Final Answer:
a) The transition matrix P for the Markov chain with states as nonnegative integers is defined by its entry Pij representing the probability of transitioning from state i to state j.
b) The properties of a Markov chain include the Markov property, which states that the probability of transitioning to any particular state depends solely on the current state and not on the sequence of events that preceded it. Additionally, the transition probabilities must sum to 1 for each state.
c) The πij in the context of the transition matrix represents the probability of transitioning from state i to state j in one time step.
d) The steady-state equations for the Markov chain involve solving the system of equations where the product of the transition matrix P and the steady-state vector is equal to the steady-state vector itself.
Step-by-step explanation:
a) In a Markov chain, the transition matrix P is a square matrix with each entry Pij representing the probability of transitioning from state i to state j in a single time step.
b) The Markov property asserts that the future state of the system depends only on its current state, not on the sequence of events that led to the current state. The transition probabilities in the matrix must satisfy the property that the sum of probabilities for transitions from any given state is equal to 1.
c) The πij values correspond to the transition probabilities in the matrix. Specifically, πij denotes the probability of moving from state i to state j in one time step. These probabilities are crucial for understanding the dynamics of the Markov chain and how the system evolves over time.
d) To derive the steady-state equations, we set up a system of equations where the product of the transition matrix P and the steady-state vector is equal to the steady-state vector itself. Solving this system provides the values of the steady-state probabilities for each state. This equilibrium distribution represents the long-term behavior of the Markov chain.