4. (A general betting wheel o) Consider a wheel with sectors. If the wheel pointer lands on sector i, the payoff obtained is 7, for every unit bet on that sector The chance of landing on sector i is pi, i = 1,2,, Let , be the fraction of one's capital bet on sector i We requires 1 and a; 20 for i = 1,2,

(a) Show that the optimal growth strategy is obtained by solving max jal (-) P, Inja

(b) Assuming that a > 0 for all i= 1.2, ,, show that the optimal values must satisfy Pkk +1- Pj 1+1 = 0 for all k = 1,2... .n Show that in this case a solution is

a, p; for i =

(c) Assume that 1,2.11

(d) For the wheel given in Example 155, find the optimal solution and determine the corresponding optimal growth rate