A gambler with an initial finite capital of d > 0 dollars starts to play a dollar slot machine. At each play, either his dollar is lost or is returned with some additional number of dollars. Let Xi be his change of capital on the ith play. Assume that {Xi; i=1, 2, .. .} is a set of IID rv s taking on integer values {-1, 0, 1, .. .}. Assume that E [Xi] <>0. The gambler plays until losing all his money (i.e., the initial d dollars plus subsequent winnings).
(a) Let J be the number of plays until the gambler loses all his money. Is the WLLN sufficient to argue that limn→∞ Pr{J > n} = 0 (i.e., that J is a rv) or is the SLLN necessary?
(b) Find E [J]. Hint: The fact that there is only one possible negative outcome is important here.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.