Please respond with clear writing and show work. Exercise 9.3.1 The following ma

Question

Please respond with clear writing and show work.

Exercise 9.3.1 The following matrix is the single-step transition probability matrix of a discrete-time Markov chain which describes the weather. State 1 represents a sunny day, state 2 a cloudy day, and state 3 a rainy day. Sunny Cloudy Rainy Sunny 0.7 0.2 0.1 Cloudy 0.3 0.5 0.2 Rainy 0.2 0.6 0.2 (a) What is the probability of a sunny day being followed by two cloudy days? (b) Given that today is rainy, what is the probability that the sun will shine the day after tomorrow? c) What is the mean length of a rainy period?

Explanation / Answer

a) The path here required is:

S --> C --> C

Therefore the probability here is computed as:

Prob = Psc *Pcc = 0.2*0.5 = 0.1

Therefore 0.1 is the required probability here.

b) Given today is rainy, the sun will shine the day after tomorrow in the following ways:

Therefore the total probability here is computed as: = 0.04 + 0.18 + 0.14 = 0.36

Therefore 0.36 is the required probability here.

c) Let the mean length of a rainy period be X. Then, we have here:

X = 0.2*1 + 0.6*1 + 0.2*(1 + X)

X = 1 + 0.2X

X = 1/0.8 = 1.25

Therefore 1.25 days is the mean length of a rainy period here.

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