Markov Process Problem
Any help is greatly appreciated. I'm completely lost.
Answer
We know that $P=P \Pi $. Also note that $P$ is a $1\times n$ matrix aka a row vector. So the k-th component of $P$ is $P$ times the k-th column of $\Pi$ :
$$ p_k =\sum _{i=1}^n p_i \pi _{ik} = \sum _{i=1}^n p_i \alpha = \alpha \sum _{i=1}^n p_i =\alpha 1=\alpha $$ since $P$ is a probability distribution and the sum of its components is one.
The interpretation is that if the probability of reaching the state $x_k$ from any state is the constant $\alpha$, then in the stationery distribution, which is the same as the probability distribution in the long run, the probability of being in the state $x_k$ is also $\alpha$.
1.6K
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 875 views
- $24.00
Related Questions
- Discrete Math
- How many balanced lists of n left and n right parentheses are there?
- Why does this spatial discretization with n intervals have a position of (n-1)/n for each interval?
- Recursive Set
- Graph theory question on Euler circuit, Euler path, Hamilton circuit, and Hamilton path
- [Discrete Mathematics] Big-O Notation
- Discrete Math/ Set theory Question
- Discete Math