Homework Log
Stat 567:Statistical Reliability
All homework is due on the Monday following the Lecture
date on which it is assigned unless otherwise specified. Make sure to identify
all homework with the Lecture number (ordinal).
To avoid confusion homework from different classes should
be handed in distinct packages. All pages of your homework should have your
name and Lecture number. Pages within a Lecture homework package should
be numbered sequentially.
First Lecture (August 21, 1996)
Second Lecture (August 26, 1996)
- Show that the cumulative hazard function of an increasing
hazard (failure) rate (IFR) distribution is convex (i.e., holds water).
- Show that if T in a positive random variable with an
exponential distribution then:
- Show that if T is a nonnegative random variable then
where f(t) is the density function and S(t) is the survival
function.
- Show that the Weibull distribution is:
- IFR if the shape parameter is greater than 1
- DFR if the shape parameter is less than 1
- Exponential if the shape parameter is equal to 1
- Take a look at Military
Reliability Standards and Handbooks and RAC
Engineering Services: Reliability Prediction to learn about MIL-STD-217
and other military reliability standards. (Thanks to Phillip Yates for
finding the first link.)
- Check the web site: Weibull
Data for Parts
Third Lecture (August 28, 1996)
- Read Pages 12 to 37 from the text. Send me e-mail on
what you found to be the less clear point in these pages.
- Send me e-mail on what you found to be the less clear
point of my lectures so far.
Fourth Lecture (September 4, 1996)
- For the ball bearing data of Example 2.1 on Page 37 of
your textbook, obtain 95% confidence intervals for S(50) and H(50).
- For the the ball bearing data of Example 2.1 on Page
37 of your textbook plot the empirical survivor function using the plotting
points derived on Page 40.
- Contribute a new study question for one of the
first four units of this Lecture and e-mail it to me.
Fifth Lecture (September 9, 1996)
- Take a look at the homepage of the course Graphical
Data Analysis out of the University of Montana
- Use JMP to do Weibull and lognormal plots of the ball
bearings data of Example 2.1 on Page 37 of your textbook.
You can download the JMP file containing these data at the bottom of the
Lecture summary.
- Identify the standard survivor function G that leads
to the Gumbel and normal QQ plot.
Sixth Lecture (September 11, 1996)
- Let A1, A2, A3, and A4
be nested events with A1 being the smallest and A4
being the largest. Show that
P(A1)=P(A1|A2)P(A2|A3)P(A3|A4)P(A4)
- Analyze the rat diet data supplied in Lecture. Write
a professional quality report. Attach a cover
letter directed at management. The report itself should be directed at
the scientists who are not statisticians. The report should have:
- Summary and Conclusions
- Data Analysis
- Appendix with the raw data and other things that interfere
with the smooth flow of the Data Analysis section of the report. (Due
September 23, 1996)
Seventh Lecture (September 16, 1996)
- In the analysis below we determine the effect on reliability
of removing Cause 9 failures. Do a similar analysis with the Cause 6 failures.
Notice that you can download a JMP file with the data by going to the end
of this page.
- Write a professional quality
report documenting the data analysis below and your analysis of the
homework above. As usual, your report should include a cover letter to
management. (Due October 7, 1996.)
- Write a proposal for the final project. What experiment
are you going to do? Who are you going work with? (Due September 30,
1996.)
Eighth Lecture (September 18, 1996)
Ninth Lecture (September 23, 1996)
- Verify that the Weibull MLEs of the ball bearing data
presented in Example 3.2, Page 57 of your textbook
satisfy Equations (3.8) and (3.9) on Page 55.
- Verify the variance-covariance matrix of the Weibull
MLE estimates of the ball bearing data presented in Example 3.2, Page 57
of your textbook satisfy the equations on
top of Page 56.
- Submit two new study questions for this or a prior lesson.
Remember that these questions could end up in the exams
Tenth Lecture (September 25, 1996)
- Verify the calculations leading to the numerical values
of estimates and standard errors on Table 3.2, Page 58 of your textbook.
- Estimate the 5% percentile and its standard error for
the data used in Table 3.2, Page 58. Assume a Weibull distribution and
the definition of percentiles used in the textbook (based on the survivor
function rather than on the cumulative distribution function).
- Assuming a Weibull model for the data of Example 2.1
and 3.2 (and of Table 3.2), find the MLE of S(60) and its standard
error. Compare your results with those you get without the Weibull assumption
using the empirical survivor function.
Eleventh Lecture (September 30, 1996)
- Read the material in the notes. The likelihood ratio
method will open a whole new world of statistical application outside the
traditional linear model and normal distribution framework. You need to
understand the logic behind the likelihood ratio method to be able to use
this method correctly.
Twelfth Lecture (October 2, 1996)
- Read the notes and examples carefully. The midterm exam
is fast approaching and you need the results of this sections to understand
most of the rest of this course and many other areas of statistical application.
- Review the instruction on writing
Lecture reports. Then read carefully Anne Freeman's
rat study report handed in Lecture. Try to understand what made hers a
professional quality report but also look for areas where it can be improved.
Put her report aside and don't look at it anymore. Reanalyze the rat study
data and rewrite your rat study report. I will scan your report to see
if it is too similar to Anne's report. If it is - it will affect your
grade very negatively. Remember that the goal is for you to learn to
write a data analysis report as a professional statistician would. And
for you to have something to show as an example of your writing to a prospective
employers. Hint: A good report is like good short story - full of substance
and with nothing extraneous. This assignment is due on Monday, October
14.
Thirteenth Lecture (October 9, 1996)
- Use the SAS JMP software to obtain the PP and SP plots
for the cord data of Example 2.3 on Page 46 of your
textbook. Hand these plots in. Use the JMP screen shots shown in this class's summary as a guide to obtaining
these plots.
- At the end of Chapter 3 in you textbook the confidence
interval for S(53) obtained by way of the PL estimator (0.54, 0.83)
is compared to the confidence interval for S(53) obtained using
a Weibull fit and the Delta method (0.58, 0.76). Show the calculations
leading to these confidence intervals. Which interval do you prefer and
why?
Fourteenth Lecture (October 11, 1996)
- Use the Nonlinear Fit platform of JMP to
obtain the MLEs for mu and sigma of a lognormal model
applied to the:
- Data on the number of revolutions to failure for each
of 23 ball bearings. (Example 2.1, Page 37 of your
textbook)
- Data on the strengths of 48 pieces of weathered braided
cord. (Example 2.3, Page 46 of your textbook)
Recall that you can download the JMP files for these data
sets if you go at the end of this Lecture summary.
This homework is due on October 23.
Fifteenth Lecture (October 14, 1996)
Sixteenth Lecture (October 21, 1996)
- Use JMP's Nonlinear Fit platform to fit the model:
to the following data:
Optain likelihood ratio confidence intervals for this
parameters.
Seventeenth Lecture (October 23, 1996)
Review Lecture (October 28, 1996)
- Rewrite the appliance report for a new grade.
- If you want to, you can also rewrite the rat data report
- yes again! I will change the grade of the report you rewrote.
Eighteen Lecture (October 30, 1996)
Nineteenth Lecture (November 4, 1996)
- Read my notes carefully because it is going to get harder
soon
Twentieth Lecture (November 6, 1996)
Twenty First Lecture (November 11, 1996)
- Read my notes carefully. The theory is necessary to understand
generalized residuals and their use
Twenty Second Lecture (November 13, 1996)
- Review all the examples in which I have used JMP's Nonlinear
Fit platform
Twenty Third Lecture (November 18, 1996)
Twenty Fourth Lecture (November 20, 1996)
- Read the material in the notes
Twenty fifth Lecture (November 25, 1996)
- Read the Lecture notes and the web guide for this Lecture.
Make sure you understand how I have used JMP since I have used it in an
unconventional way to obtain the values of maximized likelihoods.
Twenty Six Lecture (November 27, 1996)
Twenty Seventh Lecture (December 2, 1996)
- Use Markov chain methods to calculate the availability
of a system with independent, identically distributed (i.i.d.) exponential
up times and i.i.d. exponential down times. Assume the up and down times
are independent.
- Draw the state diagram for a three component system with
one repair person. Derive the steady state equations. Assume independent
exponential life and repair times.
Twenty Eighth Lecture (December 4, 1996)
Do you have something to
tell me?
