State the Chapman-Kolmogorov equation and mention the relation between higher and lower transition probabilities, it establishes.
Given the following transition matrix of a Markov chain with three states 1, 2 and 3:
find the matrix of three-step transition probabilities and, hence, obtain the transition probability
A symmetric random walk starts at x = 0. Find the probabilities that the walk
(i) s at x = 0 after 10 steps;
(ii) s at x =1 after 5 steps;
(iii) s at x = –3 after 9 steps.
Let's tackle this step-by-step: --- ### Part 1: Chapman-Kolmogorov Equation #### Definition: The Chapman-Kolmogorov equation gives a relationship between *n-step* and *k-step* transition probabilities in a Markov chain. If $P^{(n)} = [p_{ij}^{(n)}]$ is the matrix of *n-step transition probabilities*, then the Chapman-Kolmogorov equation states: $$ p_{ij}^{(n+m)} = \sum_{k} p_{ik}^{(n)} \cdot p_{kj}^{(m)} $$ That is, the probability of transitioning from state $i$ to state $j$ in $n+m$ steps is the sum over all intermediate states $k$ of the probability of going from $i$ to $k$ in $n$ steps and then from $k$ to $j$ in $m$ steps. --- ### Part 2: Transition Matrix Given: $$ P = \begin{bmatrix} \frac{1}{10} & \frac{1}{10} & \frac{4}{5} \\ \frac{1}{5} & \frac{1}{5} & \frac{3}{5} \\ \frac{3}{10} & \frac{3}{10} & \frac{2}{5} \end{bmatrix} $$ We want to find $P^{(3)}$ (three-step transition matrix), and specifically, the element $p_{32}^{(3)}$. We'll do this step by step: --- ### _______ _________ ____ __________ ______.
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State the Chapman-Kolmogorov equation and mention the relation between higher and lower transition probabilities, it establishes.
Given the following transition matrix of a Markov chain with three states 1, 2 and 3:
find the matrix of three-step transition probabilities and, hence, obtain the transition probability
A symmetric random walk starts at x = 0. Find the probabilities that the walk
(i) s at x = 0 after 10 steps;
(ii) s at x =1 after 5 steps;
(iii) s at x = –3 after 9 steps.
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