4. (25 total points) A researcher is doing eye-tracking research where (using an eye-tracker) they are observing the location of a person's eyes every second in one of 4 locations on a screen. A picture of the possible locations a person's eyes could be looking at are shown below. For simplicity, a person can only switch their stare from one zone to another after 1 second has passed and it is possible to stare in the same zone for more than one second. Zone 1 Zone 2 Zone 3 Zone 4 a) (6 points) Assume this setup can be modeled as a discrete time Markov chain. Please (1) clearly define the state space and (2) draw the Markov chain diagram. Assume all states are directly accessible to each other and have non-zero probabilities. b) (4 points) Given the person being observed has their stare in zone 3 after 20 seconds of running the study, write the expression to determine how many seconds will go by (on the average) for the person to eventually move their eyes to stare in zone 2. c) (4 points) Write the expression to determine the probability of the person fixating their stare in zone 4 for 5 seconds in a row. d) (4 points) Write the expression to determine the expected amount of time it will take for someone staring in zone 3 to return back to zone 3. e) (7 points) It turns out the best/most accurate way to predict where a person's stare from one zone of the screen to another is to create a second order discrete time Markov model instead of the first order model that you defined in part a. Please (1) clearly define the state space for a second order Markov model that uses the last two locations of where the person was staring, and (2) draw the Markov chain diagram. 4. (25 total points) A researcher is doing eye-tracking research where (using an eye-tracker) they are observing the location of a person's eyes every second in one of 4 locations on a screen. A picture of the possible locations a person's eyes could be looking at are shown below. For simplicity, a person can only switch their stare from one zone to another after 1 second has passed and it is possible to stare in the same zone for more than one second. Zone 1 Zone 2 Zone 3 Zone 4 a) (6 points) Assume this setup can be modeled as a discrete time Markov chain. Please (1) clearly define the state space and (2) draw the Markov chain diagram. Assume all states are directly accessible to each other and have non-zero probabilities. b) (4 points) Given the person being observed has their stare in zone 3 after 20 seconds of running the study, write the expression to determine how many seconds will go by (on the average) for the person to eventually move their eyes to stare in zone 2. c) (4 points) Write the expression to determine the probability of the person fixating their stare in zone 4 for 5 seconds in a row. d) (4 points) Write the expression to determine the expected amount of time it will take for someone staring in zone 3 to return back to zone 3. e) (7 points) It turns out the best/most accurate way to predict where a person's stare from one zone of the screen to another is to create a second order discrete time Markov model instead of the first order model that you defined in part a. Please (1) clearly define the state space for a second order Markov model that uses the last two locations of where the person was staring, and (2) draw the Markov chain diagram