4 This problem concerns 2-dimensional random walk on the square lattice Z2 (consisting of points (z(1), (2)) with (1) and (2) both integers). Let X = (X(1),X (2)) be a random vector which takes the values (1,0), (-1,0), (0, 1), (0, -1) with equal probabilities. For j = 1,2, the marginal p.m.f. of X() takes value 1 with probability, -1 with probability , and 0 with probability Let X be i.i.d. with the same distribution as X, and let Sn = X1 +...+X. Then S represents the position after n steps of a random walker on Z2, starting from (0,0), whose random steps are equally likely to be any of the four unit vectors. (a) The expectation of a random vector Y = (Y(1), y(2)) is given by EY = (EY (1), EY (2)) and the variance is given by Var(Y) = E[(- EY). (Y - EY)] with the dot indicating the dot product. Show that, for the random walk, ES= (0,0) and Var(5m) = n. (b) If the random walk is instead 1-dimensional (probability of steps left or right) or 3- dimensional (probability of steps north, south, east, west, up, down), what is the expected position and the variance of the walk after n steps