IME6080 - Final Exam
Total time: 30 minutes
Name: _
Problem 1 (6 points):
Time to failure of an electronic device is Weibull distributed where =2000 hrs, and =2.
a)
Find the mean (), and standard deviation () of the distribution.
b)
What are the chances t

2.3.2
Special Staged Censored Data Sets
Sometimes experiments are run in stages, and special stopping rules are used to
decide when to stop testing. Two very common forms are discussed here, both of
which are special forms of type II, failure-censored tes

Stating Specications
To be truly meaningful, reliability and condence levels must be considered an
integral part of a reliability specication, and vice versa. For example, a
specication that merely states that an automotive sunroof must be able to be
cycl

3.7.1
Logistic and Log-Logistic Distributions
The logistic distribution is a two-parameter location-scale distribution with
distribution function
expz
ty
where z
Ft
3:22
1 expz
s
with location parameter 1 < y < 1 and scale parameter 0 < s.
The shape of

2
Preliminaries, Denitions, and Use
of Order Statistics in Reliability
Estimation
Have you heard the story about a plant engineer who was made a
reliability engineer? The engineers lack of training in the eld of reliability
never seemed to matter until on

Direct integration of f t requires the use of either complex numerical
methods or functional approximations of f t. Alternatively, tabulated values of
cumulative probabilities of the standard normal distribution have been developed
and are widely availabl

To estimate Ht, we make use of Eq. (2.10):
H^ t ln R^ t
Example 2-4:
2:27
Use of empirical reliability estimates on complete data sets
Recorded failures for a sample of size n 9 occur at t 60, 150, 299, 550, 980,
1270, 1680, 2100, and 2400 Kcycles. Develo

c.
Find t0:90 (90th percentile of the survival distributionthe design life
corresponding to 90th% reliability).
d. Given that a unit has survived 40 hr, what is the probability that it will
survive 30 more hr? Hint: Use a 70 hr and b 40 hr in the followin

FIGURE 4-15
Adequacy of linear t of hazard plot.
constant (see Figure 4-15). A convex t is indicative of an increasing hazard-rate
function; a concave t is indicative of a decreasing hazard-rate function.
Example 4-8:
Worked-out example (multiply failure

TABLE 4-4 Possible Estimators of the Exponential MTTF Parameter
n
P
r
P
ti
(1) MTTF i1
n
MTTF is underestimated if some of the
observations are censored.
r
P
ti
(3) MTTF i1
r
MTTF is underestimated since censoring
times are ignored in the numerator.
ti
(2

and by the central limit theorem, ln Yn will be approximately normally distributed
for large n.
The model described by Eq. (3.4) can be used to model a wide variety of
failure phenomena. The lognormal distribution is a popular distribution for
modeling fa

The three-parameter Weibull distribution is used to model phenomena that
take a minimum time to evolve, such as failures due to fatigue, corrosion, creep,
and other degradation phenomena. The distribution is often used to correct for a
poor t of the data

Furthermore, the median of Wi , denoted F^ i, is found using
F^ i
i
i n 1 iF2n1i;2i;0:50
2A:11
The demonstration of this result is founded on the general relationship
between the F-distribution and the beta distribution, which is described in the
followi

1A.1.1
Types of FMEA
There are ve main types of FMEA as depicted in Figure 1-11. They are
1.
2.
3.
4.
5.
1A.1.2
Concept FMEA
Design FMEA
Process FMEA
Service FMEA
Equipment FMEA
Design FMEA (DFMEA)
The design FMEA is used to identify and correct potential

TABLE 2-5 Table of Median Ranks Based on the Use of Microsoft1 Excel to Evaluate Eq. (2.20)
Copyright 2002 Marcel Dekker, Inc.
FIGURE 2-6 Instance of the use of both the mean and median rank plotting method on
Weibull probability paper.
2.2.4
Beta-Binomia

FIGURE 1-2 Classication of potential failure modes according to their severity and
degree of loss of product function.
organizations are using sound engineering design and design verication methods. Accordingly, when we speak of design deciencies, we are

heading is summarized in the cell corresponding to the intersection of the selected
row and column.
2.1.2
Population Moments
Population Mean (m) or MTTF
The population mean, m, or mean time-to-failure (MTTF) is given by
1
m
tf tdt
01
Rtdt
2:11
0
1
Proof o

With the action plan, the designer should be sure to provide detailed
supporting facts that give traceability to the action plan.
Hierarchical Nature of FMEA Information
The information reported in an FMEA is hierarchically arranged, allowing one or
more

FIGURE 1-7 Reliability bathtub curve; overall instantaneous failure rate is sum of hazard
rates associated with (1) weak items, (2) stress-related, and (3) wearout.
insignicant levels beyond the wear-in period. This relationship is used to
account for a s

The median, tM is found using FtM 0:5 tM =a ) tM 0:5a, which
is the midpoint of the interval 0; a.
The reliability functions for this uniform distribution are charted Figure
2-3, for a 1:0.
Example 2-2
The failure rate on a new steering pump design is est

According to Eq. (3.11), the conditional probability PT > t
DtjT > t PT > Dtthe unconditional probability of failure over the next
Dt time units. Accordingly, we state that the exponential distribution is memoryless since the probability of failure in th

1
A Modern View of Reliability
Concepts and Design for Reliability
True Story
ANZ Corporation, a second-tier automotive supplier of trim products, was
informed by its customer, an OEM (original equipment manufacturer, e.g.,
GM, Ford, or Toyota), that its

7.4
Failure Rate Estimates - MTTF Estimates
The same estimates * is valid for all the test procedures.
* = T/n
Where: T = Total operational time of all test units;
n = Number of failures.
Of course, for each class of tests, the total operating time T shou

FIGURE 4-3
Weibull plot of data in Table 4-2.
Graphical Estimation (see Figure 4-3)
b^ 1:60:
y^ 130K revs:
Copyright 2002 Marcel Dekker, Inc.
Inverse Rank Regression
The results of an inverse rank regression of ln t (y-values) versus
doubln R ln ln 1=1 F^

Setting Up Rank Regression Models
Accordingly, we suggest the following regression procedure to estimate m and s:
1.
Create two columns of information:
a. Ordered failures, ti
b. F^ ti using adjusted ranks, if data set contains censored observations
Only

pfm ps.tex
Problem Set on Probabilistic Failure Models
14
15. Weibull distribution and the failure rate function. (p.27) In table 1 are listed the
lifetimes of 15 out of a population of 20 mechanical switches. The term lifetime refers to the
number of cyc

pfm ps.tex
Problem Set on Probabilistic Failure Models
5
5. Many identical components. (p.19) (Weibull distribution). A device has a large number
of identical components. The device fails as soon as the rst component fails. The failure rate
function for t

24
Problem Set on Probabilistic Failure Models
pfm ps.tex
Solution to problem 10. (p.10)
(a) The probability of failure is:
F
= Probability of failure = Prob(sy L)
=
0
pL ()py (sy ) dsy
0
sy
=
e
0
=
(82)
e
d
(83)
dsy d
(84)
0
+
(85)
The probability of

Solution to Assignment #5; IME 6080; Due Tuesday March 8, 2011
Example 20: An incoming part from our supplier has 0.01 chance of being defective. A lot containing 100 units has just
arrived.
a) What is the probability of no defective items in the lot?
t

Use Software R to do Survival Analysis and
Simulation. A tutorial
Mai Zhou
Department of Statistics, University of Kentucky
c
GPL
2.0 copyrighted
In this short tutorial we suppose you already have R (version 1.5.0 or later) installed, and know
how to star