Expected Value and Poisson Distributions

Transcription

1 1. I.K. Kim is one of the top player on the LPGA tour. One of the statistics in golf is the measurement of how accurate a player drives ball when teeing off. I.K. s driving accuracy is This means that when she tees off she hits her tee shot in the fairway 76% of the time. The more accurate off the tee a player is, the more times her tee shot lands in the fairway. a. Construct a probability distribution table for the number of times that I.K. s tee shots hit the fairway given that she has six more holes to play. Assume that each tee shot is independent of any of the others and the driving accuracy doesn t change for any of the tee shots. Round each probability to four decimals. b. Find the mean of this probability distribution and write a sentence to interpret the meaning of the mean. c. Find the standard deviation of this probability distribution. d. A friend of yours creates a game for you. He wants to consider the number of times that I.K. hits her tee shot in the fairway on her last six holes. If she hits fewer than four fairways on her last six tee shots, then you owe your friend $25. If she hits five fairways, then your friend owes you $5. And, if she hits all six fairways, your friend owes you $15. Otherwise, your friend owes you nothing and you don t owe your friend anything. Find your expected winnings (or losses) from playing this game. You may want to complete a probability distribution table where X is the amount of winnings (or losses). 2. Suppose that at a particular small regional airport, there is on average 18 small aircraft that land there per month. Let X be the number of small aircraft that land at the airport per month and assume a Poisson distribution. a. What are the possible values of X? b. Find the probability that there is exactly 15 small aircraft that land at the airport during the month. c. Find the probability that at most 15 small aircraft land at the airport d. Find the probability that at least 15 small aircraft land at the airport e. It would be unusual to have fewer than small aircraft land at the airport f. It would be unusual to have more than small aircraft land at the airport

2 1. I.K. Kim is one of the top player on the LPGA tour. One of the statistics in golf is the measurement of how accurate a player drives ball when teeing off. I.K. s driving accuracy is This means that when she tees off she hits her tee shot in the fairway 76% of the time. The more accurate off the tee a player is, the more times her tee shot lands in the fairway. a. Construct a probability distribution table for the number of times that I.K. s tee shots hit the fairway given that she has six more holes to play. Assume that each tee shot is independent of any of the others and the driving accuracy doesn t change for any of the tee shots. Round each probability to four decimals. Let X be the number of fairways hit. X = 0, 1, 2, 3, 4, 5, 6 X P(X) 0 binompdf(6,.76,0) = binompdf(6,.76,1) = binompdf(6,.76,2) = binompdf(6,.76,3) = binompdf(6,.76,4) = binompdf(6,.76,5) = binompdf(6,.76,6) = b. Find the mean of this probability distribution and write a sentence to interpret the meaning of the mean. μ = np = (6)(0.76) = 4.56 When randomly selecting any 6 tee shots, the average number of fairways that I.K. Kim would hit is 4.56 c. Find the standard deviation of this probability distribution. σ = npq = (6)(0.76)(0.24) = 1.05

3 d. A friend of yours creates a game for you. He wants to consider the number of times that I.K. hits her tee shot in the fairway on her last six holes. If she hits fewer than four fairways on her last six tee shots, then you owe your friend $25. If she hits five fairways, then your friend owes you $5. And, if she hits all six fairways, your friend owes you $15. Otherwise, your friend owes you nothing and you don t owe your friend anything. Find your expected winnings (or losses) from playing this game. You may want to complete a probability distribution table where W is the amount of winnings (or losses). Let W be the winnings of playing this game. W = $25, $0, $5, $15 W P(W) -$25 P(W = $25) = P(X < 4) = binomcdf(6,.76,3) = $0 P(W = $0) = P(X = 4) = binompdf(6,.76,4) = $5 P(W = $0) = P(X = 5) = binompdf(6,.76,5) = $15 P(W = $0) = P(X = 6) = binompdf(6,.76,6) = Now, calculated the expected value of the game. W P(W) W P(W) -$ ( $25)(0.1539) = $ $ ($0)(0.2882) = $ $ ($5)(0.3651) = $ $ ($15)(0.1927) = $ The expected value of the game μ = [W P(W)] = $ So, if you were to play this game over and over again, on average you would win $ per game.

4 2. Suppose that at a particular small regional airport, there is on average 18 small aircraft that land there per month. Let X be the number of small aircraft that land at the airport per month and assume a Poisson distribution. a. What are the possible values of X? X = 0,1,2, b. Find the probability that there is exactly 15 small aircraft that land at the airport P(X = 15) = poissonpdf(18,15) = c. Find the probability that at most 15 small aircraft land at the airport during the month. P(X 15) = poissoncdf(18,15) = d. Find the probability that at least 15 small aircraft land at the airport during the month. P(X 15) = 1 P(X < 15) P(X 15) = 1 P(X 14) P(X 15) = 1 poissoncdf(18,14) = e. It would be unusual to have fewer than 10 small aircraft land at the airport Unusually low number of landings = μ 2σ μ = 18 σ = μ = 18 = 4.24 So, μ 2σ = = 9.52 Thus, fewer than 10 landings would be unusual. f. It would be unusual to have more than 26 small aircraft land at the airport Unusually high number of landings = μ + 2σ = = Thus, more than 26 landings would be unusual

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