Question

A study conducted by the Urban Energy Commission in a large metropolitan area indicates the probabilities that homeowners within the area will use certain heating fuels or solar energy during the next 10 years as the major source of heat for their homes. The following transition matrix represents the transition probabilities from one state to another.

		Elec.  Gas    Oil    Solar
	Electricity Natural Gas Fuel Oil Solar Energy	0.60 0.05 0.10 0 0.15 0.85 0.10 0.09 0.10 0.02 0.75 0.09 0.15 0.08 0.05 0.82

Among homeowners within the area, 20% currently use electricity, 35% use natural gas, 40% use oil, and 5% use solar energy as the major source of heat for their homes. In the long run, what percentage of homeowners within the area will be using solar energy as their major source of heating fuel? (Round your answer to one decimal place. Assume the trend continues.)

Answer 1

let fixed probability vector is W = [a b c d]
therefore from relation: WT =W
0.6a+0.05b+0.1c+0d=a…..(1)
0.15a+0.85b+0.1c+0.09d=b…..(2)
0.1a+0.02b+0.75c+0.09d=c…..(3)
0.15a+0.08b+0.05c+0.82d=d…..(4)
solving above: a=0.097; b=0.407; c=0.184 ;d=0.313

  In the long run percentage of homeowners within the area will be using solar energy as their major source of heating fuel =31.3%