Michelle on a visit to a popular Las Vegas casino, plays a fair game. In which if she bets $ k-dollars, if she wins she will receive $ 2k-dollars with probability  frac{1}{2 }  and could lose said capital with probability frac{1}{2 } . She has taken several courses of Probability in school and believes that if she follows the following strategy she could win a lot of money: She bets on the first attempt $ 1-dollar if she loses she will bet $ 2-dollars on the next attempt. So if she loses on the first n-tries will bet 2n on the (n + 1) -try. She will stop playing the game the instant she first wins on any attempt Let Xn be the random variable that denotes the net earnings that Michelle has immediately after the nth attempt has finished playing this game. We also have that X0 = 0

a) Show that {Xn} n≥1 is a Martingale

b) It proves that this game will almost certainly end in a finite time.

c) Find the Expected Time in this game finish.

d) Get an expression that describes the net earnings Michelle will have before the end of the game.

e) Get the most money Michelle can lose by playing this game.