Mathematics
QUESTION 4 Show that span {(1,2,-1,0),(1,1,0,1),(0,0, 1,1)} where =(2,5, -5,1) by finding scalars k,/ and m such that =k(1,2,-1,0) + /(1,1,0,1)+m(0,0,-1,1). k= 1 = m=
Consider the random walk W = (Wn)nzo on Z where Wn Wo + X + + Xn and X, X2,... are independent, identically distributed random variables with 3 3 1 P(Xn 1) P(Xn = 1) P(Xn = 2) 8' 4 We define the hitting times T := = inf{n 20: W = k}, where inf):= +[infinity]. For k, m 0, let x(m) be the probability that the random walk visits the origin by time m given that it starts at position k, that is, (m) := Xk = P(To m | Wo = k). (0) (a) Give x for k 0. For m 1, by splitting according to the first move, show that (m) 3 (m-1) 3 (m-1) 1 Ik + l 8 k-1 (m-1) = + X k+2 (Vk > 1) 8 4 (m) and co = 1. [5 marks] For k0, let x be the probability that the random walk ever visits the origin given that it starts at position k, that is, x= P(To