[15 pts] The probability of winning a round of a certain casino game is p, independent of all previous rounds. If a gambler bets an amount c on a round and wins, he or she leaves with a total of 2c (the net gain is c). If the gambler loses, the net gain is c. A gambler named Robin Erbody decides to implement the following strategy: i. He will bet c on the first round. ii. He will stop betting when he wins a round. iii. Each time he loses a round, he will bet twice as much on the next round. Let X denote the number of rounds Robin bets on before he stops. What type of random variable is X ? Let Y denote Robin's net winnings when he nally stops betting. Prove Y E c. That is, show that by using this strategy, Robin is guaranteed to walk away with twice the amount he started with. Let Z denote the amount of money Robin will need in order to implement this strategy. For instance, if Robin wins on the first round, he will only need c dollars to implement the strategy, but if he wins on the second round, he will need a total of 3c dollars to cover his bets in both rounds. Express Z as a function ofX. Prove that E(Z) = co . Give an intuitive interpretation of this result. [10 pts] Let S be a set with n distinct elements. Suppose elements are drawn from S randomly in successive independent trials with repiacement until some element is drawn for the second time. LetX denote the number of draws which take place. a. What are the possible values ofX? b. Recall that the survival function associated with X is defined by sx (x) a\" P(X > x) for all real x. Find an explicit formuia for sx(x). c. Verify that y0ur formula from Part b satises 3;; (1) = 1 , and explain this result intuitively. d. Use your formula from Part b to find formulas for the cdf FX and pmffx