Consider the random variable Where each Bi is obtained as a result of the toss of a fair coin We let n 4 and consider X 4 (a) Calculate the E X 4 I 1 , E X 4 I 2 , E X 4 I 4 (b) Let Z i E X 4 I i Is Z i , I 1 4, a martingale (c) Now define V i B i i And Is Vi a martingale (d) Can you convert Vi into a martingale by an appropriate transformation (e) Can you convert Vi into a martingale by changing the probabilities associated with a cointoss X,

Question: Consider the random variable: Where each Bi is obtained as a result of the toss of a fair coin: We let n = 4 and

Consider the random variable:

Where each Bi is obtained as a result of the toss of a fair coin:

We let n = 4 and consider X₄.

(a) Calculate the E [X₄| I₁], E [X₄| I₂], E [X₄| I₄]

(b) Let

Z_i = E [X₄| I_i]

Is Z_i, I = 1. . . 4, a martingale?

V_i = B_i + ?_i

And

Is Vi a martingale?(d) Can you convert Vi into a martingale by an appropriate transformation?(e) Can you convert Vi into a martingale by changing the probabilities associated with a cointoss?

. X,

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