Exponential DistributionsPrinter lifetime The lifetime of

Problem 5E In exercise, determine which are probability density functions and justify your answer.

Problem 56E Spacecraft components A component of a spacecraft has both a main system and a backup system operating throughout a flight. The probability that both systems fail sometime during the flight is 0.0148. Assuming that each system separately has the same failure rate, what is the probability that the main system fails during the flight?

Problem 2E In exercise, determine which are probability density functions and justify your answer.

Problem 3E In exercise, determine which are probability density functions and justify your answer.

Problem 4E In exercise, determine which are probability density functions and justify your answer.

Problem 1E In exercise, determine which are probability density functions and justify your answer.

Problem 6E In exercise, determine which are probability density functions and justify your answer.

Problem 7E In exercise, determine which are probability density functions and justify your answer.

Problem 8E In exercise, determine which are probability density functions and justify your answer.

Problem 11E Verify that the functions in Exercise are probability density functions for a continuous random variable X over the given interval. Determine the specified probability.

Problem 12E Verify that the functions in Exercise are probability density functions for a continuous random variable X over the given interval. Determine the specified probability.

Problem 9E Let f be the probability dendity function for the random variable L in Example 2f. Explain the meaning of each integral.

Problem 10E Let f(X) be the uniform distribution for the random variable X in Example 11. Express the following probabilities as integrals. a. The probability that the arrow points either between South and West or between North and West. b. The probability that the arrow makes an angle of at least 2 radians.

Problem 13E Verify that the functions in Exercise are probability density functions for a continuous random variable X over the given interval. Determine the specified probability.

Problem 14E Verify that the functions in Exercise are probability density functions for a continuous random variable X over the given interval. Determine the specified probability.

Problem 16E Verify that the functions in Exercise are probability density functions for a continuous random variable X over the given interval. Determine the specified probability.

Problem 15E Verify that the functions in Exercise are probability density functions for a continuous random variable X over the given interval. Determine the specified probability.

Problem 17E In Exercise, find the value of the constant c so that the given function is a probability density function for a random variable over the specified interval.

Problem 18E In Exercise, find the value of the constant c so that the given function is a probability density function for a random variable over the specified interval.

Problem 19E In Exercise, find the value of the constant c so that the given function is a probability density function for a random variable over the specified interval.

Problem 20E In Exercise, find the value of the constant c so that the given function is a probability density function for a random variable over the specified interval.

Problem 22E Find the value of c so that is a probability density function for the random variable X over [ 0, 1], and find the probability P(0.25 ? X ? 0.5).

Problem 24E Suppose f is a probability density function for the random able X with mean ?. Show that its variance satisfies

Problem 25E Compute the mean and median for a random variable with the pr bility density functions in Exercise.

Problem 23E Show that if the exponentially decreasing function is a probability density function, then A = c.

Problem 21E Let Find the value of c so that f is a probability density function. If f is a probability density function for the ran-dom variable X, find the probability P(1 ? X ? 2).

Problem 26E Compute the mean and median for a random variable with the pr bility density functions in Exercise.

Problem 27E Compute the mean and median for a random variable with the pr bility density functions in Exercise.

Problem 28E Compute the mean and median for a random variable with the pr bility density functions in Exercise.

Problem 29E Exponential Distributions Digestion time The digestion time in hours of a fixed amount of food is exponentially distributed with a mean of 1 hour. What is the probability that the food is digested in less than 30 minutes?

Problem 31E Exponential Distributions Lifetime of light bulbs A manufacturer of light bulbs finds that the mean lifetime of a bulb is 1200 hours. Assume the life of a bulb is exponentially distributed. a. Find the probability that a bulb will last less than its guaran-teed lifetime of 1000 hours. b. In a batch of light bulbs, what is the expected time until half the light bulbs in the batch fail?

Problem 32E Exponential Distributions Lifetime of an electronic component The life expectancy in years of a component in a microcomputer is exponentially distrib-uted, and 1/3 of the components fail in the first 3 years. The company that manufactures the component offers a 1 year war-ranty. What is the probability that a component will fail during the warranty period?

Problem 30E Exponential Distributions Pollinating flowers A biologist models the time in minutes until a bee arrives at a flowering plant with an exponential distri-bution having a mean of 4 minutes. If 1000 flowers are in a field, how many can be expected to be pollinated within 5 minutes?

Problem 34E Exponential Distributions Car accidents The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. Based on historical data, an insurance company expects that 30% of high-risk drivers will be involved in an accident during the first 50 days of the calendar year. In a group of 100 high-risk drivers, how many do you expect to be involved in an accident during the first 80 days of the calendar year?

Problem 33E Exponential Distributions Lifetime of an organism A hydra is a small fresh-water ani-mal, and studies have shown that its probability of dying does not increase with the passage of time. The lack of influence of age on mortality rates for this species indicates that an exponential distribution is an appropriate model for the mortality of hydra. A biolo-gist studies a population of 500 hydra and observes that 200 of them die within the first 2 years. How many of the hydra would you expect to die within the first six months?

Problem 36E Exponential Distributions Airport waiting time According to the U.S. Customs and Border Protection Agency, the average airport wait time at Chicago's O'Hare International airport is 16 minutes for a traveler arriving during the hours 7-8 A.M., and 32 minutes for arrival during the hours 4-5 P.M. The wait time is defined as the total processing time from arrival at the airport until the completion of a passen-ger's security screening. Assume the wait time is exponentially distributed. a. What is the probability of waiting between 10 and 30 minutes for a traveler arriving during the 7-8 A.M. hour? b. What is the probability of waiting more than 25 minutes for a traveler arriving during the 7-8 A.M. hour? c. What is the probability of waiting between 35 and 50 minutes for a traveler arriving during the 4-5 P.M. hour? d. What is the probability of waiting less than 20 minutes for a traveler arriving during the 4-5 P.M. hour?

Problem 35E Exponential Distributions Customer service time The mean waiting time to get served after walking into a bakery is 30 seconds. Assume that an expo-nential density function describes the waiting times. a. What is the probability a customer waits 15 seconds or less? b. What is the probability a customer waits longer than one minute? c. What is the probability a customer waits exactly 5 minutes? d. If 200 customers come to the bakery in a day, how many are likely to be served within three minutes?

Problem 37E Exponential Distributions Printer lifetime The lifetime of a $200 printer is exponentially distributed with a mean of 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 100 printers, how much should it expect to pay in refunds?

Problem 38E Exponential Distributions Failure time The time between failures of a photocopier is exponentially distributed. Half of the copiers at a university require service during the first 2 years of operations. If the university purchased 150 copiers, how many do you expect to require service during the first year of their operation?

Problem 39E Normal Distributions Cholesterol levels The serum cholesterol levels of children aged 12 to 14 years follows a normal distribution with mean ? = 162 mg/dl and standard deviation ? = 28 mg/dl. In a popu-lation of 1000 of these children, how many would you expect to have serum cholesterol levels between 165 and 193? between 148 and 167?

Problem 40E Normal Distributions . Annual rainfall The annual rainfall in inches for San Fran-cisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the proba-bility that next year's rainfall will exceed 17 in.?

Problem 41E Normal Distributions Manufacturing time The assembly time in minutes for a com-ponent at an electronic manufacturing plant is normally distributed with a mean of ? = 55 and standard deviation ? = 4. What is the probability that a component will be made in less than one hour?

Problem 42E Normal Distributions Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean µ = 22,000 miles and ? = 4,000 miles. a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles? b. What is the minimum number of miles you would expect to find as the lifetime for 90% of the tires?

Problem 43E Normal Distributions Height The average height of American females aged 18-24 is normally distributed with mean ?= 65.5 inches and ? = 2.5 inches. a. What percentage of females are taller than 68 inches? b. What is the probability a female is between 51" and 5'4" tall?

Problem 45E Normal Distributions Length of pregnancy A team of medical practitioners deter-mines that in a population of 1000 females with ages ranging from 20 to 35 years, the length of pregnancy from conception to birth is approximately normally distributed with a mean of 266 days and a standard deviation of 16 days. How many of these females would you expect to have a pregnancy lasting from 36 weeks to 40 weeks?

Problem 46E Normal Distributions Brain weights In a population of 500 adult Swedish males, medical researchers find their brain weights to be approximately normally distributed with mean ?= 1400 gm and standard deviation ? = 100 gm. a. What percentage of brain weights are between 1325 and 1450 gm? b. How many males in the population would you expect to have a brain weight exceeding 1480 gm?

Problem 47E Normal Distributions Blood pressure Diastolic blood pressure in adults is normally distributed with µ = 80 mm Hg and ? = 12 mm Hg. Ina random sample of 300 adults, how many would be expected to have a diastolic blood pressure below 70 mm Hg?

Problem 48E Normal Distributions Albumin levels Serum albumin in healthy 20-year-old males is normally distributed with ?, = 4.4 and ? = 0.2. How likely is it for a healthy 20-year-old male to have a level in the range 4.3 to 4.45?

Problem 49E Normal Distributions Quality control A manufacturer of generator shafts finds that it needs to add additional weight to its shafts in order to achieve proper static and dynamic balance. Based on experimental tests, the average weight it needs to add is µ = 35 gm with ? = 9 gm. Assuming a normal distribution, from 1000 randomly selected shafts, how many would be expected to need an added weight in excess of 40 gm?

Problem 44E Normal Distributions Life expectancy At birth, a French citizen has an average life expectancy of 81 years with a standard deviation of 7 years. If 100 newly born French babies are selected at random, how many would you expect to live between 75 and 85 years? Assume life expectancy is normally distributed.

Problem 50E Normal Distributions Miles driven A taxicab company in New York City analyzed the daily number of miles driven by each of its drivers. It found the average distance was 200 mi with a standard deviation of 30 mi. Assuming a normal distribution, what prediction can we make about the percentage of drivers who will log in either more than 260 mi or less than 170 mi?

Problem 51E Normal Distributions Germination of sunflower seeds The germination rate of a particular seed is the percentage of seeds in the batch which successfully emerge as plants. Assume that the germination rate for a batch of sunflower seeds is 80%, and that among a large population of n seeds the number of successful germinations is normally distributed with mean . a. In a batch of n = 2500 seeds, what is the probability that at least 1960 will successfully germinate? b. In a batch of n = 2500 seeds, what is the probability that at most 1980 will successfully germinate? c. In a batch of n = 2500 seeds, what is the probability that between 1940 and 2020 will successfully germinate?

Problem 52E Normal Distributions Suppose you toss a fair coin n times and record the number of heads that land. Assume that n is large and approximate the discrete random variable X with a continuous random variable that is normally distributed with . If n = 400, find the given probabilities.

Problem 54E You roll a pair of six-sided dice, and the random variable X assigns to each outcome the sum of the number of dots showing on each face, as in Example 2e. a. Find the set of possible outcomes. b. Create a probability bar graph for X. c. What is the probability that X = 8? d. What is the probability that X ? 5? X > 9?

Problem 53E A fair coin is tossed four times and the random variable the number of tails that appear in each outcome. a. Determine the set of possible outcomes. b. Find the value of X for each outcome. c. Create a probability bar graph for X, as in Figure 8.21. What is the probability that at least two heads appear in the four tosses of the coin? Figure 8.21 probability bar graph for the random variable X when tossing a fair coin three times.

Problem 55E Three people are asked their opinion in a poll about a panic brand of a common product found in grocery stores. They answer in one of three ways: "Like the product brand" (L), “Dislike the product brand" (D), or "Undecided" (U). For each come, the random variable X assigns the number of L's that appear. a. Find the set of possible outcomes and the range of X. b. Create a probability bar graph for X. c. What is the probability that at least two people like the product brand? d.. What is the probability that no more than one person dislikes the product brand?