(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=4) p, P(Z₁ = -3)=q=1-p. (i) Find the mean and variance of Zi, and hence find the mean and variance of X, the position of the particle after n steps. (ii) Find the probability distribution of Xn, and state the range of Xn. Explain all the steps in your derivation of the distribution. (iii) Calculate the following probabilities in terms of p and q: P(X10 -2), P(X = 1). = (iv) Calculate the probabilities up, u7, 414 in terms of p and q, and hence find the probability that the particle returns to the origin for the first time after 14 steps.