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(ii) Let W be the number of steps of size +5 in the first n steps. Then Wn ~ B(n, p), so P(Wn = a) = (" )pag-a, a =0,1, ...,n. If there are a steps of +5 in the first n steps, and hence n - a steps of -3, then Xn = 5a + (-3)(n - a) = 8a - 3n. So for a = 0, 1 . . ., n, P(Xn = 80 - 3n) = P(Wn = a) = (" ) paq"-a. If & = 80 - 3n, then a = (3n + x)/8 and n - a = (5n - x)/8, so P(X, = x) =( ((3n + 2)/8 ) p(3n+x)/8,(5n-2)/8 and the range of Xg is {-3n, -3n + 8, -3n + 16, ..., 5n - 16, 5n - 8, 5n}.\f(iv) Using the formula from part (ii), un = P(Xo = 0) = 1, us = P(Xs = 0) = (3 ) 2395 -56p395, U16 = P(X16 = 0) = (5 ) 15910 = 8008p5q10. The event Xn = 0 is possible only when n is divisible by 8, and the return probabilities fun} and {f, } are related to one another by (8.1). By definition, up = 1 and fo = 0, so us = f8, U16 = f16 + faug. Hence fs = us = 56p*q", f16 = u16 - fsus = 8008p6q10 - (56p3q5)2 = 4872p"q10.A particle executes an unrestricted random walk starting at the origin. The ith step, Za, has the following distribution: P(Zi = 5) = p, P(Z: = -3) = q =1-p. ii Find the probability distribution of X, ' , and state the range of Xn. Explain all the steps in your derivation. iii Find in terms of p and q the values of P(X10 = -2), P(X10 = 2), E(X10), V(X10). iv Find in terms of p and q the probabilities uo, us and u16, and hence find the probability that the particle returns to the origin for the first time after 16 steps.sample question part :i the distribution of Xn for an unrestricted random walk starting at the origin, for which the ith step, Z,, has the following distribution: P(Zi = 2) = p, P(Z: =-1) = q =1 -p. Find the mean and variance of Zi. part i sample question part i Hence find the mean and variance of Xn. answer( ; The mean and variance of Z, can be found using a method similar to that used for a simple random walk: E(Z;) = 2xp+ (-1) xq = 2p - 9. E(Z? ) = 22 x p+ (-1)2 x q = 4p + 9. so V(Z.) = E(Z?) - (E(Z.))2 = 4p+q - (2p - q)2 part i sample = 4p - 4p- +q - q2 + 4pq answer = 4p(1 - p) + q(1 - q) + 4pq = 4pq + pq + 4pq = 9pq. for (Xn) Since Xn = Z1+ Z2 + ... + Zn; E(Xn) = n(2p - q), and since 21, Z2, . ..; Zn are independent, V(X, ) = n x 9pq = Inpq.[b] A particle executes an unrestricted random walk on the line starting at the origin. The ith step, Zi, has the following distribution: P(Zi :4) :33, P13; 2 3)=q=1p. (i) Find the mean and variance of Hi, and hence nd the mean and variance of X\