Homework 9 (for lectures on 4/2)

Transcription

1 Spring 2015 MTH122 Survey of Calculus and its Applications II Homework 9 (for lectures on 4/2) Yin Su Problems: 1. Suppose X, Y are discrete random variables with the following distributions: X Y Compute the expected value and variance for each random variable above. 2. (Asset investment) You have two options to invest $ in the stock market. There are two nice stocks available, A and B. Suppose the current price of A and B are both $ 100. You are planning to buy 100 shares of all A, or 100 shares of all B. The possible payoff tables are shown: Price of A in a month Price of B in a month (1) Compute your probability distribution of your TOTAL profit from A or B: Profit of A in a month Profit B in a month (2) Compute the expected profit in a month respectively. Which stock has higher average profit? () Compute the variance of the profit from A and B respectively. Which stock has higher variance? (4) If you are assumed to be risk-averse, which stock will you buy?. (Road service package) BBB is a vehicle road service company. It provides emergency road service (such as towing, jump start) for the customers. BBB offers a road service package as follows: If you pay $ 50, then you will be able to request free road services at most four times in the next 65 days. Now suppose it cost the company BBB $ 100 for each road service. The probability of number of service request is as follows: Number of service requests or up

2 (1) Compute the average cost of road service for BBB in each package contract. (2) Compute the average profit for BBB by selling a road service package. () Instead of signing a package contract with BBB, a driver can choose not to participate in any contract. So he has to pay the service fee every time he needs road service. Assume he chooses not to buy the package. Then each time he requests a road service, he has to pay 200 dollars. The probability of the number of requests is the same as the above table. Try to compute his average expense on road service in the next 65 days if he doesn t buy the package. (4) Based on the computation, if he wants to minimize his average expense on road service, should he buy the service package from company BBB? 4. There are 10 balls in a box. Among these balls, one is red, four are blue and all others are black. Draw a random ball from the box. (1) Let X be the color of the ball you drew. Find the probability distribution table of X : X red blue black (2) A game is designed as follows: You draw a random ball from the box, and the color of the ball will correspond to some payoff. The payoff table is shown below: Color of the ball Profit in dollars red 12 blue 2 black 6 Let Y be the payoff in one game. Compute the average payoff if you play this game once. Are you going to earn money or lose money on average? Also compute the variance Var[Y ]. () Assume you decide to double the bet. In other words, the payoff table will be doubled for all colors: Color of the ball Profit in dollars red 24 blue 4 black 12 Let Z be the payoff in one game when payoff doubles. Do you think the average payoff will double too? I.e. E[Z] = 2E[Y ]? Explain why. Do you think the variance will double too? I.e. Var[Z] = 2Var[Y ]? Explain why. 5. (Side bet in three card poker game) A popular poker table game in casino is called the three card

3 poker. In each game, there are two mode of play, namely Ante and Play, and Pair Plus. You can check the general rules online: We only consider the side bet Pair Plus here. In the game, three cards in a 52-card deck are dealt to the player and the dealer. The player will win if the pattern he gets is better than or equal to a pair. (This side bet has nothing to do with the dealer s hand cards.) The winning patterns and payoff table are shown as follows. Suppose we also know the probability of the occurrence of each outcome (You don t need to know how this is computed. Actually higher probability theory is needed in the computation. We will just assume this is the correct probability. This result is also shown on the above Wikipedia page.) Hand card Pattern Payoff of occurrence Straight flush Three of a kind Straight Flush A Pair Others Now evaluate the average profit if you play this side bet once. Is the game against you? 6. (Game of Roulette Wheel, simplified) In the game of Roulette wheel, there is a layout with 7 single numbers (1 through 6, and 0) which correspond to a Roulette Wheel having identical numbers as the layout. The Dealer spins the Roulette Wheel in one direction and a small ball in the opposite direction. The ball will stop at one number (You can check the rules in Seneca Casino: com/sites/senecaniagaracasino.com/files/how-to-play/ roulette.pdf) Suppose now there are two options to play:

4 (a) Mode of Straight up: place you bet on exactly one number out of the 7 numbers on the number when the wheel stops. This mode pays 5 to 1. (You will win $ 5 or you will lose $ 1 in one game). (b) Mode of odd or even: Select between odd or even numbers on the number when the wheel stops. In this mode, 1,, 5,, 5 are odd, and 2, 4,, 6 are even. But 0 is treated as neither odd or even. This mode pays 1 to 1. (You will win $ 1 or you will lose $ 1.) Now let s evaluate the profit of these two games. Finish the following questions: (1) If we play straight up mode, what is the probability that a single number from 0 to 6 occurs in a game? (2) If we play straight up mode, the probability table of your profit is Profit in Straight up 5 (Win) 1 (Lose) () What s the expected return if we play straight up once? (4) If we play odd and even, compute the probabilities that odd occurs, even occurs or neither (0 occurs): Number occur Odd Even 0 (5) Find the probability table for your potential profit: Profit in odd and even 1 (Win) 1(Lose) (6) Compute the expected profit if we play odd and even once. (7) Comparing these two modes, which one has higher average profit (or lower average lost)? Does any of there two mode favor the player?

5 Homework 9 Answer 1. E[X ] = 5 and Var[X ] = 1.2. E[Y ] = 1.4 and Var[Y ] = (1) Profit of A in a month / 1/ Profit B in a month (2) E[A] = 1000 and E[B] = Both stocks have the same average profit. () Var[A] = and Var[B] = (4) Buy A because A has small variance.. (1) Average cost for BBB is 5. (2) Since a package contract will earn 50 for the company BBB, the average profit will be 50 5 = 15. () Average expense for the customer is 70. (4) He should buy the package. 4. (1) Probabilities are 0.1, 0.4, 0.5; (2) E[Y ] = 1; lose money; Var[Y ] = ; () E[Z] = 2 so average doubles; Var[Z] = 12 so variance doesn t double. 5. Average is It is against the player. 6. (1) 1/7. (2) Probabilities are 1/7 and 6/7. () 1/7. (4) Probabilities: 18/7, 18/7, 1/7. (5) Probabilities: 18/7, 19/7. (6) Expected value: 1/7. (7) Average loss is the same for both games. Both games favor the dealer.