Suppose that a point starts from the origin (i.e. (0, 0)), and on any move is equally likely to go one unit up, down, left, or right, independently of previous moves. Let X1, X2, X3, X4 be random variables representing the number of moves up, down, left, and right respectively in a sequence of n moves. a. In a sequence of n = 8 moves, what is the probability that X1 = 2, X2 = 3, X3 = 1, X4 = 2? b. If n = 8, and X1 = 2, X2 = 3, X3 = 1, and X4 = 2, what is the location of the point after the 8 moves? Please give your final answer in x-y coordinates (i.e. (3, -2)). c. Give an expression for the probability that the point is at the origin after a sequence of 8 moves. You do not have to give a numerical answer, just an expression. d. After n moves, the point will have traveled a distance of n units "on the ground", but we want the "as the crow flies" distance. Let D represent the Euclidean distance of the point from the origin. So, if the coordinates of the point are (x, y) after n moves then D = vx2 + y2. Give an expression for D2 in terms of X1, Xz, X3, X4 and show that E[D?] = n