between Brand A and Brand B. If a customer uses Brand A, there's a 70% chance they'll use it again next time (p11=0.7) and 30% they'll switch to B (p12=0.3). If they use Brand B, there's a 20% chance they'll switch to A (p21=0.2) and 80% they'll stick with B (p22=0.8).

Inputs for the vector of stable probabilities calculator:

• p11 = 0.7
• p21 = 0.2

The calculator finds s1 = 0.2 / (0.3 + 0.2) = 0.4, s2 = 0.3 / (0.3 + 0.2) = 0.6. In the long run, 40% of customers will be using Brand A and 60% Brand B.

Example 2: Weather Model (Simplified 3-State)

A simple weather model has three states: Sunny (1), Cloudy (2), Rainy (3). Transition probabilities:

• If Sunny: 60% Sunny, 30% Cloudy, 10% Rainy (p11=0.6, p12=0.3, p13=0.1)
• If Cloudy: 10% Sunny, 80% Cloudy, 10% Rainy (p21=0.1, p22=0.8, p23=0.1)
• If Rainy: 20% Sunny, 20% Cloudy, 60% Rainy (p31=0.2, p32=0.2, p33=0.6)

Using the 3×3 vector of stable probabilities calculator with these inputs, we get (approximately) s1=0.26, s2=0.51, s3=0.23. Long-term, it's sunny 26% of the time, cloudy 51%, and rainy 23%.

How to Use This Vector of Stable Probabilities Calculator

1. Select Matrix Size: Choose between a 2×2 or 3×3 transition matrix using the dropdown.
2. Enter Probabilities: Input the transition probabilities pij (probability of going from state i to state j).
   • For 2×2: Enter p11 and p21. p12 and p22 will be calculated as 1-p11 and 1-p21 respectively.
   • For 3×3: Enter p11, p12, p21, p22, p31, p32. p13, p23, p33 will be calculated to ensure rows sum to 1.
   • Ensure probabilities are between 0 and 1.
3. Calculate: The calculator updates automatically, but you can click "Calculate" to ensure the latest values are used.
4. View Results: The "Primary Result" shows the vector of stable probabilities (s1, s2, or s1, s2, s3). Intermediate results show derived probabilities. The table and chart also visualize the matrix and stable vector.
5. Interpret: The stable probabilities tell you the long-run proportion of time the system spends in each state.

Key Factors That Affect Vector of Stable Probabilities Results

1. Transition Probabilities (pij): These are the most direct factors. Small changes in pij values can significantly alter the stable state distribution, especially in matrices close to being non-regular.
2. Matrix Regularity: The method used assumes the Markov chain is regular (ergodic). If the chain is not regular (e.g., has absorbing states or is periodic), a unique stable vector may not exist or the interpretation changes. Our vector of stable probabilities calculator is designed for regular chains.
3. Number of States: The complexity of the system and the calculations increase with the number of states.
4. Initial State (for short-term): While the stable vector describes the long-run behavior independent of the start, the initial state vector determines how quickly the system approaches this steady state.
5. Time Scale: The transition probabilities are defined for a specific time step. The stable vector is the limit as the number of steps goes to infinity.
6. Data Accuracy: The pij values are often estimated from data. Inaccurate estimates will lead to an inaccurate vector of stable probabilities.

Frequently Asked Questions (FAQ)

Q: What is a Markov chain? A: A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The key characteristic is the "Markov property": the probability of transitioning to any particular state depends only on the current state and not on the sequence of events that preceded it.

Q: What does "stable probabilities" mean? A: It refers to the probabilities of finding the system in each state after it has run for a very long time, such that these probabilities no longer change with further steps. Our vector of stable probabilities calculator finds these.

Q: Does every Markov chain have a vector of stable probabilities? A: Not every Markov chain has a unique stable vector that is independent of the initial state. Regular (or ergodic) Markov chains do. Chains with absorbing states or periodic chains have different long-run behaviors. This vector of stable probabilities calculator assumes regularity.

Q: What if my matrix is not regular? A: If the matrix is not regular, the system might not converge to a single steady state, or the long-run behavior might depend on the starting state. The calculator might still give a result if the equations are solvable, but its interpretation as a unique stable vector might be incorrect.

Q: How do I know if my matrix represents a regular Markov chain? A: A sufficient condition for a Markov chain to be regular is that some power of its transition matrix P (P^k for some k >= 1) has all strictly positive entries.

Q: Can the stable probabilities be 0 or 1? A: Yes, in some cases, especially if there are near-absorbing states or very strong transitions, some stable probabilities can be very close or equal to 0 or 1, although for a regular chain with all positive entries in some power, they will be strictly between 0 and 1.

Q: How is the vector of stable probabilities calculator useful? A: It helps predict long-term behavior in systems like market share dynamics, population genetics, queuing theory, and weather patterns, assuming they can be modeled as regular Markov chains.

Q: What if the sum of probabilities in a row of my input is not 1? A: The calculator derives the last probability in each row to ensure the sum is 1 based on the preceding inputs for that row. You only input n-1 probabilities for an n-state row.

Related Tools and Internal Resources

• Matrix Multiplication Calculator: Useful for seeing how state vectors evolve over steps (s_new = s_old * P).
• Basic Probability Calculator: For understanding individual probability concepts.
• Eigenvalue and Eigenvector Calculator: The stable vector is related to the eigenvector corresponding to the eigenvalue 1 of the transpose of P.
• System of Linear Equations Solver: The core of finding the stable vector involves solving a system of linear equations.
• Introduction to Markov Chains: Learn more about the theory behind the vector of stable probabilities calculator.
• Steady-State Analysis Techniques: Explore different methods for analyzing long-run system behavior.