Question

8 Total Question 3 A clothing company changes its sales depending on if the weather is clear or raining. A weather forecasting company has told them that if it is clear today, it is 80% likely it will be clear tomorrow. However, if it is raining today, it is 65% likely to be raining tomorrow. 3(a). What is the transition matrix for this Markov chain? /1 mark 3(b). The weather forecast for next Monday says it is 20% likely to be clear Taking that as your initial distribution, what is the probability it will rain next Tuesday? /2 marks

Answer 1

(a)

Let the states be C and R denoting that it is clear or raining today respectively.
The transition probability from C to C is 0.8
The transition probability from C to R is 1 - 0.8 = 0.2
The transition probability from R to R is 0.65
The transition probability from R to C is 1 - 0.65 = 0.35

The transition matrix is,

P= 0.8 0.35 0.35 0.2 0.65

(b)

Probability that it will rain next Tuesday = Probability that it will rain on Monday * transition probability from R to R + Probability that it will be clear on Monday * transition probability from C to R

= (1 - 0.2) * 0.65 + 0.2 * 0.2

= 0.56

(c)

Let $\pi$ = [a b] be the long run proportions of clear or raining weather

Then,  $\pi$P = $\pi$ and a + b = 1

which gives,

0.8a + 0.35b = a => 0.2a = 0.35b => b = 0.5714286 a

0.2a + 0.65b = b

a + b = 1 => a + 0.5714286 a = 1

=> 1.5714286 a = 1

=> a = 1/ 1.5714286 = 0.6364

Thus, the long run proportions of clear weather is 0.6364