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Explain how a transition matrix models change over time.
It shows the probabilities of moving from one state to another at each time step, allowing us to track how the system evolves.
Explain the concept of a steady state in Markov chains.
It's the long-term distribution where the probabilities of being in each state remain constant, even after multiple transitions.
How can you predict future states using a transition matrix and state vector?
By repeatedly multiplying the transition matrix by the state vector, you can project the probabilities of being in each state at future time intervals.
Why is the sum of entries in each column of a transition matrix equal to 1?
Because each column represents the probabilities of transitioning from a single state to all possible states, and these probabilities must add up to 100%.
What does it mean if a transition matrix has an inverse?
It means you can predict past states by multiplying the inverse of the transition matrix by the current state vector.
What is the significance of the entries in a transition matrix?
Each entry represents the probability of transitioning from one state to another in a single time step.
Why is matrix multiplication order important when using transition matrices?
Because matrix multiplication is not commutative, so changing the order can lead to incorrect results.
Explain how Markov chains can be used to model real-world scenarios.
Markov chains can model systems that transition between states based on probabilities, such as customer behavior, weather patterns, or population dynamics.
What is the purpose of finding the steady-state vector?
To determine the long-term distribution of states, which can help predict the eventual outcome of a system.
How does the initial state vector affect the future states in a Markov chain?
The initial state vector determines the starting point, and the transition matrix then dictates how the probabilities evolve over time from that starting point.
How do you construct a transition matrix from a given scenario?
Identify the states, determine the probabilities of transitioning between them, and arrange these probabilities in a matrix where columns sum to 1.
How do you predict the state vector after a certain number of transitions?
Raise the transition matrix to the power of the number of transitions, then multiply by the initial state vector.
How do you find the steady-state vector?
Repeatedly multiply the transition matrix by an initial state vector until the state vector converges, or solve the equation TX = X.
How do you predict a past state vector?
Find the inverse of the transition matrix, then multiply it by the current state vector.
How do you determine if a matrix is a valid transition matrix?
Check that all entries are non-negative and that the sum of entries in each column equals 1.
How do you set up a Markov chain model for a given problem?
Define the states, determine transition probabilities, create the transition matrix, and define the initial state vector.
How do you find the inverse of a 2x2 matrix?
For a matrix egin{bmatrix} a & b \ c & d end{bmatrix}, the inverse is frac{1}{ad-bc} egin{bmatrix} d & -b \ -c & a end{bmatrix}.
How do you multiply two matrices?
Multiply rows of the first matrix by columns of the second matrix, summing the products to get each entry in the resulting matrix.
How do you interpret the entries in a transition matrix?
Each entry represents the probability of transitioning from the state represented by the column to the state represented by the row.
How do you determine the long-term behavior of a system using a Markov chain?
Find the steady-state vector, which represents the distribution of states that the system will eventually converge to.
What is the formula for a general 2x2 transition matrix?
How to find the state vector after 'n' transitions?
Multiply the transition matrix T by itself 'n' times, then multiply by the initial state vector X:
How do you find the previous state vector?
Multiply the inverse of the transition matrix by the current state vector:
What is the condition for a matrix to be a valid transition matrix?
The sum of entries in each column must equal 1, and all entries must be non-negative.
How do you find the state vector after one transition?
Multiply the transition matrix (T) by the current state vector (X):
What is the formula to find the steady-state vector?
Repeatedly multiply the transition matrix by the state vector until the state vector converges. Alternatively, solve TX = X.
How do you calculate the state vector after two transitions?
Multiply the transition matrix by itself, then multiply the result by the initial state vector:
What is the formula to predict a future state?
Multiply the transition matrix by the current state vector:
What formula do you use to predict a past state?
Multiply the inverse of the transition matrix by the current state vector:
What is the formula to find the probability of staying in state A?