(b) A particle executes an unrestricted random walk on the line starting at the origin. The ith step, Zi, has the following distribution: P(Z =2) = p, P(Z = -4)= q=1-p. (i) Show that the probability distribution of X, is n = = ( (4n + 2)/6) p(n+2)/01 (2n-1)/0, P(Xn = 2) for r {-4n, -4n+6,-4n+ 12,..., 2n - 12, 2n - 6, 2n}. Explain all of the steps in your derivation. (ii) Calculate in terms of p and q the values of V(X10), P(X1014) and P(X10 = -14). (iii) Calculate in terms of p and q the probabilities o, u, and us, where un is the probability that the particle returns to the origin, not necessarily for the first time, after n steps. (iv) Hence find the probability that the particle returns to the origin for the first time after 6 steps. A [5] [3] [2]