A solar heating system can recharge (store heat) in a single sunny day enough to provide for the next two days. In otherwords, the only times that the solar heat must be augmented by someother heat source is when there are three of more consecutive cloudy days. During the heating season of 120 days, theweather patterns are historically that if a day is sunny, there isa 50% chance the next day will be sunny; otherwise it iscloudy. If a day is cloudy, there is only a one-third chancethe next day will be sunny; otherwise it will cloudy again.
a) If today is sunny, how long (in days) willit be until the heating system requires back-up fuel?
b) If yesterday was sunny and today is cloudy,what is the expected number of days that the house can go untilrequiring back-up fuel?
c) If the system is running on back-up fuel,how many more consecutive days will it continue to operate onback-up before the solar heating can take over again?
d) If the back-up heating is needed today, onaverage how long will it be until it is needed again?
e) If the heating season is 120 days long andback-up fuel costs $10 a day what will the season's typicalfuel cost be?
Generate a Matlab simulation to answer thesequestions. Next produce a state diagram, and its associated probability matrix. Find the steady state probability matrixand note that 1/π4 is the same as the simulatedanswer to part d above.