The time between failures for an electrical appliance is exponentially distributed with a mean of 25 months. What is the probability that the next failure will not occur before 30 months have elapsed?
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- 7. Assume that there is a 9% rate of disk drive failure in a year. a. If all your computer data is stored on a hard disk drive with a copy stored on a second hard disk drive, what is the probability that during a year, you can avoid catastrophe with at least one working drive? b. If copies of all your computer data are stored on three independent hard disk drives, what is the probability that during a year, you can avoid catastrophe with at least one working drive? a. With two hard disk drives, the probability that catastrophe can be avoided is nothing. (Round to four decimal places as needed.)1. Three hundred identical cathode ray tubes (CRTS) placed into service simultaneously on January 1, 1976, experienced the following numbers of failures through December 31, 1988: 1983 Year Number of failures 1984 1985 1986 1987 1988 13 19 16 34 21 38 Assume that there were no failures before 1983. a. Based on these data, estimate the cumulative distribution function (CDF) of a CRT chosen at random. Using the results of part (a), estimate the probability that a CRT chosen at random 13.1 Reliability of a Single Component 753 b. Lasts more than 5 years. c. Lasts more than 10 years. d. Lasts more than 12 years. e. That has survived for 10 years fails in the 11th year of operation.7. A piece of equipment has a lifetime T (measured in years) that is a continuous ran- dom variable with cumulative distribution function F(t) = 1 – e/10 – (1/10) e¯/10 for all t> 0. a. What is the probability density function of T? b. What is the probability that a piece of equipment survives more than 20 years? c. What is the probability that a piece of equipment survives more than 10 years but fewer than 20 years? d. What is the probability that a piece of equipment survives more than 20 years given that it has survived for 10 years? 8. For the equipment mentioned in Problem 7, a. Derive the failure rate function r(1), and draw a graph of the function. b. Without using the failure rate function, determine the probability that a piece of equipment that has survived 20 years of operation fails in the 21st year. c. Does r(20) accurately estimate your answer to part (b)? Why or why not?
- 2. Suppose that the cumulative distribution function of the lifetime of a piece of operating equipment is given by F(t) = 1 – e-0.61 – 0.6te¬0.61 where t is measured in years of continuous operation. a. Determine the reliability function. b. Determine the failure rate function. c. What is the probability that this piece of equipment fails in the first year of operation? d. What is the probability that this piece of equipment fails in the fifth year of operation? e. What proportion of the equipment surviving four years fails in the fifth year? (Calculate without using the failure rate function.) f. Does r(4) closely approximate the answer to part (e)? Why or why not? g. What proportion of the equipment surviving four years fails in the first month of the fifth year? (Calculate using the failure rate function.)A product is composed of four parts. In order for the product to function properly in a given situation, each of the parts must function. Two of the parts have a .96 probability of functioning, and twohave a probability of .99. What is the overall probability that the product will function properly?A typical retail transaction consists of several smaller steps, which can be considered components subject to failure. A list of such components might include: Let the eight probabilities of success be .92, .94, .99, .99, .98, .97,.95, and .96. What is the reliability of the system; that is, the probabilitythat there will be a satisfied customer? If you were the storemanager, what do you think should be an acceptable value forthis probability? Which components would be good candidatesfor backup, which for redesign?
- Lily believed that a prospective insured's building might be considered a highly protected risk. Which one of the following buildings is most likely to be considered a highly protected risk? A small building located in the middle of a very rural area A masonry noncombustible industrial complex that relies on its own water supply An office building without any public fire protection A large fire-resistive building with automatic sprinkler systemsThe number of yearly breakdowns for the power generator observed over the past 20 years at Orlando Utilities is presented in the following data: Number of breakdowns 0 1 2 3 4 5 6 Number of years in which breakdowns occurred 3 3 5 3 4 1 1 For Orlando Utilities, the expected number of breakdowns = (round your response to two decimal places).A product is composed of four parts. In order for the product to function properly in a given situation, each of the parts must function. Two of the parts have a 0.96 probability of functioning, and two have a probability of 0.99. (Leave your answers in 4 decimal places).
- The International Organization for Standardization is the first globally recognized standard related to risk management: a) True b) Falsea factory that produces a small number of items per day. In about 30% of working days the factory produces 7 units of the product, in 45% of working days it produces 8 units, and the rest of working days it produces 9 units. After production is completed, each unit is thoroughly inspected. Each unit fails inspection with probability ??. If two or more units fail inspection on the same day, the factory closes for a week to re-calibrate equipment. Say the factory opens today after being closed for a week, write a mathematical expression to calculate the probability that the factory will remain open at least 30 days before closing again.The company "Best Manufacturers Ltd" is applying a typical Failure Mode and Effect Analysis. For a specific failure mode, the value for "Severity" is equal to 8, the value for "Occurrence" is equal to 2 and the value for "Detection" is equal to 6, respectively. If the policy of the company is to take a further action for all failure modes with a Risk Priority Number (RPN) greater than or equal to 96, what should the company do with the aforesaid failure mode: Take NO further action because the RPN is NOT above the threshold imposed by the company. Take further action because the value for Severity is life-threatening. Take further action because the value for Occurrence is life-threatening. Take further action because the value for Detection is life-threatening. Take further action because the value for RPN is above the threshold imposed by the company. Revisit the specific failure mode and re-calculate the RPN.