Chapter 09 Exercises collection

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Chapter 9 Estimation of reliability parameters from experimental data 9.1 Failure times The failure time data (5.2, 6.8, 11.2, 16.8, 17.8, 19.6, 23.4, 25.4, 32.0, 44.8 minutes) are exponentially distributed as 𝐹𝑋(𝑥) = 1 − 𝑒−𝑥 𝜃 Make a probability plot and estimate the parameter, θ. 9.2 Catalytic converter test Twenty units of a catalytic converter are tested to failure without censoring. The times to failure (in days) are the following Tab1e 9.1. Times of fai1ures 2.6 3.2 3.4 3.9 5.6 7.1 8.4 8.8 8.9 9.5 9.8 11.3 11.8 11.9 12.7 12.3 16.0 21.9 22.4 24.2 l. Plot on exponential paper and determine whether the failure rate is increasing or decreasing with time. 2. Plot the results on Weibull paper and estimate its parameters. 3. Find the method-of-moments estimates of the Weibull parameters. 9.3 Confidence bounds Suppose that the time to failure T (years) of a certain item is an exponential random variable with probability density function: ℎ(𝑡) = 𝜆𝑒−𝜆𝑡 , 𝑡 > 0 l. If we have a sample of 3 observations on T, i.e. E = {t1 = l, t2 = 2.8, t3 = 2.2} , find the 95% upper confidence bound and the 90% confidence interval for A using frequentist statistics. 2. Find the 95% upper confidence bound and the 90% confidence interval for A using frequentist statistics and using the information in l. What are the corresponding Bayesian quantities? 9.4 Remission times Suppose that the remission time, in weeks, of leukaemia patients that have undergone a certain type of chemotherapy treatment is an exponential random variable having an unknown mean θ. A group of twenty such patients are being monitored and, at present, their remission times are (in weeks) 1.2, 1.8* ' 2.2, 4.1, 5.6, 8.4, 11.8*, 13.4*, 16.2, 21.7, 29*, 41, 42*, 42.4*, 49.3, 60.5, 61*, 94, 98, 99.2*, where an * next to the data…