Introduction to Stochastic Process

Question in the picture.Introduction to Stochastic Process.MATH 447 in McGill.4.26 In a lottery game, three winning numbers are chosen uniformly at random from {15 , 100}, sampling without replacement. Lottery tickets cost $1 and allow a player to pick three numbers. If a player matches the three winning numbersthey win the jackpot prize of $1,000. For matching exactly two numbers, theywin $15. For matching exactly one number they win $3. (3) Find the distribution of net winnings for a random lottery ticket. Show thatthe expected value of the game is —70.8 cents. (b) Parlaying bets in a lottery game occurs when the winnings on a lottery (C) ticket are used to buy tickets for future games. Hoppe (2007) analyzes theeffect of parlaying bets on several lottery games. Assume that if a playermatches either one or two numbers they parlay their bets, buying respec-tively 3 or 15 tickets for the next game. The number of tickets obtainedby parlaying can be considered a branching process. Find the mean of theoffspring distribution and show that the process is subcritical. See Exercise 4.19. Let T denote the duration of the process, that is, thelength of the parlay. Find P(T = k), for k = l, ,4. (d) Hoppe shows that the probability that a single parlayed ticket will ulti- mately win the jackpot is approximately p / (l — m), where p is the probabil—ity that a single ticket wins the jackpot, and m is the mean of the offspringdistribution of the associated branching process. Find this probability andShow that the parlaying strategy increases the probability that a ticket willultimately win the jackpot by slightly over 40%.

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