Problem: Fitting distributions to failure time data
Consider the failure time data for the asset below which was observed from installation up to time 94.1 hours.
|
Failure limes (hours)
|
Censored?
|
|
68.4
|
|
|
73.7
|
|
|
40
|
|
|
37.2
|
|
|
94.1
|
|
|
63.9
|
|
|
67.8
|
|
|
50.5
|
|
|
45.1
|
|
|
41
|
|
|
84.6
|
|
|
47.7
|
|
|
49.8
|
|
|
94.1
|
|
|
94.1
|
|
|
63.1
|
|
|
16.4
|
|
|
77.7
|
|
|
55.7
|
|
|
90.9
|
|
Do the following:
i) Fill in the column labelled "Censored?", which is true of the asset has survived until the end of the observation period and false if the asset has failed during the observation period.
ii) Construct an empirical CDF for each asset. Please carry out the (simple) computations by hand and use the Product-Limit (Kaplan-Meier) estimator plotting positions. Please plot (in Excel or MATLAB) the empirical CDF using the values you computed by hand, as well as the upper and lower confidence interval boundaries.
iii) Construct a Weibull Probability Plot for the asset using Excel or MATLAB. Comment on what you expect to see if a Weibull is a good fit for this data.
iv) Estimate q and 13 by using linear rectification and add the fitted Weibull CDF to the same plot as the empirical CDF. Comment on the quality of the fit.
v) Use the parameters you found in part d):
a. Find an expression in t for the hazard of the system and plot it from 0 < t 5 100. Is the asset more likely to fail as it ages? How do you know?
b. Find the mean time to failure
c. If the asset has survived up to time to = 20 hours, find the conditional reliability and use this to find the probability that the asset will survive another 10 hours.