机率论习题.doc

4 - 2 4. Suppose that Determine the following probabilities:(a) (b) (c) (d) (e) Determine x such that .5. Suppose that Determine the following probabilities:(a) (b) (c) (d) (e) (f) Determine x such that .6. The probability density function of the time of failure of an electronic component in a copier (in hours) is. Determine the probability that (a) A component lasts more than (超過)3000 hours before failure.(b) A component fails in the interval from 1000 to 2000 hours.(c) A component fails before 1000 hours.(d) Determine in the number of hours at which 10% of all components have failed.(每小時中，一台影印機用到故障的時間)8. The probability density function of the length of a hinge for fastening a door is millimeters. Determine the following:(隨機變數為一個緊固門的鉸鏈的長度)(a) (b) (c) If the specifications for this process are from 74.7 to 75.3 millimeters, what proportion of hinge meets specifications?(如果這個過程的規格是從74.7到75.3毫米，鉸鏈符合規則的機率?)9. The probability density function of the length of a metal rod is meters.隨機變數為一個金屬桿(棒)的長度(單位:公尺)(a) If the specifications for this process are from 2.25 to 2.75 meters, what proportion of the bars fails to meet the specifications?(b) Assume that the probability density function is for an interval of length 0.5 meters. Over what value should the density be centered to achieve the greatest proportion of bars within specifications?(a)如果這個過程的規格是從2.25到2.75米，桿(棒)未能符合規格的機率？ (b)假設機率密度函數為，區間長度0.5為公尺。在密度應該為什麼值才能達到桿(棒)最巨大 的比例在規格之內的？4 - 312. Suppose the cumulative distribution function of the random variable X isDetermine the following : (a) (b) (c) (d) .14. The probability density function of the time customers arrive at a terminal (in minutes after 8:00 A.M.) is Determine the probability that:(隨機變數為顧客在到達終端(8:00上午以後)的時間)(a) The first customers arrive by 9:00 A.M.(b) The first customers arrive between 8:15 A.M. and 8:30 A.M.(c) Two or more customers arrive before 8:40 A.M. among five that arrive at the terminal. Assume customer arrivals are independent. (d) Determine the cumulative distribution function and use the cumulative distribution function to determine the probability that the first customers arrives between 8:15 A.M. and 8:30 A.M.(c)兩個或多個顧客在五個之中在8:40上午之前到達。 假設顧客到來是獨立的。 (d)計算累積分配函数並且使用累積分布函數計算一位顧客在8:15上午和8:30到達上午之間機率？17Determine the cumulative distribution function for the distribution in Exercise 4-418. Determine the cumulative distribution function for the distribution in Exercise 4-6. Use the cumulative distribution function to determine the probability that a component lasts more than 3000 hours before failure.(利用累積分配函數(CDF)計算)19. Determine the cumulative distribution function for the distribution in Exercise 4-9. Use the cumulative distribution function to determine the probability that a length exceeds 2.7 meters.(利用累積分配函數(CDF)求長度超過2.7公尺的機率)22Determine the probability density function for each of the following cumulative distribution functions.4 - 424. Suppose Determine the mean and variable of.25. Suppose Determine the mean and variable of.26. Suppose that Determine the mean and variable for.27. Suppose that contamination particle size (in micrometers) can be modeled as Determine the mean of.(致汙物顆粒大小(單位:微米)30. The probability density function of the weight of packages delivered by a post officeispounds.郵局提供的包裹重量的機率密度函数(a) Determine the mean and variance of weight.(b) If the shipping cost is $2.50 per pound, what is the average shipping cost of a package?(c) Determine the probability that the weight of a package exceeds 50 pounds.31. Integration by part in required. The probability density function for the diameter of a drilled hole in millimeters is mm. although the target diameter is 5 millimeters, vibrations, tool wear, and other nuisances produce diameters larger than 5 millimeters. (a) Determine the mean and variable of the diameter of the holes.(b) Determine the probability that a diameter exceeds 5.1 millimeters. 一個鑽孔的直徑。 雖然目標直徑是5毫米，振動、刀具磨損和其他討厭導致直徑大於5毫米。 (a)計算鑽孔的直徑的平均數和變異數。 (b)計算直徑超出5.1毫米的機率。4 - 535. The thickness of a flange on an aircraft component is uniformly distribution between 0.95 and 1.05 millimeters.航空器中耳輪緣的厚度均勻分布在0.95和1.05毫米中。(a) Determine the cumulative distribution function of flange thickness.(b) Determine the proportion of flanges that exceeds 1.02 millimeters.(c) What thickness is exceeded by 90% of the flanges?(d) Determine the mean and variable of flange thickness.36.Suppose the time it takes a data collection operator to fill out an electronic from for a database is uniformly between 1.5 and 2.2 minutes.(操作員填好表格時間均勻分布在1.5和2.2分鐘)(a) What is the mean and variance of the time it takes an operator to fill out the form?(b) What is the probability that it will take less than two minutes to fill out the form?(c) Determine the cumulative distribution function of the time it takes to fill out the form.38.An adult can lose or gain two pounds of water in the course of a day. Assume that the change in water weight is uniformly distributed between minus two and plus two pounds in a day. What is the standard deviation of you weight over a day? 一位成人在一天之內失去二磅的水。 假設水重量上符從均勻分配在負二和正二之間。 你一天重量的標準差？39.A dolphin show is scheduled to start at 9:00 A.M., 9:30 A.M., and 10:00A.M. Once the show starts, the gate will be closed. A visitor will arrive at the gate at a time uniformly distributed between 8:30 A.M. and 10:00 A.M. Determine(海豚秀預定開始時間在上午9:00， 上午9:30和上午10:00開始,秀開始前門是關著。 訪客每次到達時間均勻分佈在上午8:30之間至上午10:00之間)(a) The cumulative distribution function of the time (in minutes) between arrival and 8:30A.M.(b) The mean and variance of the distribution in the previous part.(c) The probability that a visitor waits less than 10 minutes for a show.(d) The probability that a visitor waits more than 20 minutes for a show.4 - 6 41.Use Appendix Table III to determine the following probabilities for the standard normal random variable Z:(a) (b) (c) (d) (e) .44. Assumehas a standard normal distribution. Use Appendix Table III to determine the value for z that solves each of the following:(a) (b) (c) (d) 45.Assumeis normal distributed with a mean of 10 and a standard deviation of 2. Determine the following:(a) (b) (c) (d) (e) .47.Assumeis normal distributed with a mean of 5 and a standard deviation of 4. Determine the following:(a) (b) (c) (d) (e) .49.The compressive strength of samples of cement(水泥樣品的耐壓強度) can be modeled by a normal distribution with a mean of 6000 kilograms per square centimeter and standard deviation of 100 kilograms per square centimeter.(a) What is the probability that samples strength is less(低於) that 6250?(b) What is the probability that samples strength is between 5800 and 5900?(c) What strength is exceeded(超過) by 95% of the samples?51.An article in Knee Surgery Sport Traumatol Arthrosc, “ Effect of provider volume on resource utilization for surgical procedures,” (2005, Vol. 13, pp. 273-279) showed a mean time of 129 minutes and a standard deviation of 14 minutes for ACL reconstruction surgery at high- volume hospitals (with more than 300 such surgeries per year).在膝蓋手術體育Traumatol Arthrosc的一篇文章， 提供者容量對外科手術時間的影響，顯示129分鐘的平均時間和14分鐘的標準差在ACL重建手術的高量醫院(超過每年300次這樣手術)。(a) What is the probability that your ACL surgery at a high-volume hospital requires a time more than two standard deviation above the mean?(超過2倍標準差)(b) What is the probability that your ACL surgery at a high-volume hospital is completed in less than(低於) 100 minutes?(c) The probability of a completed ACL surgery at a high-volume hospital is equal to(等於) 95% at what time?(d) If your surgery requires 199 minutes, what do you conclude about the volume of such surgeries at your hospital? Explain.如果你的手術要199分鐘，在你的醫院你如何斷定關於這樣的手術的量？53.The line width for semiconductor manufacturing(半導體的線寬度製造) is assumed to be normally distributed with a mean of 0.5 micrometer and a standard deviation of 0.05 micrometer.(a) What is the probability that a line width is greater than(大於) 0.62 micrometer?(b) What is the probability that a line width is between 0.47 and 0.63 micrometer?(c) The line width of 90% of samples is below(低於)what value?54.The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a standard deviation of 0.1 fluid ounce.碳酸化合飲料的填裝罐頭使用一臺自動化的填充機填充容量符從常態分配平均數12.4標準差0.1(a) What is the probability a fill volume(填充容量) is less than(低於) 12 fluid ounces?(b) If all cans less than 12.1 or greater than 12.6 ounces are scrapped(廢棄), what proportion of cans is scrapped?(罐頭被作廢的比例)(c) Determine specifications that are symmetric(對稱) about the mean that include 99%of all cans.55.In the pervious exercise, suppose that the mean of the filling operation(填裝操作) can be adjusted easily, but the standard deviation remains at 0.1 ounce.(a) At what value should the mean be set so that 99.9% of all cans exceed 12 ounces?(b) At what value should the mean be set so that 99.9% of all cans exceed 12 ounces if the standard deviation can be reduced to 0.05 fluid ounce?59.In an accelerator center, an experiment needs a 1.41 cm thick aluminum cylinder (/mumu/target/Solenoid _Coil.pdf). Suppose that the thickness of a cylinder has a normal distribution with a mean 1.41 cm and a standard deviation of 0.01 cm.在加速器中心，實驗需要1.41 cm厚實的鋁圓筒。 假設圓筒的厚度符從常態分佈平均數1.41 cm和標準差0.01 cm。(a) What is the probability that a thickness is greater than 1.42 cm?(b) What thickness is exceeds by 95% of the samples?(c) If the specifications require that the thickness is between 1.39 cm and 1.43 cm, what proportion of the samples meet specifications(圓筒符合規格的機率)?61.The life of semiconductor laser at a constant power(在固定功率下半導體雷射的生存時間) is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours.(a) What is the probability that a laser fails before 5000 hours?(在5000小時前失效)(b) What is the life in hours that 95% of the lasers exceed?(c) If three lasers are used in a product and they are assumed to fail independently, what is the probability that all three are still operating(操作) after 7000 hours?62.The diameter of the dot produced by a printer(點的直徑由印表機印出) is normally distributed with a mean diameter of 0.002 inch and a standard deviation diameter of 0.0004 inch.(a) What is the probability that the diameter of a dot exceeds(超過) 0.0026 inch?(b) What is the probability that a diameter is between 0.0014 and 0.0026 inch?(c) What standard deviation of diameter is needed so that the probability in part(b) is 0.995?64.Measurement error that is normally distributed with a mean zero and a standard deviation of 0.5 gram is added to the true weight of a sample. Then the measurement is rounded to the nearest gram. Suppose that the true weight of a sample is 165.5 grams.(然後測量環繞到最近的克,假設樣品的真實重量是165.5克)(a) What is the probability that the rounded result is 1.67 grams?(真實值在167附近(多或少)的機率？)(b) What is the probability that the rounded result is 1.67 grams or greater? (真實值超過167的機率？)4 - 7 65.Suppose thatis a binomial random variable with(a) Approximate the probability thatis less than or equal to 70.(b) Approximate the probability thatis greater than 70 and less than 90.(c) Approximate the probability.66.Suppose thatis a Poisson random variable with(a) Compute the exact(精確的) probability is less than 4.(b) Approximate the probability thatis less than 4 and compare to the result in part(a).(c) Approximate the probability that.67.Suppose thatis a Poisson distribution with a mean of 64. Approximate the probabilities:(a) (b) (c).69.There were 49.7 million people with some type of long-lasting condition or density living in the United States in 2000. This represented 19.3 percent of the majority(多數) of civilians aged five and over . A sample of 1000 persons is selected at random.(a) Approximate the probability that more than 200 persons in the sample have a disability.(無能力)(b) Approximate the probability that between 180 and 300 people in the sample have a disability.70.Phoenix water is provided to approximately 1.4 million people, who are served through more than 362,000 accounts. All accounts are metered and billed monthly. The probability that an account has error in a month is 0.001, and accounts can be assumed to be independent.鳳凰城的水提供給大约140萬人民，通過超過362,000個帳戶服務。所有帳戶測量和每月開帳單。每個月帳戶有錯誤的機率是0.001,並且帳戶之間是獨立的。(a) What is the mean and standard deviation of the number of account errors each month?(b) Approximate the probability of fewer than 350 errors in a month.(c) Approximate a value so that the probability that the number of errors exceeding this value is 0.05.(超過多少值的機率為0.05)(d) Approximate the probability that more than 400 errors per month in the next two months. Assume that results between months are independent.73.Suppose that the number of asbestos particles in a sample of 1 squared centimeter of dust(在一平方公分的塵土中,石棉微粒的顆數)is a Poisson random variable with mean of 1000. What us the probability that 10 squared centimeter of dust contains more than 10,000 particles?75.Hits to a high-volume Web site(命中到一個大容積網站) are assumed to follow a Poisson distribution with a mean of 10,000 per day. Approximate each of the following:(a) The probability of more than 20,000 hits in a day.(b) The probability of less than 9900 hits in a day.(c) The value such that the probability that the number of hits in a day exceed the value is 0.01.(超過多少值的機率為0.01)(d) Approximate the expected number of days in a year (365 days) that exceed 10,200 hits.(e) Approximate the probability that over a year (365days) more than 15 days each have more than 10,200 hits.4 - 8 77.Supposehas an exponential distribution with mean equal to 10. Determine the following:(a) (b) (c) (d) Find the value ofsuch that .78.Supposehas an exponential distribution with a mean of 10. Determine the following:(a) (b) (c) Compare the results in part(a) and (b) and comment on the role of the lack of memory property. 79.Suppose the counts recorded by a Geiger counter on a Poisson process with an average of two counts per minute. (假設在Poisson過程用蓋格計數器計數記錄,平均每分鐘2次)(a) What is the probability that there are no counts in a 30-second interval?(b) What is the probability that the first count occurs in less than 10 seconds?(c) What is the probability that the first count occurs between 1 and 2 minutes after start-up?80.Suppose that the log-ons to a computer network follow a Poisson process with an average of 3 counts per minute.(假設在Poisson過程用電腦網路註冊,平均每分鐘3次)(a) What is the mean time between counts?(b) What is the standard deviation of the time between counts?(c) Determinesuch that the probability that at least one count occurs before timeminute is 0.95.82.The life of automobile voltage regulators(汽車電壓調節器的壽命) has an exponential distribution with a mean life of six years. You purchase an automobile that is six years old, with a working voltage regulator, and plan to own if for six years.(a) What is the probability that the voltage regulator fails during your ownership(所有權)?(b) If your regulator fails after you own the automobile three years and it is replaced, what is the mean time until the next failure? (如果你擁有汽車3年後,你的調節器故障並且替換它,直到下一個故障的平均時間？)83.Suppose that time to failure (in hours) of fans in a personal computer(個人電腦風扇失效的時間) can be modeled by an exponential distribution with(a) What proportion of the fans will last at least 10,000 hours?(b) What proportion of the fans will last at most 7000 hours?85.The time between arrivals of taxis at the busy intersection is exponentially distributed with a mean of 10 minutes.(在繁忙的十字路口,計程車到達前來的時間)(a) What is the probability that you wait longer than one hour for a taxi?(b) Suppose you have already been waiting for one hour for a taxi, what is the probability that one arrival within the next 10 minutes?(假設你已經用一小時等到一台車,下一台車是在10分鐘後的機率？)(c) Determinesuch that the probability that you wait more thanminutes is 0.10.(d) Determinesuch that the probability that you wait less thanminutes is 0.90.(e) Determinesuch that the probability that you wait less thanminutes is 0.50.86.The number of stock sightings on a route in South Carolina follows a Poisson process with a mean of 2.3 per year.(正要上市股票的數量符從Poisson過程)(a) What is the mean time between sightings?(b) What is the probability that there are no sightings within three months (0.25 years)? (3個月內沒有上市股票的機率嗎)(c) What is the probability that the time until the first sighting exceeds six months?(d) What is the probability that no sightings within three years?88.The distance between major cracks in a highway follows an exponential distribution with a mean of 5 miles.(一條高速公路中大裂縫的距離)(a) What is the probability that there are no major cracks in a 10-mile stretch of the highway?(b) What is the probability that there are two major cracks in a 10-mile stretch of the highway?(c) What is the standard deviation of the distance between major cracks?(d) What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection?(e) What is the probability that there are no major cracks in two separate 5-mile stretches of the highway?(f) Given that there no cracks in the first 5 miles inspected, what is the probability that there are no major cracks in the next 10 miles inspected?89.The lifetime of mechanical assembly in a vibration test is exponentially distributed with a mean of 400 h