- Let (X) be a random walk such that Xo = 0, and for all n,...

 

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- Let (X) be a random walk such that Xo = 0, and for all n, P(Xn+1 - Xn= 1)=1/3 and P(Xn+1 - Xn = -1) = 2/3. (a) Define Mn = 2x. Prove that (Mn) is a martingale. == = (b) Define 7 = min{n: n > 1, |X| 100}. Prove that r is a stopping time. (c) Assume that P(T < ) = 1 and the optional stopping theorem can be applied to Mr. Show that the probability of (Xn) reaching 100 before -100 is 1 2100+1 - Let (X) be a random walk such that Xo = 0, and for all n, P(Xn+1 - Xn= 1)=1/3 and P(Xn+1 - Xn = -1) = 2/3. (a) Define Mn = 2x. Prove that (Mn) is a martingale. == = (b) Define 7 = min{n: n > 1, |X| 100}. Prove that r is a stopping time. (c) Assume that P(T < ) = 1 and the optional stopping theorem can be applied to Mr. Show that the probability of (Xn) reaching 100 before -100 is 1 2100+1