If the animal is in the woods on one observation, then it is three times as like

Question

If the animal is in the woods on one observation, then it is three times 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 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 four times as likely to be in the meadows as in the woods, find the state vector XX that represents this information?

(3) Using the state vector determined in the preceding part as the initial state vector, find the probability that the animal is in the meadow on the third subsequent observation.

(4) If the probability that the animal will be the meadow at a specific point in time is 0.05, how many subsequent observations must be made before the probability that it is in the meadow exceeds 0.2?

Explanation / Answer

Let Si, i = 1, 2, denote the state i,

where i = 1 and i = 2 are for meadows and woods, respectively.

If Tij denotes the transition from Sj to Si,
transition probabilities are given as follows.

     Probability(T11) = 1/3

     Probability(T21) = 1- Probility(T11) = 1 - 1/3 = 2/3

     Probability(T12) = 1/4 = 1- Probability(T22) = 1 - 3/4 = 1/4

     Probability(T22) = 3/4

(1) So, the Markovian transition matrix, M, is

     1/3     1/4

     2/3     3/4

2) Probability(T22) =4/5

Prob(T12) = 4/5 = 1- Prob(T22) = 1/5

1/3     1/5

2/3 4/5

M3 = M2·M

M2=

11/45 17/75

34/45 58/75

M3=

157/675 259/1125]

518/675 866/1125

3)=518/675=0.767

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