I am not sure how to use a given values to form a initial state vector.
There are 30% of customers from Company A switch to company B every month, and 35% customers from Company B switch to company A every month.
From the information above, the stochastic matrix is:
$$S=\begin{pmatrix}0.7&0.35\\ 0.3&0.65\end{pmatrix}$$
If I know that the start of year when Company A has 250,000 customer and Company B has 125,000, how can I write an initial state vector x0 ?
Do I just simply put these 2 value together to form a state vector ? if not is there any formula to calculate the initial state vector ?
Thank you.
Since Markov chains deal with probability the entries of the initial state vector $x^0$ should sum to $1$. So, this is just the proportions of customers from company $A$ and $B$.
$x^0: \begin{pmatrix} 2/3\\ 1/3 \end{pmatrix}$
I believe the transition probability matrix you have should be $$\begin{matrix} &A&B\\ A&.7&.3\\ B&.35&.65 \end{matrix}$$
because the rows of the transition probability matrix should sum to $1$ (since the total probability of moving from state $A$ to state $A$ and $A$ to $B$ should be $1$).
An easy way to read the transition probability matrix is by looking at the row, let's say company $A$, and the probability that the customer will switch from $A$ to $B$ in the next month is $.3$