Every day, a friend chooses at random a coffee shop that is directly connected t

Question

Every day, a friend chooses at random a coffee shop that is directly connected to the shop visited on the previous day. That is, if he was at shop 3 yesterday, he can only be at 2 or 4 today with equal probabilities; see the figure below with shops marked by 1 - 4. Give the long run mean frequency of visits to the coffee shop 2. Suppose that the colleague buys an espresso in each coffee shop he visits. For concreteness, suppose that an espresso in the store i costs c_i dollars, where c_1 = 2.00, c_2 = 2.50, c_3 = 3.00, c_4 = 3.00. Approximately, on average, how much our friend will spend on espresso during a whole year?

Explanation / Answer

Long run mean frequency of visits to the coffee shop 2 : 1-2 ,3-2, 4-2 are three pssibilities .That is he can visit coffee shop 2 today if he has visited coffee shops 1,3 or 4 yesterday. Long run mean = 1*1 + 1*1/2 + 1* 1/2 = 1 + 1/2 +1/2 = 2.Long run mean frequency is 3 1-2-(1 or 3 or 4) -{2 - (2 or 4) - (2 or 3)} - (1or 3 or 4) -{(1 or 3 or 4 ) - ( 2 or 3)} -{(1 or 3 or 4) - (2 or 4)} -2 - (2 or 3) -(2 or 4)- (1 or 3 or 4) -------- sequence continues upto 365 days( nonleap year) Average expenditure = {2 +2.5 + ( 2 + 3 +3 )/3 + (2.5 + 3)/2 + (2.5 +3)/2 } * 73 = [2 +2.5 + 2.67 + 2.75 +2.75] * 73 = 12.67*73 = $ 924.91 (365/5 = 73 as sequence repeats after 5 terms)

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