Transition Matrix

What is 'X1(bar)', 'Pij(n,m)', 'pij(n.m)', 'P', 'p', 'q' according to the transition matrix?

I'm not very familiar with how to write in subscripts or superscripts.

Also, the difference between 'Pij' and 'pij'

 

\( \underline{X_1} \) is a row vector after first transition.
\( \require{enclose}P_{ij}^{(n,m)} \) where did you get that notation from?
\( \require{enclose}p_{ij}^{(n,m)} \) is the probability of being in state j at time m that was being in state I at time n.
P is the notation for transition matrix.
p is transition probability(rather \( p_{ij} \) )
\( q_k \) is initial probability distribution or initial row vector.

 

X1(bar) is which row?

I know that. Which element is it? I'm kinda confused.

Which row vector?

Also, I wanted to know the difference between Pij and pij.

 

1) If you multiply q with P then you get so.
2) is the probabilities in matrix.
3) I meant initial proportion of States.
I didn't get \(P_{ij} \)

 

What's q? How is it different from qk?

 

I understood what pij^(n,m) means. What is pij then?

 

\( p_{ij} \) is the probability to get to state j from i.
It is in matrix P placed at \( i^{th} row & j^{th} column \)

 

\( q_k \) is initial probability for being in state k.

 

In how many steps?

 

Where is it in the matrix?

 

\( P= \pmatrix{ p_{11} & p_{12} \\ p_{21} & p_{22} } \)

\( p_{12} \) is present in first row &2nd column ...

 

\( q_k \) is not in matrix......

It's initial distribution

For example,
If a group of persons has distribution as
State A=20%
State B=40%
State C=40%
Then
\( q_k = \pmatrix{ q_A=.2 & q_B=.4 & q_C=.4 } \)