Long-run share of rainy days

Question

Each day the weather is sunny, cloudy, or rainy, and tomorrow depends only on today: - From sunny: to sunny 0.5, to cloudy 0.3, to rainy 0.2. - From cloudy: to sunny 0.4, to cloudy 0.4, to rainy 0.2. - From rainy: to sunny 0.3, to cloudy 0.3, to rainy 0.4. In the long run, what fraction of days are rainy?

Accepted Answer

We want the steady shares (\piS, \piC, \piR) that stop changing day to day. Each share equals its total inflow: (share of every origin) times (chance it leads here). Focus on rainy, since a rainy day arrives from all three states (0.2, 0.2, 0.4): \piR = 0.2\,\piS + 0.2\,\piC + 0.4\,\piR \implies 0.6\,\piR = 0.2\,\piS + 0.2\,\piC. The right side is 0.2\,(\piS + \piC), and since the three shares add to 1 we have \piS + \piC = 1 - \piR. Substitute: 0.6\,\piR = 0.2\,(1 - \piR) = 0.2 - 0.2\,\piR \implies 0.8\,\piR = 0.2 \implies \piR = 	frac14. So one quarter of days are rainy. (The other two sha…

Long-run share of rainy days

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