5. A pressure transducer regulates a climate control system in a factory. Thetransducer fails according to an exponential distribution with rate one failureevery five years on average.a. What is the cumulative distribution function of the time until failure?b. What is the probability that a transducer chosen at random functions for eightyears without failure?754 Chapter Thirteen Reliability and Maintainabilityc. What is the probability that a transducer that has functioned for eight years con-tinues to function for another eight years?6. For the pressure transducer mentioned in Problem 5, use the failure rate functionto estimate the likelihood that a transducer that has been operating for six yearsfails in the seventh year. How close is the approximation to the exact answer?

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Answered: 5. A pressure transducer regulates a…

A First Course in Probability (10th Edition)

10th Edition

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Author:Sheldon Ross

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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.A new, miracle diabetes drug that diminishes major symptoms of diabetes has been approved by the FDA. Health care professionals and researchers believe that the new drug will prolong lifespan of diabetes patients. If the population is in steady state and the incidence is constant, what will the effect of this new drug be on the prevalence of diabetes in the population? Explain.
1. ABC inc. stock is currently selling for $30, one year from today the stock price can either increase by 20% or decrease by 15%. The probability of an increase in the stock price is equal to 0.3. The one-year risk-free rate is 5% What is the value of a European put that expires in one year with an exercise price of $24. 2. Graphically, show the value and the profit and loss of the following butterfly position: Long in a call with an exercise price of $30, short in 2 calls with an exercise price of $45, and long in a call with an exercise price of 60. All calls are written on the same stock and have the same maturity. 3. "Early exercise of an American option on a stock that does not pay any dividend is not optimal regardless of whether the option is a Call or a Put". True, False, or Uncertain. Explain.
Product failure behavior. An article in Hotwire (Dec. 2002) discussed the length of time till failure of a product pro- duced at Hewlett-Packard. At the end of the product’s life- time, the time till failure is modeled using an exponential distribution with mean 500 thousand hours. In reliability jargon, this is known as the “wear-out” distribution for the product. During its normal (useful) life, assume the prod- uct’s time till failure is uniformly distributed over the range 100 thousand to 1 million hours. a. At the end of the product’s lifetime, find the probability that the product fails before 700 thousand hours. b. During its normal (useful) life, find the probability that the product fails before 700 thousand hours. c. Show that the probability of the product failing before 830 thousand hours is approximately the same for both the normal (useful) life distribution and the wear-out distribution
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4. When the Titanic sank in 1912, there were over 1,300 passengers aboard. A random sample of 50 passengers that were aboard is taken. The amount they paid for the voyage and whether or not they survived is recorded. A model is fit to predict the probability they survived when the ship sank is obtained. Survived Logistic Fit of Survived By Fare 1.00 0.75 0.50- 0.25- 50 4 Parameter Estimates 100 Term Intercept -1.404254 0.4353289 Fare 0.01246686 0.0097945 For log odds of Yes/No Fare Estimate Std Error ChiSquare 10.41 1.62 150 200 Prob> ChiSq 0.0013 0.2031 No Yes a) Suppose someone paid a fare of 25 pounds. Predict the probability they survived. b) How much would someone pay to have at least a 65% chance of survival?
2 The operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 100 cfs (cubic feet per second). (a) Find the probability that the demand will exceed 250 cfs during the early afternoon on a randomly selected day. (Round your answer to four decimal places.) (b) What water-pumping capacity, in cubic feet per second, should the station maintain during early afternoons so that the probability that demand will exceed capacity on a randomly selected day is only 0.02? (Round your answer to two decimal places.) cfs
5. A pressure transducer regulates a climate control system in a factory. The transducer fails according to an exponential distribution with rate one failure every five years on average. a. What is the cumulative distribution function of the time until failure? b. What is the probability that a transducer chosen at random functions for eight years without failure? irteen Reliability and Maintainability c. What is the probability that a transducer that has functioned for eight years con- tinues to function for another eight years?
KOW 1 2 3 4 5 6 Decade Count 1851 1860 1861 1870 1871 1880 1881-1890 1891 - 1900 1901-1910 NO 6 1 7 5 00 8 4 1911-1920 8 1921-1930 5 LO 9 1931 1940 00 8 10 1941 - 1950 10 11 1951-1960 12 1961-1970 13 1971-1980 14 1981 1990. 4 15 1991-2000 16 2001 - 2010 9 17 2011-2020 саоттоа со 6 6 18 19 20 21 22 61° L var3 var4
Еxample 4 The daily discharge of a river can be modeled using log-normal distribution with a mean of 1300 m/sec with a standard deviation of 500 m/sec. a. What is the probability that the discharge shall exceed 2000 m³/sec. b. What is the probability that the discharge shall exceed 1300 m³/sec. c. What is the conditional probability that the discharge is less than 2000 m/sec given it is more than the mean 1300 m/sec. d. If a bank is to be made across the river to avoid flooding with the design daily discharge that shall be exceeded once in 10 years. Calculate the design daily discharge
Define on empirical way and a statistical way to choose the best forecasting model for your time series data. Consider the following situations: 1. The data has no trend or seasonality 2. The data has trend but not seasonality 3. The data presents trend and seasonality but negligible noise 4. The data presents trend, seasonality and significant noise.
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