Question: 2.12 Let the random variable Sn be defined as follows: Sn 5 0 X n 5 0 n k51 Xk n $ 1 where Xk

2.12 Let the random variable Sn be defined as follows: Sn 5 0 X n 5 0 n k51 Xk n $ 1 where Xk is the kth outcome of a Bernoulli trial such that P½Xk 5 1 5 p and P½Xk 521 5 q 5 1 2 p, and the Xk are independent and identically distributed. Consider the process fSnjn 5 1; 2; ...g.

a. For what values of p (relative to q) is fSng a martingale?

b. For what values of p is fSng a submartingale?

c. For what values of p is fSng a supermartingale?

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