Question

Emma has noticed a small red fox living in the park near her apartment. She takes a walk at the same time every day and has observed the fox in three different areas: in the woods, in the meadow, and by the pond.

If it is in the woods on one observation, then it is twice as likely to be in the woods as in the meadow on the next observation but not by the pond.

If it is in the meadow on one observation, then it is equally likely to be in any of the three locations on the next observation.

If it is by the pond on one observation, there is a 0.5 probability it will be by the pond on the next observation and will otherwise be in the woods.

When Emma went for a walk today, the red fox was in the woods.

a. Define the states and construct the transition matrix for this Markov chain.

b. Find the initial distribution vector for this Markov Chain.

c. Determine the probability that the red fox is in each of the three areas tomorrow.

If this trend continues, what is the probability the red fox will be in the meadow in the long run?

Show all work to support your answer. Correct answers without supporting work will not receive credit.

Answer 1

Answer:-

Given That:-

If it is by the pond on one observation, there is a 0.5 probability it will be by the pond on the next observation and will otherwise be in the woods. When Emma went for a walk today, the red fox was in the woods.

Given,

Emma has noticed a small red fox living in the park near her apartment. She takes a walk at the same time every day and has observed the fox in three different areas: in the woods, in the meadow, and by the pond.

a. Define the states and construct the transition matrix for this Markov chain.

The states are

1) Woods

2) Meadow

3) Pond

Now from the given information markov chain is

	Woods	Meadow	Pond
Woods	2/3	1/3	0
Meadow	1/3	1/3	1/3
Pond	1/2	0	1/2

b. Find the initial distribution vector for this Markov Chain.

Let initial distribution is

x=y

3x=2x+z

x=z

x=z

So initial distribution

c. Determine the probability that the red fox is in each of the three areas tomorrow.

If this trend continues, what is the probability the red fox will be in the meadow in the long run?

Show all work to support your answer. Correct answers without supporting work will not receive credit.

probability of being an woods,Meadow and pond tomorrow are   repetively.

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