Please solve all parts of the following question. Please show all work and all steps. Please assist in the following questions using your own solutions, and do not use solutions that were already posted on coursehero or chegg.

1.) Let Xn is a simple random walk (p = 1/2) on {0, 1, , 100} with absorbing boundaries. Suppose X₀ = 50. Let T = min{j : Xj = 0 or 100}. Let Fn denote the information contained in X1, , Xn.

(a) Verify that Xn is a martingale.

(b) Find P(X_T = 100).

(c) Let Mn = X²_n n. Verify that M_n is also a martingale.

(d) It is known that M_n and T satisfy the assumptions of Optimal Sampling Theorem and hence

E(M_T ) = E(M₀). Use this fact to find out E(T).