Question

9.2.10

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

here transition mtrix T =
	electricity	Gas	Oil	Solar
electricity	0.6	0.15	0.1	0.15
Gas	0.05	0.85	0.02	0.08
Oil	0.1	0.1	0.75	0.05
Solar	0	0.09	0.09	0.82

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

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