Question

15. An individual possesses r umbrellas which he employs in going from his home to office, or vice versa. If he is at home (resp., the office) at the beginning (resp., end) of a day and it is raining, then he will take an umbrella with him to the office (resp., home), provided there is one to be taken. If it is not raining, then he never takes an umbrella. Assume that, independent of the past, it rains at the beginning (resp., end) of a day with probability p. i.)Define a Markov chain with r1 states which will help us to determine the proportion of time that the man gets wet. (Note: He gets wet if it is raining, and all umbrellas are at his other location.) (Hint: Let Xnumber of umbrellas at his present location.) i.)Show that the limiting probabilities are given by tq r+q where q = 1-p. (Hint: You need only show the given solution satisfies and use uniqueness.) iii) What fraction of time does the man get wet? (Hint: How does enter?) iv.) Whenr 3, what value of p maximizes the fraction of time he gets wet? (Hint: Use (iii) withr 3 and maximize.)

Answer 1

a) w define a Maxkov chain itn ttI States which siit help us dateimine tne Pspaon whose ídonotes the number or um 6nel (as ot, his Presen1 夂oCofren now use detox m ine the elemants, op the transitPtobabitities mai e at time n he is at his hous2 affice (house) ahe has one umbrella at his esesetocatton and p-i a the otheu ie:, is does not ain and (ie, it sains ard he takes one umbrallo from his Present oCotion to the othoxt)

Oo toe Knols nas b) Solving there, use g24 tne probabititieg P41-P