Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

$12$ am

Home

Home

Home

Home

Home

Home

Home

$1$ am

Home

Home

Home

Home

Home

Home

Home

$2$ am

Home

Home

Home

Home

Home

Home

Home

$3$ am

Home

Home

Home

Home

Home

Home

Home

$4$ am

Home

Home

Home

Home

Home

Home

Home

$5$ am

Home

Home

Park

Home

Home

Home

Home

$6$ am

Home

Park

Park

Grocery

Home

Home

Home

$7$ am

Home

Park

Home

Home

Park

Home

Home

$8$ am

Office

Home

Office

Office

Park

Home

Home

$9$ am

Office

Office

Office

Office

Home

Restaurant

Park

$10$ am

Office

Office

Office

Office

Home

Restaurant

Park

$11$ am

Office

Office

Office

Office

Office

Grocery

Park

Noon

Office

Office

Office

Office

Office

Grocery

Park

$1$ pm

Office

Office

Office

Office

Office

Home

Park

$2$ pm

Office

Office

Office

Office

Office

Home

Park

$3$ pm

Office

Office

Office

Office

Office

Home

Home

$4$ pm

Office

Office

Office

Office

Office

Home

Home

$5$ pm

Office

Office

Office

Office

Office

Park

Home

$6$ pm

Home

Office

Grocery

Restaurant

Movie

Park

Home

$7$ pm

Home

Home

Grocery

Restaurant

Movie

Home

Home

$8$ pm

Home

Home

Home

Restaurant

Movie

Restaurant

Home

$9$ pm

Home

Home

Home

Home

Home

Restaurant

Home

$10$ pm

Home

Home

Home

Home

Home

Restaurant

Home

$11$ pm

Home

Home

Home

Home

Home

Home

Home

Table: User A$$s Hourly Location Log

Consider a smart thermostat $($e$.$g$.,$ Nest by Google$).$ The thermostat shuts down the air conditioning system when users are not at home, which should help to save energy. Based on a predictive model, and given the location of the users, the thermostat determines when it should turn on the air conditioning so the temperature is comfortable as soon as the users arrive at their homes $($and remains nice until they leave again$).$

Suppose that a particular user $($User A$)$ lives alone in a house. User A$$s hourly location log is provided in the table. This log is available to a mobile application that interfaces with the Nest thermostat.

Design a markov chain with $2$ states Home and not Home. Consider that the day of the week does not matter. Consider data for all $7$ days of the week with equal probability of occurrence when you build the Markov chain. Compute the stationary probability matrix.

Assume that it takes $15$ minutes for Nest to get the house temperature to $75$ degrees, regardless of the starting temperature. Given that it is $9$ am and User A is not at home, if the AC restarts at $12$:$45$ pm$,$ what is the probability that User A will arrive home in time with the temperature at $75$ degrees? $($Write your answer as a decimal and round to the nearest hundredth.$)$