Any help with any of this would be much appreciated

11:02 5G Done stat304_assig1_2023 STAT304/STAT804 Assignment 1 Due by 25th August 2023 5pm 1. In the game Snakes & Ladders, a player has a piece which starts on the first tile. The player rolls a die and moves his/her piece on the board along the numbered tiles. If the piece lands on the head of a snake, it is immediately placed at the snake's tail. If the piece lands at the foot of a ladder, it is immediately placed at the top of the ladder. To win the game, the player's piece must land exactly on the last tile. If it does not, the remaining steps in the player's turn are played backwards. On the next roll the piece is moved forward again. Once the piece falls on the last tile, the game is over and stays in that state of completion forever. Consider the following board and assume that the die only has three faces (i.e., one can roll only 1,2, and 3). END II START (a) Denote as {Xn}ner, T = {0, 1, ...}, the position of the player's piece on the board. What is the state-space of { X}ner, S? (Hint: There are less than 9 states) (1 mark) (b) Draw a state diagram and write down the one-step probability transition matrix P. (1 mark) (c) Let Xn be the position of the player's piece at the nth turn. Is {X, }ner, T = {0, 1, 2, . .. } a finite-state, stationary, Markov chain? (2 marks) (d) Classify the states as transient, recurrent, or absorbing. List the communication classes, and indicate which communication classes are closed or not closed. Is the chain irreducible? (2 marks) A11:02 5G Done stat304_assig1_2023 classes, and indicate which communication classes are closed or not closed. Is the chain irreducible? 8 2 of2 (2 marks) (e) Let fi be the probability that the process eventually revisits State i given that it starts in State i. Derive fi for i e S. (4 marks) (f) How many times, on average, is the process in State i given that it starts from State i, for i E S? (1 mark) (g) On average, how many moves are there in a game with one player? (4 marks) A