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In the realm of Markov chains, the transition matrix is a core component. It is a square matrix describing the probabilities of moving from one state to another in a system. For a matrix with two states, like the one in our exercise, the transition matrix is expressed as:\[P = \begin{bmatrix} 0.5 & 0.3 \0.5 & 0.7 \end{bmatrix}\]This matrix demonstrates the probabilities of transitioning between these two states:- The first row indicates the probability of moving from state 1 to itself (0.5) and to state 2 (0.3).- The second row shows the probability of moving from state 2 to state 1 (0.5) and to staying in state 2 (0.7).Understanding how to read and interpret these probabilities is crucial for predicting future distributions of a population within these states.

The concept of a state vector is fundamental to Markov chains. It represents the current distribution of populations among the different states. In our exercise, the initial state vector is given as:\[\mathbf{x}_{0} = \begin{bmatrix} 0.5 & 0.5 \end{bmatrix}\]This vector can be understood as:- The first element (0.5) signifies that 50% of the population is in state 1.- The second element (0.5) indicates that 50% of the population is in state 2.The state vector evolves as we apply the transition matrix over consecutive steps. This evolution reflects how the population shifts between different states over time.

Matrix multiplication is a mathematical method used for transforming the state vector using the transition matrix. This operation helps us observe how populations shift over states through multiple transitions.When we multiply the initial state vector by the transition matrix, we calculate the new state vector after one step.For example: \[\mathbf{x}_1 = \mathbf{x}_0 P = \begin{bmatrix} 0.5 & 0.5 \end{bmatrix} \begin{bmatrix} 0.5 & 0.3 \0.5 & 0.7 \end{bmatrix} = \begin{bmatrix} 0.5 & 0.5 \end{bmatrix}\]By repeating the multiplication process, in this case multiplying \(\mathbf{x}_1\) with \(P\) again, we obtain \(\mathbf{x}_2\). This shows us the population distribution after two steps. Note the consistency in this exercise, where \(\mathbf{x}_1\) remains identical to \(\mathbf{x}_0\), and thus \(\mathbf{x}_2\) also remains \(\begin{bmatrix} 0.5 & 0.5 \end{bmatrix}\).

Population distribution within the context of Markov chains refers to the way populations are apportioned among various states at each step. The state vector after each multiplication gives us this distribution.Initially, the distribution is equal as per \(\mathbf{x}_0 = \begin{bmatrix} 0.5 & 0.5 \end{bmatrix}\). After multiple transitions dictated by the transition matrix, the distribution may change, reflecting shifts in population between the states.In our exercise, the final state after two steps—\(\mathbf{x}_2 = \begin{bmatrix} 0.5 & 0.5 \end{bmatrix}\)—shows that the population distribution does not change and remains equally divided between the two states. This suggests that, in this scenario, the distribution stabilizes quickly after each transition.