Question

If the animal is in the woods on one observation, then it is twice as likely to be in the woods as the meadows on the next observation. If the animal is in the meadows on one observation, then it is four times as likely to be in the meadows as the woods on the next observation.

Assume that state 1 is being in the meadows and that state 2 is being in the woods.

(1) Find the transition matrix for this Markov process.

P =

(2) If the animal is initially in the woods, what is the probability that it is in the woods on the next three observations?

(3) If the animal is initially in the woods, what is the probability that it is in the meadow on the next three observations?

Answer 1

@Directrix

Answer 2

@jim_thompson5910

Answer 3

The transition matrix is going to be a 2x2 matrix where the ijth entry is the probability of moving from state j into state i. I found the entries to be:

p11 = 4/5
p12 = 2/3
p21 = 1/5
p22 = 1/3

{4/5 2/3}
{1/5 1/3}

To solve the next to parts you simply apply this transition matrix onto the two initial state vectors n times, where n is how many years in the future you want to know about. So for the woods it looks like:

P^3 * S2 Where S2 is the column vector with entries 0 1 (no animal in state 1, 1 animal in state 2)

Similarly for part c it should look like:

P^3 * S1, and here S1 is the column vector 1 0


I will admit I am not incredibly familiar with Markov Processes but I hope this helps at least a little. Do not take my word for it definitely work through it again yourself.

Answer 4

Here is a useful write up on this topic

http://www.worldatlas.com/img/areamap/continent/caribbean_map.gif

Answer 5

I think yours doesnt work because the rows don't add up to 1

Answer 6

It could very well be that the source I learned it from had different conventions, but where I read it had that the columns added up to one. If you just flip mine on its side it should work for you though.

Answer 7

that does not work either

Answer 8

What about it does not work?