3. Weather in Whistler during the ski seasons shows the following longterm patte

Question

3. Weather in Whistler during the ski seasons shows the following longterm pattern. It alternates between sunny (Su), cloudy (C1), and snowy (Sn) days. A sunny day stays sunny half times and turns into a cloudy and snowy weather at equal chances. A cloudy day never stays cloudy and always turns snowy the next day. A snowy day never turns sunny and has equal chances of staying snowy and turning cloudy. (a) Setup a probability transition matrix for Whistler weather by defining the probability Su(t) vector as x Sn(t) (b) We know that a Markov matrix always has an eigenvalue 1, Find the corresponding eigenvector. If it is unique, then it is the unique equilibrium (stationary) probability distribution Te. (Hint: since fe is a probability distribution here, one should scale its components so that they sum to 1.)

Explanation / Answer

(a) Probability Transition Mattrix (P) is as folows:

(b) For eigen value lambda = 1. for Markov process,

use P-I =0, where I is an identity matrix (3,3)

Eigen vector can be computed as

0.8164966 0.8164966 0.8164966
0.4082483 -0.4082483 -0.4082483
0.4082483 -0.4082483 -0.4082483

Next State    Current State Sunny [Su(0)] Cloudy [Cl(0)] Snowy [Sn(0)] Sunny [Su(1)] 0.5 1.0 0 Cloudy [Cl(1)] 0.25 0 0.5 Snowy  [Sn(1)] 0.25 0 0.5

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