Please solve stochastic processes problem attached, and show all work. Thank you.

After solving the book fiasco, you realize that you're famished and are craving some gummy sour worms from the snack machine on the 4th floor of Mohler, so you decide to make your way there. When you get in the elevator and start to rise, you hear a strange noise beneath your feet. All of a sudden you see a panel in the floor that you never noticed before. When you open it up, you find that the elevator is being raised by a team of sloths who literally push it up by climbing on each other's backs in a sort of sloth pyramid formation. (That explains why its such a slow elevator!) You feel awful for the sloths, but they say they are perfectly happy to work there since they are more than fairly compensated in the form of bags of gummy sour worms. Their compensation depends on the changes they have to make in the state of the elevator. The elevator locks in place whenever it is not being moved up or down, so for all intents and purposes you consider the state of the elevator as changing at discrete times, say t = 1,2,3, .... The state of the elevator is the floor that it is on, which can be 1, 2, 3, or 4. Based on historical data, if the elevator is on floor 1, then with probability 0.2 an occupant wants it to be raised to floor 2, with probability 0.3 an occupant wants it to be raised to floor 3, and with probability 0.5 an occupant wants it to be raised to floor 4. These and the remaining probabilities are shown in the following matrix whose rows and columns correspond respectively to floors 1, 2, 3, and 4: 0.0 0.2 0.3 0.5 0.3 0.0 0.2 0.5 0.4 0.1 0.0 0.5 0.8 0.1 0.1 0.0 (a) (6 points) Write a system of equations that you can use to determine the long-run probability that the last transition of the elevator was to floor j. This should be a single set of equations that you can solve for all j E {1, 2, 3, 4} simultaneously. (b) (4 points) Solve your system from part (a) and state the long-run probabilities. (c) (4 points) Suppose that every time the elevator transitions to floor 1, the sloths are paid 1 bag of sour gummy worms; similarly, they are paid 2 bags for transitions to floor 2, 3 bags for transitions to floor 3, and 4 bags for transitions to floor 4 What is the long-run expected average number of bags of sour gummy worms that the sloths are paid per transition