Question: 4. (10 points) Let Sn = So + > Xi (n2 1) i= 1 where X1; X2, ... is a sequence of i.i.d. random variables

4. (10 points) Let Sn = So + > Xi (n2 1) i= 1 where X1; X2, ... is a sequence of i.i.d. random variables independent of the random variable So such that P[X/ = +1] = P[X = =1] = (1 =1). (a) Compute the expected value E[S,, ] for all n 2 0. (b) Show that E[Sn+1 | S,] = 5, (n 20). Carefully justify each step of your computations. (c) Suppose that (S,),=0,1,2... represents the profit and loss from $1 bets of a gambler with initial capital So who is repeatedly playing a fair game with 50% chances to win or lose her stake. What are the interpretations of the results in (a) and (b)

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