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 X, 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: [4] [5] P(X10 -2), P(X = 1). [2] (iv) Calculate the probabilities uo, 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. [4]