## Transfer Matrix Interpretation

Standard deviation, mean, variance, z-scores, t-tests, etc.
jk22
Posts: 10
Joined: Sat Jun 26, 2010 10:39 am
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### Transfer Matrix Interpretation

Hi, nice to meet you.

I'm trying to understand the formalism of transition matrices in probability :

a) if I have a state determined by (s1,...sr), where si is in {h,t} (head tail), the probability of head were p, and tail : q=1-p.

what Does the matrix : $M=\left(\begin{array}{ccccc} p & q & 0 \ldots &0&\\ p & 0 & q \ldots &0&\\\vdots & \vdots & \ddots &q\\p & 0 & \ldots &0 \end{array}\right)$

mean :

s1 becomes head (p), and s2-sr becomes tail (q) ?

b) how to prove that : $p(1,q,\ldots q^{r-1})(\mathbb{1}-M)^{-1}\left(\begin{array}{c}1\\\vdots\\ 1\end{array}\right)=\sum_{m=1}^\infty p^m(mr+r(r+1)/2)$ ?

thx.

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Transfer Matrix Interpretation

Hi, nice to meet you.

I'm trying to understand the formalism of transition matrices in probability :

a) if I have a state determined by (s1,...sr), where si is in {h,t} (head tail), the probability of head were p, and tail : q=1-p.

what Does the matrix : $M=\left(\begin{array}{ccccc} p & q & 0 \ldots &0&\\ p & 0 & q \ldots &0&\\\vdots & \vdots & \ddots &q\\p & 0 & \ldots &0 \end{array}\right)$

mean :

s1 becomes head (p), and s2-sr becomes tail (q) ?

b) how to prove that : $p(1,q,\ldots q^{r-1})(\mathbb{1}-M)^{-1}\left(\begin{array}{c}1\\\vdots\\ 1\end{array}\right)=\sum_{m=1}^\infty p^m(mr+r(r+1)/2)$ ?

thx.
is the last row in M supposed to just be [p,0,0...,0]?

is M supposed to be a Markov transition matrix?

is $p(1,q,\ldots q^{r-1})$, $p$ times the vector $(1,q,\ldots q^{r-1})$?

jk22
Posts: 10
Joined: Sat Jun 26, 2010 10:39 am
Contact:

### Re: Transfer Matrix Interpretation

Hello,

yes, I read it like that. M is said to be a stochastic matrix.

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
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### Re: Transfer Matrix Interpretation

Hello,

yes, I read it like that. M is said to be a stochastic matrix.

then how can the bottom row be [p,0,...0]

shouldn't each row add to 1?

jk22
Posts: 10
Joined: Sat Jun 26, 2010 10:39 am
Contact:

### Re: Transfer Matrix Interpretation

Hi,

if it were a sub-stochastic matrix, it wouldn't need this condition ?

thanks.

Martingale
Posts: 333
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Transfer Matrix Interpretation

Hi,

if it were a sub-stochastic matrix, it wouldn't need this condition ?

thanks.
Did you remove an absorbing state?