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.(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

Answer:

a) The transition matrix for this Markov process will be:

$M = \left[\begin{array}{ccc}2/3 &1/4 \\1/3 &3/4}\end{array}\right]$

b) If the animal is initially in the woods, probability that it is in the woods on the next three observations =  0.602

c) If the animal is initially in the woods, probability that it is in the meadow on the next three observations =  0.398

Step-by-step explanation:

Let the state be $S_{i}$

There are two states, i = 1,2

Let $T_{ij}$ be transition between the states, i = 1,2; j = 1,2

The transition matrix for a markov process is:

$M = \left[\begin{array}{ccc}T_{11} &T_{12} \\T_{21} &T_{22}\end{array}\right]$

Based on the analogies in the exercise, the transition probabilities will be given as:

$T_{11} = 2/3\\T_{22} = 3/4\\T_{21} = 1 - T_{11} = 1 - 2/3\\T_{21} = 1/3\\T_{12} = 1 - T_{22} = 1 - 3/4\\T_{12} = 1/4$

a) The transition matrix for this Markov process will be:

$M = \left[\begin{array}{ccc}2/3 &1/4 \\1/3 &3/4}\end{array}\right]$

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

To do this, let us find M³ to be able to find any probability on the next three observations.

$M^{3} = \left[\begin{array}{ccc}2/3 &1/4 \\1/3 &3/4}\end{array}\right]\left[\begin{array}{ccc}2/3 &1/4 \\1/3 &3/4}\end{array}\right]\left[\begin{array}{ccc}2/3 &1/4 \\1/3 &3/4}\end{array}\right]$

$M^{3} = \left[\begin{array}{ccc}203/432 &229/576 \\229/432 &347/576}\end{array}\right]$

If the animal is initially in the woods, probability that it is in the woods on the next three observations = $M_{22} ^{3}$ = 347/576

$M_{22} ^{3}$ = 0.602

3) If the animal is initially in the woods, probability that it is in the meadow on the next three observations = $M_{12} ^{3}$ = 229/576

$M_{12} ^{3}$ = 0.398