Statistics 533
Spring 2014
Final Examination
Name
Exam Instructions
When asked to explain something, provide an explanation that could be understood by someone who
does not have formal training in statistical methods. Your explanations should be clear and concise.
Generally, you must show all of your work in order to get partial credit.
When you are asked to derive an expression you must show all steps of your derivation.
Do not use your calculator until you have otherwise completed the entire exam. When you are
asked to compute a quantity you do not need to do all of the computations to a final answer. Just
show that you know how to do the problem. I am not interested in how well you can use your
calculator.
Students may use two sheets (8.5 by 11 inches, both sides) of paper containing equations or notes.
Students may have up to 120 minutes (2 hours) to complete the exam.
There are 23 questions or question parts in the exam. Students must choose and mark three
(3) question parts as “Do Not Grade.”
As soon as possible after the exam is completed, it should be scanned by the proctor and returned
to the ISU Testing Center, by one of the suggested methods (upload, email or FAX).
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1. Failure is caused by a simple one-step chemical reaction, the rate of which can be described
by the Arrhenius relationship. Suppose that time to failure T ∼ LOGNOR(µ, σ) at an given
level of temperature.
(a) Derive an expression for the time acceleration factor relating life at two different levels
of temperature.
(b) Show why it is that the simple linear regression relating log life and reciprocal Kelvin
temperature, implied by the Arrhenius rate reaction model, will have constant σ (i.e., σ
will not depend on temperature).
2. The manufacturer of computer hard disks reports in its promotional literature a “MTBF”
figure. This figure is obtained by taking a sample of units from each days production, putting
the units on test for a period tc (perhaps one week), the pooling the available data over many
months, and computing
TTT
MTBF =
r
where TTT is the total time on test of all units that were tested and r is the observed number
of failures (typically a very small number of drives). Reported figures are typically numbers
like MTBF=100 years. Comment on the usefulness and validity of the use of this figure for
characterizing the reliability of disk drives.
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3. Consider the following block system diagram.
R3
R1
R5
R2
R6
R4
Derive an expression for the reliability of the system as a function of the component reliabilities R1 , R2 , . . . , R6 , assuming independence of the failure times of the individual components.
4. Give an example of a system in which one might expect component life times to be positively
dependent. Explain why the component lifetimes would be dependent.
b where θb is the ML estimator of a scalar pa5. Suppose that you have an expression for Var(θ),
rameter θ. Use
p the delta method to obtain an expression for the approximate (large-sample)
b
variance of θ.
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6. One important reason to use Bayesian methods is to allow the formal incorporation of engineers’ prior information into an analysis, providing better precision. What is another important reason?
7. Maximum likelihood is the standard method of fitting a probability distribution to censored
data. An alternative that has been suggested is to use ordinary least squares to fit a line to
the points on a probability plot. Give two reasons why this is a bad thing to do, in general.
8. A sample of n units was put into operation at time 0. The failure time of these units can be
described by an exponential cdf
t
Pr(T ≤ t; θ) = F (t; θ) = 1 − exp −
, t > 0.
θ
The units will be monitored until a specified number of r (0 < r < n) of failures have been
observed.
(a) Write down an expression for the log likelihood of the resulting data.
(b) Write down an expression for the first derivative of the log likelihood in part 8a with
respect to θ.
(c) Derive an expression for the ML estimator of θ.
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9. The following table gives data on reported complaints about the operation of a particular
product for units that were all sold on a particular day (day 1) at ten different locations
where the company sells this product. Data were collected for these units for a period of two
weeks (14 days).
Location
1
2
3
4
5
6
7
8
9
10
Number sold Day complaint was received
10
3, 4, 9
14
1,2, 7
7
2, 9,
5
3
12
1, 4, 12
15
2, 2, 7
10
2, 5
13
1, 3, 6, 10
9
2, 3, 7
5
1
(a) Compute an estimate of the mean cumulative number of complaints per unit sold, as a
function of days since purchase.
(b) Make a simple plot of the MCF. What you you say about its shape and the likely reason
for the shape.
(c) What is the interpretation of this sample MCF and how is it related to the rate of occurrence of complaints?
(d) Give one or more reasons why the complaint rate might vary across locations.
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10. To quantify the reliability of an automobile, and engineer has divided the automobile into
eight components that can be assumed to have failure times that are independent.
• Four tires, all of which must be operating for the automobile to operate. The reliability
of each tire is RT .
• Two headlights, one of which must be operating for the automobile to operate. The
reliability of each headlight is RH .
• A drivetrain, which must be operating for the automobile to operate. The reliability of
the drivetrain is RD .
• An electrical system, which must be operating for the automobile to operate. The reliability of the electrical system is RE .
Of course, each of these components could be looked at as a system with its own components.
(a) Draw a block diagram for the automobile system, as described above.
(b) Derive an expression for the reliability of the system as a function of RT , RH , RD , and RE .
11. When an automobile headlight burns out and fails, it might make sense, depending on the
shape of the hazard function, to replace both headlights at the same time. Based on your
knowledge of the shape of the hazard function of the life distribution of a headlight, provide
an argument for or against this practice.
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12. MTTF is a sensible metric for reliability for some applications but is inappropriate for other
applications. Briefly explain.
13. A sample of units was put into operation at time 0. The failure time of these units can be
described by a Weibull cdf
" #
β
t
Pr(T ≤ t; η, β) = F (t; η, β) = 1 − exp −
, t > 0.
η
Information about failures in this sample was recorded only after a known time ts > 0. If any
units failed before time ts there was no record of the failure times or even of the existence of
such units. We know only the number of units that survived until ts .
(a) Derive an expression for cdf of the future failure times for the units that survived until
ts and will be monitored after that time.
(b) What is the name of the distribution of the future failure times for units that survived
until ts ?
14. A failure-time distribution for the random variable T has a hazard function h(t) = θt, t > 0.
Derive an expression for the corresponding cdf for T .
15. Consider the use of a Weibull distribution to describe a time-to-failure process. Explain why
doing a Bayesian analysis with a specified joint prior distribution on the unknown parameters
η and β of a Weibull distribution is not a model that implies that η and β vary from unit to
unit in the population.
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