Question

5.7 Seven pumps have failure times (in months) of 15.1, 10.7, 8.8, 11.3, 12.6, 14.4 and 8.7 Assume an exponential distribution. Assume the dataset is complete and all units have been operated to failure a) Find a point estimate of the MTTF b) Estimate the reliability of a pump for t 12 months c) Calculate the 95% two-sided confidence interval for λ, using the X2 distribution. Note: s,if the distribution is exponential, the 2T 2T distribution of the MTTF can be approximated by the Chi-squared distribution, where r is the number of failures and T is the accumulated hours in a functioning iaie A radar system uses 650 similar electronic devices. Each device has a failure rate of 0.00015 per month. If all these devices operate independently, what is the probability that there are no failures in a given year? 5.8

Answer 1

5.7)

a) The estimated MTTF is

b) Here $\lambda =1/MTTF=0.0858$ .

The reliability after 12 months is $R\left ( 12 \right )=e^{-12\times 0.0858}={\color{Blue} 0.3571}$ .

c) The accumulated hoiurs is $T=7\times 11.657=81.6$ .

The $\left ( 1-\alpha \right )100\%$ a sided confidence interval for $\lambda$ is

$\frac{\chi ^2_{\alpha /2}\left ( 2r \right )}{2T}\leqslant \lambda \leqslant \frac{\chi ^2_{1-\alpha /2}\left ( 2r \right )}{2T}$

Use R to calculate the CI. Here $\alpha =0.05,r=7$ .

qchisq(0.025,14)/2/81.6
qchisq(1-0.025,14)/2/81.6

${\color{Blue} 0.034\leqslant \lambda \leqslant 0.16}$

5.8) The probability of no failure (reliability)of a device is

$R\left ( 12 \right )=e^{-12\times 0.00015}=0.9982$ .

The probability of no failure in 650 devices is $0.9982^{650}={\color{Blue} 0.3104}$