236 CHAPTER 5 Discrete Probability Distributions Statistical Literacy and Critical Thinking 1.Internet Traffic Data Set 27 “Internet Traffic” includes 9000 arrivals of Internet traffic at the Digital Equipment Corporation, and those 9000 arrivals occurred over a period of 19,130 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use Formula 5-9 to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of m, x, and e that would be used in that formula? 2. Internet Arrivals For the random variable x described in Exercise 1, what are the possible values of x? Is the value of x = 4.8 possible? Is x a discrete random variable or a continuous random variable? 3.Internet Traffic For the distribution described in Exercise 1, find the probability of exactly 2 arrivals in one thousandth of a minute. 4.Probability if 0 For the random variable x described in Exercise 1, use Formula 5-9 to evaluate P102. For any Poisson probability distribution, simplify Formula 5-9 for the case of x = 0. Hurricanes. In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability. 5. Hurricanes a. Find the probability that in a year, there will be 7 hurricanes. b. In a 118-year period, how many years are expected to have 7 hurricanes? c. How does the result from part (b) compare to the recent period of 118 years in which 14 years had 7 hurricanes? Does the Poisson distribution work well here? 6. Hurricanes a. Find the probability that in a year, there will be no hurricanes. b. In a 118-year period, how many years are expected to have no hurricanes? c. How does the result from part (b) compare to the recent period of 118 years in which there were 2 years without any hurricanes? Does the Poisson distribution work well here? 7. Hurricanes a. Find the probability that in a year, there will be 3 hurricanes. b. In a 118-year period, how many years are expected to have 3 hurricanes? c. How does the result from part (b) compare to the recent period of 118 years in which 17 years had 3 hurricanes? Does the Poisson distribution work well here? 8. Hurricanes a. Find the probability that in a year, there will be 10 hurricanes. b. In a 118-year period, how many years are expected to have 10 hurricanes? c. How does the result from part (b) compare to the recent period of 118 years in which 4 years had 10 hurricanes? Does the Poisson distribution work well here? In Exercises 9–16, use the Poisson distribution to find the indicated probabilities. 9. Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births. a. Find the mean number of births per day. b. Find the probability that in a single day, there are 16 births. 5-3 Basic Skills and Concepts
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