Kelly makes a series of n bets, each of which she has probability p of winning, independently. Initially, she has zo dollars. Let X, be the amount she has immediately after her jth bet is settled. Let f be a constant in (0,1), called the betting fraction. On each bet, Kelly wagers a fraction f of her wealth, and then she either wins or loses that amount. For example, if her current wealth is $100 and f 0.25, then she bets $25 and either gains or loses that amount. (A famous choice when p > I/2 is f = 2p-1, which is known as the Kelly criterion.) Find E(%) (in terms of n, p,f,20).
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