Probability and risk analysis: an introduction for engineers ; with ... 7 tables
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2006
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Literaturverz. S. [275] - 278 |
| Physical Description: | X, 281 S. graph. Darst. |
| ISBN: | 3540242236 9783540242239 |
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MARC
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| 020 | |a 9783540242239 |9 978-3-540-24223-9 | ||
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| 100 | 1 | |a Rychlik, Igor |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Probability and risk analysis |b an introduction for engineers ; with ... 7 tables |c Igor Rychlik ; Jesper Rydén |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a X, 281 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 500 | |a Literaturverz. S. [275] - 278 | ||
| 650 | 4 | |a Engineering mathematics | |
| 650 | 4 | |a Probabilities | |
| 650 | 4 | |a Reliability (Engineering) | |
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Record in the Search Index
| _version_ | 1804135965881532416 |
|---|---|
| adam_text | Contents
1 Basic
Probability
.......................................... 1
1.1 Sample Space, Events,
and Probabilities
.................... 4
1.2
Independence...........................................
8
1.2.1
Counting
variables................................. 10
1.3
Conditional Probabilities and the Law of Total Probability
... 12
1.4
Event-tree Analysis
...................................... 15
2
Probabilities in Risk Analysis
.............................. 21
2.1
Bayes
Formula
.......................................... 22
2.2
Odds and Subjective Probabilities
......................... 23
2.3
Recursive Updating of Odds
.............................. 27
2.4
Probabilities as Long-term Frequencies
..................... 30
2.5
Streams of Events
....................................... 33
2.6
Intensities of Streams
.................................... 37
2.6.1
Poisson
streams of events
........................... 40
2.6.2
Non-stationary streams
............................ 43
3
Distributions and Random Variables
....................... 49
3.1
Random Numbers
....................................... 51
3.1.1
Uniformly distributed random numbers
.............. 51
3.1.2
Non-uniformly distributed random numbers
........... 52
3.1.3
Examples of random numbers
....................... 54
3.2
Some Properties of Distribution Functions
.................. 55
3.3
Scale and Location Parameters
-
Standard Distributions
..... 59
3.3.1
Some classes of distributions
........................ 60
3.4
Independent Random Variables
........................... 62
3.5
Averages
-
Law of Large Numbers
......................... 63
3.5.1
Expectations of functions of random variables
......... 65
VIII Contents
4
Fitting Distributions to Data
-
Classical Inference
......... 69
4.1
Estimates of Fx
........................................ 72
4.2
Choosing a Model for Fx
................................ 74
4.2.1
A graphical method: probability paper
............... 75
4.2.2
Introduction to
χ2
-method for goodness-of-fit tests
.... 77
4.3
Maximum Likelihood Estimates
........................... 80
4.3.1
Introductory example
.............................. 80
4.3.2
Derivation of ML estimates for some common models
.. 82
4.4
Analysis of Estimation Error
.............................. 85
4.4.1
Mean and variance of the estimation error
E
.......... 86
4.4.2
Distribution of error, large number of observations
..... 89
4.5
Confidence Intervals
..................................... 92
4.5.1
Introduction. Calculation of bounds
................. 92
4.5.2
Asymptotic intervals
............................... 94
4.5.3
Bootstrap confidence intervals
....................... 95
4.5.4
Examples
........................................ 95
4.6
Uncertainties of Quantiles
................................ 98
4.6.1
Asymptotic normality
.............................. 98
4.6.2
Statistical bootstrap
...............................100
5
Conditional Distributions with Applications
...............105
5.1
Dependent Observations
..................................105
5.2
Some Properties of Two-dimensional Distributions
...........107
5.2.1
Covariance and correlation
.........................113
5.3
Conditional Distributions and Densities
....................115
5.3.1
Discrete random variables
..........................115
5.3.2
Continuous random variables
.......................116
5.4
Application of Conditional Probabilities
....................117
5.4.1
Law of total probability
............................117
5.4.2
Bayes
formula
....................................118
5.4.3
Example: Reliability of a system
....................119
6
Introduction to Bayesian Inference
........................125
6.1
Introductory Examples
...................................126
6.2
Compromising Between Data and Prior Knowledge
..........130
6.2.1
Bayesian credibility intervals
........................132
6.3
Bayesian Inference
.......................................132
6.3.1
Choice of a model for the data
-
conditional
independence
.....................................133
6.3.2
Bayesian updating and likelihood functions
...........134
6.4
Conjugated Priors
.......................................135
6.4.1
Unknown probability
..............................137
6.4.2
Probabilities for multiple scenarios
..................139
6.4.3
Priors for intensity of a stream A
...................141
Contents
IX
6.5
Remarks on Choice of Priors
..............................143
6.5.1
Nothing is known about the parameter
θ
.............143
6.5.2
Moments of
θ
are known
..........................144
6.6
Large number of observations: Likelihood dominates prior
density
.................................................147
6.7
Predicting Frequency of Rare Accidents
....................151
7
Intensities and
Poisson
Models
............................157
7.1
Time to the First Accident
—
Failure Intensity
..............157
7.1.1
Failure intensity
...................................157
7.1.2
Estimation procedures
.............................162
7.2
Absolute Risks
..........................................166
7.3
Poisson
Models for Counts
................................170
7.3.1
Test for
Poisson
distribution
-
constant mean
.........171
7.3.2
Test for constant mean
-
Poisson
variables
............173
7.3.3
Formulation of
Poisson
regression model
..............174
7.3.4
ML estimates of
β0,
...,
βρ .........................
180
7.4
The
Poisson
Point process
................................182
7.5
More General
Poisson
Processes
...........................185
7.6
Decomposition and Superposition of
Poisson
Processes
.......187
8
Failure Probabilities and Safety Indexes
...................193
8.1
Functions Often Met in Applications
.......................194
8.1.1
Linear function
...................................194
8.1.2
Often used non-linear function
......................198
8.1.3
Minimum of variables
..............................201
8.2
Safety Index
............................................202
8.2.1
Cornell s index
....................................202
8.2.2
Hasofer-Lind index
................................204
8.2.3
Use of safety indexes in risk analysis
.................204
8.2.4
Return periods and safety index
.....................205
8.2.5
Computation of Cornell s index
.....................206
8.3
Gauss Approximations
...................................207
8.3.1
The delta method
.................................209
9
Estimation of
Quantités
....................................217
9.1
Analysis of Characteristic Strength
........................217
9.1.1
Parametric modelling
..............................218
9.2
The Peaks Over Threshold (POT) Method
.................220
9.2.1
The POT method and estimation of xa quantiles
.....222
9.2.2
Example: Strength of glass fibres
....................223
9.2.3
Example: Accidents in mines
........................224
9.3
Quality of Components
...................................226
9.3.1
Binomial distribution
..............................227
9.3.2
Bayesian approach
.................................228
X
Contents
10 Design
Loads and Extreme Values
.........................231
10.1
Safety Factors, Design Loads, Characteristic Strength
........232
10.2
Extreme Values
.........................................233
10.2.1
Extreme-value distributions
.........................234
10.2.2
Fitting a model to data: An example
.................240
10.3
Finding the 100-year Load: Method of Yearly Maxima
.......241
10.3.1
Uncertainty analysis of
Ѕт
:
Gumbel case
.............242
10.3.2
Uncertainty analysis of
ѕт
:
GEV case
................244
10.3.3
Warning example of model error
....................245
10.3.4
Discussion on uncertainty in design-load estimates
.....247
A Some Useful Tables
........................................251
Short Solutions to Problems
...................................257
References
.....................................................275
Index
..........................................................279
This book presents notions and ideas from the foundations of a statistical
treatment of risks. Bayesian methods are frequently used in that area, hence
a reasonable proportion of the presentation is devoted to such approaches.
Modern statistical tools, namely
Poisson
regression, analysis of deviance,
extreme-value theory and threshold methods are also used to identify and
solve practical problems. The knowledge of such tools facilitates the under¬
standing of the role of probability in risk analysis and proper use of outputs
given by software packages.
The book is written with a student in mind who has studied elementary
undergraduate courses in engineering mathematics, perhaps including an
introductory course in statistics. The text emphasizes an understanding of
the theory and methods presented. Hence, comments are given verbally
and an intuitive reasoning is frequent. This is not just another new textbook
on elementary probability and statistics, but focuses strongly on applica¬
tions within the field of risk and safety analysis.
242239
)
springer.com
|
| adam_txt |
Contents
1 Basic
Probability
. 1
1.1 Sample Space, Events,
and Probabilities
. 4
1.2
Independence.
8
1.2.1
Counting
variables. 10
1.3
Conditional Probabilities and the Law of Total Probability
. 12
1.4
Event-tree Analysis
. 15
2
Probabilities in Risk Analysis
. 21
2.1
Bayes'
Formula
. 22
2.2
Odds and Subjective Probabilities
. 23
2.3
Recursive Updating of Odds
. 27
2.4
Probabilities as Long-term Frequencies
. 30
2.5
Streams of Events
. 33
2.6
Intensities of Streams
. 37
2.6.1
Poisson
streams of events
. 40
2.6.2
Non-stationary streams
. 43
3
Distributions and Random Variables
. 49
3.1
Random Numbers
. 51
3.1.1
Uniformly distributed random numbers
. 51
3.1.2
Non-uniformly distributed random numbers
. 52
3.1.3
Examples of random numbers
. 54
3.2
Some Properties of Distribution Functions
. 55
3.3
Scale and Location Parameters
-
Standard Distributions
. 59
3.3.1
Some classes of distributions
. 60
3.4
Independent Random Variables
. 62
3.5
Averages
-
Law of Large Numbers
. 63
3.5.1
Expectations of functions of random variables
. 65
VIII Contents
4
Fitting Distributions to Data
-
Classical Inference
. 69
4.1
Estimates of Fx
. 72
4.2
Choosing a Model for Fx
. 74
4.2.1
A graphical method: probability paper
. 75
4.2.2
Introduction to
χ2
-method for goodness-of-fit tests
. 77
4.3
Maximum Likelihood Estimates
. 80
4.3.1
Introductory example
. 80
4.3.2
Derivation of ML estimates for some common models
. 82
4.4
Analysis of Estimation Error
. 85
4.4.1
Mean and variance of the estimation error
E
. 86
4.4.2
Distribution of error, large number of observations
. 89
4.5
Confidence Intervals
. 92
4.5.1
Introduction. Calculation of bounds
. 92
4.5.2
Asymptotic intervals
. 94
4.5.3
Bootstrap confidence intervals
. 95
4.5.4
Examples
. 95
4.6
Uncertainties of Quantiles
. 98
4.6.1
Asymptotic normality
. 98
4.6.2
Statistical bootstrap
.100
5
Conditional Distributions with Applications
.105
5.1
Dependent Observations
.105
5.2
Some Properties of Two-dimensional Distributions
.107
5.2.1
Covariance and correlation
.113
5.3
Conditional Distributions and Densities
.115
5.3.1
Discrete random variables
.115
5.3.2
Continuous random variables
.116
5.4
Application of Conditional Probabilities
.117
5.4.1
Law of total probability
.117
5.4.2
Bayes'
formula
.118
5.4.3
Example: Reliability of a system
.119
6
Introduction to Bayesian Inference
.125
6.1
Introductory Examples
.126
6.2
Compromising Between Data and Prior Knowledge
.130
6.2.1
Bayesian credibility intervals
.132
6.3
Bayesian Inference
.132
6.3.1
Choice of a model for the data
-
conditional
independence
.133
6.3.2
Bayesian updating and likelihood functions
.134
6.4
Conjugated Priors
.135
6.4.1
Unknown probability
.137
6.4.2
Probabilities for multiple scenarios
.139
6.4.3
Priors for intensity of a stream A
.141
Contents
IX
6.5
Remarks on Choice of Priors
.143
6.5.1
Nothing is known about the parameter
θ
.143
6.5.2
Moments of
θ
are known
.144
6.6
Large number of observations: Likelihood dominates prior
density
.147
6.7
Predicting Frequency of Rare Accidents
.151
7
Intensities and
Poisson
Models
.157
7.1
Time to the First Accident
—
Failure Intensity
.157
7.1.1
Failure intensity
.157
7.1.2
Estimation procedures
.162
7.2
Absolute Risks
.166
7.3
Poisson
Models for Counts
.170
7.3.1
Test for
Poisson
distribution
-
constant mean
.171
7.3.2
Test for constant mean
-
Poisson
variables
.173
7.3.3
Formulation of
Poisson
regression model
.174
7.3.4
ML estimates of
β0,
.,
βρ .
180
7.4
The
Poisson
Point process
.182
7.5
More General
Poisson
Processes
.185
7.6
Decomposition and Superposition of
Poisson
Processes
.187
8
Failure Probabilities and Safety Indexes
.193
8.1
Functions Often Met in Applications
.194
8.1.1
Linear function
.194
8.1.2
Often used non-linear function
.198
8.1.3
Minimum of variables
.201
8.2
Safety Index
.202
8.2.1
Cornell's index
.202
8.2.2
Hasofer-Lind index
.204
8.2.3
Use of safety indexes in risk analysis
.204
8.2.4
Return periods and safety index
.205
8.2.5
Computation of Cornell's index
.206
8.3
Gauss' Approximations
.207
8.3.1
The delta method
.209
9
Estimation of
Quantités
.217
9.1
Analysis of Characteristic Strength
.217
9.1.1
Parametric modelling
.218
9.2
The Peaks Over Threshold (POT) Method
.220
9.2.1
The POT method and estimation of xa quantiles
.222
9.2.2
Example: Strength of glass fibres
.223
9.2.3
Example: Accidents in mines
.224
9.3
Quality of Components
.226
9.3.1
Binomial distribution
.227
9.3.2
Bayesian approach
.228
X
Contents
10 Design
Loads and Extreme Values
.231
10.1
Safety Factors, Design Loads, Characteristic Strength
.232
10.2
Extreme Values
.233
10.2.1
Extreme-value distributions
.234
10.2.2
Fitting a model to data: An example
.240
10.3
Finding the 100-year Load: Method of Yearly Maxima
.241
10.3.1
Uncertainty analysis of
Ѕт
:
Gumbel case
.242
10.3.2
Uncertainty analysis of
ѕт
:
GEV case
.244
10.3.3
Warning example of model error
.245
10.3.4
Discussion on uncertainty in design-load estimates
.247
A Some Useful Tables
.251
Short Solutions to Problems
.257
References
.275
Index
.279
This book presents notions and ideas from the foundations of a statistical
treatment of risks. Bayesian methods are frequently used in that area, hence
a reasonable proportion of the presentation is devoted to such approaches.
Modern statistical tools, namely
Poisson
regression, analysis of deviance,
extreme-value theory and threshold methods are also used to identify and
solve practical problems. The knowledge of such tools facilitates the under¬
standing of the role of probability in risk analysis and proper use of outputs
given by software packages.
The book is written with a student in mind who has studied elementary
undergraduate courses in engineering mathematics, perhaps including an
introductory course in statistics. The text emphasizes an understanding of
the theory and methods presented. Hence, comments are given verbally
and an intuitive reasoning is frequent. This is not just another new textbook
on elementary probability and statistics, but focuses strongly on applica¬
tions within the field of risk and safety analysis.
242239
)
springer.com |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Rychlik, Igor Rydén, Jesper |
| author_facet | Rychlik, Igor Rydén, Jesper |
| author_role | aut aut |
| author_sort | Rychlik, Igor |
| author_variant | i r ir j r jr |
| building | Verbundindex |
| bvnumber | BV021990853 |
| classification_rvk | QH 170 SK 800 SK 850 |
| classification_tum | TEC 700f |
| ctrlnum | (OCoLC)180887672 (DE-599)BVBBV021990853 |
| discipline | Technik Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Technik Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV021990853 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:10:35Z |
| indexdate | 2024-07-09T20:48:53Z |
| institution | BVB |
| isbn | 3540242236 9783540242239 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015205638 |
| oclc_num | 180887672 |
| open_access_boolean | |
| owner | DE-706 DE-573 DE-91G DE-BY-TUM DE-384 DE-92 DE-11 DE-703 |
| owner_facet | DE-706 DE-573 DE-91G DE-BY-TUM DE-384 DE-92 DE-11 DE-703 |
| physical | X, 281 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| spelling | Rychlik, Igor Verfasser aut Probability and risk analysis an introduction for engineers ; with ... 7 tables Igor Rychlik ; Jesper Rydén Berlin [u.a.] Springer 2006 X, 281 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. [275] - 278 Engineering mathematics Probabilities Reliability (Engineering) Probabilistische Sicherheitsanalyse (DE-588)4322949-9 gnd rswk-swf Risikoanalyse (DE-588)4137042-9 gnd rswk-swf Probabilistische Sicherheitsanalyse (DE-588)4322949-9 s DE-604 Risikoanalyse (DE-588)4137042-9 s 1\p DE-604 Rydén, Jesper Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015205638&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015205638&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Rychlik, Igor Rydén, Jesper Probability and risk analysis an introduction for engineers ; with ... 7 tables Engineering mathematics Probabilities Reliability (Engineering) Probabilistische Sicherheitsanalyse (DE-588)4322949-9 gnd Risikoanalyse (DE-588)4137042-9 gnd |
| subject_GND | (DE-588)4322949-9 (DE-588)4137042-9 |
| title | Probability and risk analysis an introduction for engineers ; with ... 7 tables |
| title_auth | Probability and risk analysis an introduction for engineers ; with ... 7 tables |
| title_exact_search | Probability and risk analysis an introduction for engineers ; with ... 7 tables |
| title_exact_search_txtP | Probability and risk analysis an introduction for engineers ; with ... 7 tables |
| title_full | Probability and risk analysis an introduction for engineers ; with ... 7 tables Igor Rychlik ; Jesper Rydén |
| title_fullStr | Probability and risk analysis an introduction for engineers ; with ... 7 tables Igor Rychlik ; Jesper Rydén |
| title_full_unstemmed | Probability and risk analysis an introduction for engineers ; with ... 7 tables Igor Rychlik ; Jesper Rydén |
| title_short | Probability and risk analysis |
| title_sort | probability and risk analysis an introduction for engineers with 7 tables |
| title_sub | an introduction for engineers ; with ... 7 tables |
| topic | Engineering mathematics Probabilities Reliability (Engineering) Probabilistische Sicherheitsanalyse (DE-588)4322949-9 gnd Risikoanalyse (DE-588)4137042-9 gnd |
| topic_facet | Engineering mathematics Probabilities Reliability (Engineering) Probabilistische Sicherheitsanalyse Risikoanalyse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015205638&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015205638&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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