1 ) Suppose X1, X2, are independent identically distributed random variables with mean 0 and variance 1 Let S n denote the partial sum S n X 1 X n Let F n (filtration) denote the information contained in X 1 X n Please answer all parts a d of the following question Please show all work and all steps clearly for parts a d (a) Compute E X 2 n 1 Fn (b) Compute E S n X n 1 Fn (c) Compute E S 2 n 1 Fn Hint Write S 2 n 1 (S n X n 1 ) 2 (d) Verify that S 2 n n is a martingale

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Question: 1.) Suppose X1, X2, are independent identically distributed random variables with mean 0 and variance 1. Let S n denote the partial sum S n

1.) Suppose X1, X2, are independent identically distributed random variables with mean 0 and variance 1.

Let S_n denote the partial sum S_n = X₁ + .... + X_n.

Let F_n (filtration) denote the information contained in X₁ + .... + X_n. Please answer all parts a-d of the following question. Please show all work and all steps clearly for parts a-d.

(a) Compute E[X²_n+1|Fn].

(b) Compute E[S_nX_n+1|Fn].

(d) Verify that S²_n n is a martingale

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