.... Toss a coin repeatedly, with the probability of head on each toss is }, as is the probability of tail. Let X, = 1 if jth toss is a head and X, = -1 if jth toss is a tail. Define Mo = 0 and n Mn = Xj, IEnsN. (1) 2 j=1 3 This process is called a symmetric random walk. It steps up one with each head, and steps down one with each tail. 1. Show that the symmetric random walk { Mo, M1, . .. , MN} is a martingale. 2. Let & be a positive constant. Define Th = ekMin for n = 0, 1, . .. N. This is the so-called geometrical symmetric random walk. Show that {Go, G1, . . . ; GN} is not a martingale. 3. Next we "discount" the geometrical symmetric random walk by defining Sn = (Her ) Gn for n = 0, 1, . .. , N. Show that the discounted the geometrical symmetric random walk {So, S1, . .. . SN } is a mar- tingale