Markov chain, is this a typo in book?

Question

I'm just starting to learn about Markov chains, and would just like to know if this is a typo in the course literature, or if there is something that I don't understand.

The book gives an example of an Markov matrix, following:

Some winter days in Minnesota it seems like the snow will never stop. A Minnesotan's view of a winter may be described by the following transition matrix for a Markov chain where $r,$ $s,$ $c$ denotes the weather rain, snow and clear.

$ \begin{bmatrix} 0.2 & 0.6 & 0.2 \\ 0.1 & 0.8 & 0.1 \\ 0.1 & 0.6 & 0.3 \end{bmatrix} $

Here the first column is $r,$ second column, $s,$ third column $c$ and the same for the rows where first row is $r,$ second row is $s,$ third row is $c.$

where the book says that regardless what weather occurs on day $D_n$ the chance of snow is always at least $0.6$ on day $D_{n+1}$

But from my understanding the column is always the starting position, day $D_n$, so element $E_{2,3}$ is gives is $D_n=$ clear (column 3) with probability for snow (row 2) in $D_{n+1}=0.1$

Would anyone like to explain to me what I'm missing out on? Is the row the starting position?

Thanks!

The book is correct. You should be looking at $E_{3,2}$ instead of $E_{2,3 }$ — geetha290krm, Nov 07 '22 at 11:38
Okey, so the idea is that in $E_{3,2}$ the $3$ stands for the start, in this case clear, e.g. $D_n$ while $2$ stands for probability for $D_{n+1}$? — uoiu, Nov 07 '22 at 11:40
Yes, $E_{i,j}$ is the probability for a transition from $i$ to $j$. — geetha290krm, Nov 07 '22 at 11:42
Thanks geetha! Been noticing you've been answering a couple of my questions, really appreciate it! :-) — uoiu, Nov 07 '22 at 11:47
It would be best to submit this as an answer so the Q can be closed :) — Pe Dro, Nov 07 '22 at 11:54

score 1 · Accepted Answer · edited Nov 07 '22 at 12:09

1

In the transition matrix $(E_{i,j})$ the $i-$th row, $j-$th column element $E_{i,j}$ is the probability of a transition from sate $i$ to state $j$. So the book is correct. All the elements in the second column of the matrix are at least equal to $0.6$.

edited Nov 07 '22 at 12:09

uoiu

593

answered Nov 07 '22 at 12:01

geetha290krm

36,632

@zzz_ Thanks for the correction. – geetha290krm Nov 07 '22 at 12:10

Markov chain, is this a typo in book?

1 Answers1