Engineering Design Handbook - Development Guide for Reliability, Part Six - Mathematical Appendix and Glossary

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AMC PAMPHLET
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ENGINEERING DESIGN
I
-_-"-"-""'-
DEVELOPMENTGUIDE
FOR ELIABILITY
4
MATHEMATICAL JPPENDIX'
AND GLOSSARY_
D!STIUBUTIOi4 STA'TEMENT A
.;pmvcd ior public releau
;"* , .......:. )ution Unlimzit~i I -, :
HEADQUARTERS, US ARMY MATERIEL COMMAND
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At4CP 706-200J
DEPARTMENT OF THE ARMY
HEADQUARTERS UNITED STATES ARMY MATERIEL COMMAND
5001 Eisenhower Ave., Alexandria. VA 22333
AMC PAMPHLET
NO. 706-20 8 January 1976
ENGINEERING DE §(GN HANDBOOK
DEVELOPMENT GUIDE FOP RELIABILITY, PART SIX
MATHEMATICAL APP NDIX AND GLOSSARY
TABLE OI0CONTENTS
Pararaph //Page
LIST OF ILLUSTRATIONS....... ........... vii
LIST OF TABLES .................. ix
PREFACE................................ xi
CHAPTER 1. GLOSSARY
CHAPTER 2!PROJBABILITY DISI iUBUTIONS,
SOME CAUTIONS AND NAMES
2-1 Cautions .. . . . . . . . . . . . ... . . . 2-1
2-2 Naming Probability Distributlons. .............. 2-2
CHAPTER 1. BINOMIAL DISTRIBUTION
3-0 List of Symbols........................~... 3-1
3-1 Introduction ............................. 3-1
3-2 Formulas ................................ 3-2
3-3 Tables and Curves.......................... 3-2
3-5 Randomized Exact s-Conf idence Intervals .......... 3-4
3-6 Choosing a s-Confidence Level .. ............... 3-12
3-7 Examples ................................ 3-12
3-7.1 Example No. 1 .. .................... 3-42
3-7.2 Example No. 2 ........................... 3-24
----- a. -
References ................................ 3-25
**
CHAPTER 4. POISSON DISTRIBUTION ?A, 6f t I
4-0 List of Symbols ............................ 4-1 ................
4-1 Introduction ............................. 4-1 ........-..... .
4-2 Formulas ................................ 4-2
4-3 Tables and Curves.......................... 4-2 1OA~~rc.
4--4 Parameter Estimation ........................ 4-2 ~
;-~
4-5 Ranoomized Exact s-Confidence Intervals ......... 4-6 A
4-7 Example, Life Test Results.......................
{*
References............................... 4-12
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AMCP 706.2%
TABLE OF;CONTENTS (Cont'd)
J
Paragraph Page
CHAPTER 5. GAUSSIAN (s-NORMAL)
DISTRIBUTION'
5-0 List of Symbols ........................... 5-I
5-1 Introduction ............................ 5-2
5-2 Formulas ......... ......................... 5-2
5-3 Tables and Curves ........................... 5-2
5-4 Parameter Estimation, Uncensored Samples ........ 5-13
5-5 Examples .................................... 5- 13
5-6 Parameter Estimation, Censored Samples ......... 5-14
References ................................. 5-15
CHAPTER 6. PROBABILITY DISTRIBUTIONS
DERIVED FROM THE GAUSSIAN
DISTRIBUTION
6-0 List of Symbols ........................... 6-1
6-1 Tntroduction ............................... 6-2
6-2 Chi-square (x2) Distribution .................. 6-2
6-2.1 Formulas ............................... 6-2
6-2.2 Tables ................................... 6-3
6-3 Chi-squarefnu (xe /v) Distribution ............... 6-3
6-3.1 Formulas ................................. 6-3
6-3.2 Tables ................................ 6-7
6-4 Student's t-Distribution ....................... 6-7
6-4.1 Formulas ............................... 6-7
6-4.2 Tables ................................. 6-7
6-5 Fisher-Snedecor F Distribotion ................ 6-12
6-5.1 Formulas ......... /....................... 6-12
6-5.2 Tables ........ .......................... 6-13
Refetences ................................. 6-22
CHAPTER 7 'EXPONENTIAL DISTRIBUTION
7-0 List of Symbols ............................. 7-1
7-1 Introduction ............................... 7-1
7-2 Foimulas ................................... 7-I
7-3 Tables ............................ ........ 7-2
7-4 Paameter Estimation ....................... 7-2
References .............................. 7-8
CHAPTER 8. WEIBULL DISTRIBUTION
8-0 List of Symbols ........................... 8-1
8-1 Introduction ............................... 8-1
(
i
. ....... . . . ...... . .. ' , . ? " : ,
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MP 706-200
TABLE OF CONTENTS (Cont'd)
Paragraph Page
8-2 Form ulas ................................... 8-2
8-3 Tables ..................................... 8-2
8-4 Parameter Estimation ......................... 8-2
-..........
8-4.1 Graphical Method ............................ 8-7
8-4.2 Maximum Likelihood Method 8-7
8-4.3 Linear Estimation Methods................... 8-10
8-4.4 Test for Failure Rate: Increasing, Decreasing, or
Constant ................................. 8-10
8-5 Comparison With Lognormal Distribution ........ 8-10
References ................................. 8-11
CHAPTER 0. LOGNORMAL DISTRIBUTION \
9-,0 list of Symbols ......................... ' 9-1
9-1 Introduction ............................. .9-1
9-2 Formulas ................................... 9-2
9-3 Tables ....................................... 9-4
9-4 Parameter Estimation ......................... 9-4
9-4.1 Uncensored Data ......................... 9-6
9-4.2 Censored Data ............................. 9-10
References ................................. 9-10
CHAPTER 10. BETA DISTRIBUTION,
1O-0 List of Symbols .................. .......... 10--i
10-1 introduction ............................... 10 -2
10-2 Formulas ... ............................... . 10-2
10-3 Tables . .................................... 10-3
10-4 Parameter EAtimation ......................... 10-3
References ................................. 10-7
CHAPTER 11. GAMMA DISTRIBUTION '
11-0 List of Symbols ............................. 11-1
11-1 Introduction .......... .................... 11-1
11-2 Form ulas ................................... 11-1
11-3 Tables ..................................... 11-2
11-4 Parar-,zter Estimation ......................... 11-2
11-5 Gamn.a Function ........................... 11-4
References...............................11-4
CHAPTER 12. s-CONFIDENCE
12-0 List of Symbols ........... ... ........ .. 12-1
12-1 Introduction ..... ......................... 12-1
0_.,, , _- .. . -- .-- . ., W : . , .... o1, - - . , . ',-, .... - .. - - - - . -. .. ,, ...... , r: '
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AMCP 706-200
TABLE OF CONTENTS (Cont'd)
t
Paragraph Page
12-2 Continuous Randqm Variables ................. 12-3
12-3 Discrete Random Variables ..................... 12-5
12-4 Discrete Random *rables,E xact Confidence
Bounds ................. ......... 12-7
12-5 MoeCopict-9S
More Complicated s-onfidence Situations ......... 12-9
References ..... . ............... 12-9
CHAPTER 13. PLOTTING POSITIONS
13-0 List of Symbols ............................. 13-1
13-1 Introduction ........................... 13-1
13-2 Sample Cdf ................................. 13-1
13-3 Percentile Ranges ............................. 13-2
13-4 Mean ..................................... 13-2
13-5 Censored Data (Hazard Plotting) ................. 13-5
CHAPTER 14. GOODNESS-OF-FIT TESTS
14-0 List of Symbols ............................. 14-1
14-1 Introduction ............................... 14-1
14-2 Chi-square ................................. 14-1
14-2.1 Discrete Random Variables .................. 14-1
14-2.2 Continuous Random Variable ................. 14-4
14-3 Kolmogorov-Smirnoff ........................ 14-5
Reference ................................. 14-8
CHAPTER IS. TESTS FOR MONOTONIC
FAILURE RATES
Reference ................................. 15-1
CHAPTER 16. BAYESIAN STATISTICS
16-1 Introduction ............................... 16-1
16-2 Bayes Formula ............................. 16-1
16-3 Interpretation of Probability ................... 16-1
16-4 Prior Distribution Is Real and Known ............. 16-4
16-5 Empirical Bayes ............................. 16-4
16-6 Bayesian Decision Theory ..................... 16-4
16-7 Subjective Probability .................... .16-7
16-8 Recommendations ........................... 16-7
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[
AM;P 70200
TABLE OF CONTENTS (Cont'd)
Pmmraph Page
CHAPTER 17.;SAMPLING PLANS
CHAPTER 18. MISCELLANEOUS DESIGN AIDS.,
References ................................. 18-
INDEX ...................................... I- 1
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NO~
AMCP7O2
LIST OF ILLUSTRATIONS
Figure No. Title page
3-1(A) I1-sided Upper s-Confidence Limit (80%) for p....... 3-9
3- 1(B) I1-sided Upper s-Confidence Limit (90%) for p....... 3-10
3-1(C) I -sided Upper s-Confidence Limit (95%) forp ....... 3-11
3-2 Special Case for No Failures in N Trials and C =RL
*3-24
4-1 Poisson Cumulative Distribution Function ...... 4-4
5-1 Curves for Gaussian Distributions...............5-3
6-1 Chi-squaze Distribution, pdf...................6-4I
6-2 Chi-square/Degrees-of-fredom Distribution, pdf ... 6-8
6-3 Student's t Distribution ...................... 6-10
7-1 Exponential Distribution ...................... 7 -
7-2 -Reliability Nomograph for the Exponential Distribution 7-4
8_-i Weibull Distributior, pdf ...................... 8-3
8-2 Weibull Distribution, Failure Rate ............... 8-4
8-3 Weibull Distribution, Contour Plot ..... ......... 8-5
9-1 Lognormal Distribution, pdf................. 9-3
9-2 Lognormal Distribution, Failure Rate ............ 9-5
9-3 Lognormal Distribution, Contour Plot ............ 9-7
9-4 Lognormal Failure Rate, Contour Plot ............ 9-8
9-5 Lognormal Distribution, Failure-rate Mode ......... 9-9
10-1 Beta Distribution ......... .................. 10-4 .
11-1 Gamma D~istribution, pdf ...................... 11-3
11-2 Gamma Distribution, Contour Plot of pdf .......... 11-5
12-1 s-Confidence Diagram: Continuous Random Variable
6 (for well-behaved situations) ................. 12-4
12-2 s-Confidence Diagram: Discrete Random Variable ... 12-6
14-1 Random Samples of 10 from the Uniform Distribu-j
tion on 10, 1]............................ 14-8
.***'~ ~
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AMCP 705-200
LIST OF TABLES
Table No. Title Page
3-1 Binomiial Distbibution, Examples ............... 3-3
3-2 1- sided Upper s-Confiderice Limits for p (The Bi-
nomial Probability)....................... 3-5
3-3 Sample Page from a Binomial Distribution ......... 3-14
3-4(A) Neyman-shortest Unbiased 95% s-Confidence Inter-
vals for p.............................. 3-16
3-4(B) Neyman-shortest Unbiased 99% s-Confidence Inter-I
vals forp ............................... 3-20
4-1 Poisson Distribution, Examples ................ 4-3
4-2 Neyman-shortest Unbiased 95% and 99% s-Confidence
Intervals for Ap .......................... 4-9
5-1 s-Normal Cdf-gauf (z).......................5-5
5-2 Gaussian (Standard s-Normal) Cdf-gauf W)........ 5-8
6-1 Percentiles of the Chi-square (X2 ) Distribution...... 6-5
6-2 Percentiles of the Chi-squarefnu (X /nu) Distribution .6-9
6-3 Percentiles of the t-Distribution ... ............ 6-11
6--4(A) F Distribution-fisf= 99%,flsfc - 1%............ 6-14
F Distribution-flsf =97.5%, flsfc = 2.5%.........6-16
F6-4(B)
6-4(C) F Distribution-flsf= 95%, flsfc= 5% ............ 6-18
6-4(D) F Distribution-flsf =90%,ftMc = 10%........... 6-20
7-1 Tables of -x ............................. 7-3
7-2 Ratio of Upper to Lower i-Confidence Limits for the
Exponential Parameter (with equal size tails on each
side) ................................. 7-7
11-1 Gamma Function......................... 11-6
13-1 Percentile Ranges for Plotting Points ............ 13-3
14-1 Data for Example No. 2..................... 14-3
14-2 Critical Values of the Kolmogorov-Smirnoff Test
Statistic............. ................ 14-6
14-3 Data for Example No. 4 ..................... 14-7
16-1 Data and Results for Example-Trial No. I1......... 16-5
16-2 Data and Results for Example-Trial No. 2 ........ 16-5
16-3 Data and Results for Example-Trial No. 3 ........ 16-6
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PREFACE
This harxdbuok, Mathematical Appendix and Glossary, is the last in a
series of five on reliability. The series is directed largely toward the working
engineers who have the responsibility for creating and producing equipment
and systems which can be relied upon by the users in the field.
The five handbooks are:
1. Design bo Reliability, AMCP 706-196
2. Reliability Prediction. AMCP 706-197
3. Reliability Measurement, AMCP 706-198
4. Contractingf or Reliability, AMCP 706-199
5. Mathematical Appendix and Glossary, AMCP 706-200.
This handbook is directed toward reliability engineers and manegers who
need to be familiar with or need to have access to statistical tables, curves,
and techniques, or to spetial terms used in the reliability discipline.
Rekerences are given to the literatui.e fcr further informaton.
Much of the handbook content was obtained fromu many individuals,
reports, journals, books, and other literature. It is impractical here to
acknowledge the assistance of everyone who made a contribution.
The original volume was prepared by Tracor Jitco, Inc. The revision was
prepared by Dr. Ralph A. Evans of Eva.ns Associates, Durham, N.C., for the
Engineering Handbook Office of the Rems, rch Triangle Institute, prime
contractor to the US Army Materiel Command. Technical guidance and
coordination on the original draft were provided by a committee under the
direction of Mr. 0. P. Bruno, Army Materiel System Analysis Agency, US
Army Materiel Command.
The Engineering Design Handbooks fall into two basic categories, those
approved for release and sale, and those classified fo security reasons. The
US Army Materiel Command policy is to release thew. Engineering Desibn
Handbooks in accordance with current DOD Directive 7230.7, dated 18
September 1973. All unclassified Handbooks can be obtained from the
National Technical Information Service (NTIS). Procedures for acquiring
these Handbooks follow:
id