Lecture 2 - What are component and system reliability and how it can be improved?

Size: px
Start display at page:

Download "Lecture 2 - What are component and system reliability and how it can be improved?"

Transcription

1 Lecture 2 - What are compoet ad system relablty ad how t ca be mproved? Relablty s a measure of the qualty of the product over the log ru. The cocept of relablty s a exteded tme perod over whch the expected operato of the product s cosdered ad we expect the product wll fucto accordg to certa expectatos over a stpulated perod of tme. Wth the customer ad warraty costs md, we must kow the chaces of successful operato of the product for at least a certa stpulated perod of tme. Such formato helps the maufacturer to select the parameters of a warraty polcy. Techcally, relablty s the probablty of a product performg ts teded fucto for a stated perod of tme uder certa specfed codtos. Four aspects of relablty are apparet from ths defto frst, relablty s a probablty of success-related cocept; the umercal value of ths probablty s always betwee 0 ad 1. Secod, the fuctoal performace of the product had to be measured uder certa stpulated codtos. Product desg s expected to esure developmet of a product that meets or exceeds the specfed requremets uder specfed operatg codtos. For example, f the breakg stregth of a ylo cord s expected to be 1000 kg, the the predefed operatoal codtos, the cord must be able to bear weghts of 1000 kg or more. Thrd, relablty mples successful operato over a certa perod of tme (t). Although o product s expected to last forever, the tme dmeso esures satsfactory performace over at least a mmal stated perod (say, 100 hours). I the cotext of these three aspects, the relablty of the ylo cord mght be descrbed as havg a probablty of successful performace of 0.92 bearg loads of 1000 kg for 1 year uder dry codtos. It s observed that most maufacturg products go through three dstct phases (see Fgure 3-8) from product cepto to wear-out. The lfe-cycle curve of Fgure 3-8 shows the varato the falure rate as a fucto of tme dfferet phases. Covetoally the falure rate ( λ ) s plotted as a fucto of tme. Ths curve s ofte referred to as the bathtub curve; t cossts of the debuggg (fatmortalty) phase, the chace-falure phase (useful lfe phase), ad the wear-out phase. The debuggg phase, also kow as the fat-mortalty phase, exhbts a drop the falure rate as tal problems detfed durg prototype testg are removed. The chace-falure phase, betwee tmes t 1 ad t 2, s the ecoutered; falures occur

2 radomly ad depedetly. Ths phase, whch the falure rate s costat, typcally represets the useful lfe of the product delvered to ed customer. I the wear-out phase, a crease the falure rate s expected due to wear ad tear of the product. Here, after the ed of ther useful lfe, parts age ad wear out. Fgure 3-8 Bathtub Curve For the radom chace-falure phase, whch represets the useful lfe of the product or compoet, the falure rate s assumed to be costat. As a result, the expoetal dstrbuto s selected to descrbe the tme-to-falure of the product for ths phase. A expoetal dstrbuto as a memory less property ad ts probablty desty fucto s gve by λ λ t f t = e, t 0 The mea-tme-to-falure (MTTF) for the expoetal dstrbuto ca be expressed as MTTF=1/λ The relablty, at tme t, say R(t), s the probablty of the product lastg up to tme t. It ca be

3 expressed as, = 1 F ( t) R t =1- t e dt = e 0 Here, F(t) represets the cumulatve dstrbuto fucto at ay tme t. Relablty decreases expoetally wth tme (Fgure 3-9) ad the falure-rate fucto, say r(t), s gve by the rato of the tme-to-falure probablty desty fucto to the relablty fucto. We have r t f t = R t Fgure 3-9 Relablty v/s Tme Thus, assumg a expoetal dstrbuto, r (t) mplyg a costat falure rate, as show below. r t λe = = λ e Let us cosder a resster compoet, whch follows a expoetal tme-to-falure dstrbuto wth a falure rate of 8% per 1000 hr. We are terested to calculate the relablty of the resster at 5000 hr, ad also we ted to calculate the mea-tme-to-falure. Here the costat falure rate λ s obtaed as

4 λ = 008 /1000 hr = /hr Thus, the relablty for 5000 hr of survval s R t = e =e =e = Thus there s about 67% chace of survval (success) of the resster uder stpulated codtos ad stpulated tme (5000 hr). The mea (average) tme-to-falure (assumg t caot be repar) of the resster wll be MTTF=1/ λ = 1 / = 12,500 h System Relablty Let us cosder a system wth three compoets (say three resster) seres as show Fgure Fgure 3-10 Seres System Wthout loss of geeralty, f the system compoets ca be assumed to have a tme-to-falure dstrbuto as expoetal wth each compoet has a costat falure rate, we ca easly compute the relablty of -system seres. Suppose the system has compoets ad seres, each wth expoetally dstrbuted tme-to-falure wth falure rates λ 1, λ 2,, λ. The system relablty s calculated as the product of the compoet relabltes:

5 R () t = e e e S =exp - λ1t λ2t = 1 λ t Ths mples that the tme-to-falure of the system s expoetally dstrbuted wth a equvalet falure rate of = 1 gve by λ. The mea tme to falure for the system s MTTF = 1 = 1 λ Systems wth Compoets Parallel System relablty ca be mproved by placg redudat compoets parallel. The system operates as log as at least oe of the compoets operates. A three compoet parallel system s represeted as gve Fgure Fgure 3-11 A Parallel Compoet System If the tme-to-falure of each compoet follows expoetal dstrbutos, each wth a costat falure rate, λ, = 1,...,, the system relablty, assumg depedece of compoet operato (falure of oe does ot mpact falure of ay other compoet), s gve by

6 ( ) R () t = 1-1 R() t S =1 λ t ( e ) =1-1 =1 I a specal case, where all compoets have the same falure rate, λ, relablty for parallel compoet s gve by the system λ t R () t = 1 1 e S For such specfc stuato, the mea-tme-to-falure for the system wth detcal compoets parallel, ad also assumg that each faled compoet s mmedately replaced by a detcal compoet, ca be expressed as MTTF = 1... λ Systems wth Compoets Seres ad Parallel Real lfe systems ofte cosst of compoets that are mxed ad cosst of both seres ad parallel cofgurato. For such system, relablty calculato s prmarly based o the prevously dscussed cocepts, ad assumpto of compoets operatg depedetly. Parallel systems are frst collated to get a composte relablty, ad the the overall compoets are cosdered as seres to calculate system relablty. Systems ca also cosst of stadby compoet, whch operates as ad whe base compoet fals. Reader may refer to book by Amtava Mtra (2008) or Besterfeld et al (2004 ) for further detals o. K-out-of-N system (parallel system s 1 out of N system) s aother possble system cofgurato.

Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions

Estimation of Stress- Strength Reliability model using finite mixture of exponential distributions Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur

More information

Functions of Random Variables

Functions of Random Variables Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,

More information

Analysis of Variance with Weibull Data

Analysis of Variance with Weibull Data Aalyss of Varace wth Webull Data Lahaa Watthaacheewaul Abstract I statstcal data aalyss by aalyss of varace, the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad

More information

CHAPTER VI Statistical Analysis of Experimental Data

CHAPTER VI Statistical Analysis of Experimental Data Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca

More information

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg

More information

Analysis of a Repairable (n-1)-out-of-n: G System with Failure and Repair Times Arbitrarily Distributed

Analysis of a Repairable (n-1)-out-of-n: G System with Failure and Repair Times Arbitrarily Distributed Amerca Joural of Mathematcs ad Statstcs. ; (: -8 DOI:.593/j.ajms.. Aalyss of a Reparable (--out-of-: G System wth Falure ad Repar Tmes Arbtrarly Dstrbuted M. Gherda, M. Boushaba, Departmet of Mathematcs,

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:

More information

Lecture 9: Tolerant Testing

Lecture 9: Tolerant Testing Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have

More information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst

More information

Third handout: On the Gini Index

Third handout: On the Gini Index Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The

More information

Chapter 5 Properties of a Random Sample

Chapter 5 Properties of a Random Sample Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample

More information

Simulation Output Analysis

Simulation Output Analysis Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5

More information

Class 13,14 June 17, 19, 2015

Class 13,14 June 17, 19, 2015 Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral

More information

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators

More information

A New Family of Transformations for Lifetime Data

A New Family of Transformations for Lifetime Data Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. A New Famly of Trasformatos for Lfetme Data Lakhaa Watthaacheewakul Abstract A famly of trasformatos s the oe of several

More information

Bayes (Naïve or not) Classifiers: Generative Approach

Bayes (Naïve or not) Classifiers: Generative Approach Logstc regresso Bayes (Naïve or ot) Classfers: Geeratve Approach What do we mea by Geeratve approach: Lear p(y), p(x y) ad the apply bayes rule to compute p(y x) for makg predctos Ths s essetally makg

More information

X ε ) = 0, or equivalently, lim

X ε ) = 0, or equivalently, lim Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece

More information

Point Estimation: definition of estimators

Point Estimation: definition of estimators Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.

More information

Special Instructions / Useful Data

Special Instructions / Useful Data JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth

More information

Chapter 14 Logistic Regression Models

Chapter 14 Logistic Regression Models Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as

More information

Lecture 07: Poles and Zeros

Lecture 07: Poles and Zeros Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto

More information

Analysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems

Analysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems Char for Network Archtectures ad Servces Prof. Carle Departmet of Computer Scece U Müche Aalyss of System Performace IN2072 Chapter 5 Aalyss of No Markov Systems Dr. Alexader Kle Prof. Dr.-Ig. Georg Carle

More information

Chapter 8. Inferences about More Than Two Population Central Values

Chapter 8. Inferences about More Than Two Population Central Values Chapter 8. Ifereces about More Tha Two Populato Cetral Values Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers ) To vestgate whether treatmet at a youg age would yeld better results tha

More information

Module 7: Probability and Statistics

Module 7: Probability and Statistics Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to

More information

ENGI 4421 Propagation of Error Page 8-01

ENGI 4421 Propagation of Error Page 8-01 ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.

More information

Lecture Notes Types of economic variables

Lecture Notes Types of economic variables Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte

More information

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package

More information

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1

STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ  1 STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ

More information

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)

Feature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture) CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.

More information

Unsupervised Learning and Other Neural Networks

Unsupervised Learning and Other Neural Networks CSE 53 Soft Computg NOT PART OF THE FINAL Usupervsed Learg ad Other Neural Networs Itroducto Mture Destes ad Idetfablty ML Estmates Applcato to Normal Mtures Other Neural Networs Itroducto Prevously, all

More information

Likewise, properties of the optimal policy for equipment replacement & maintenance problems can be used to reduce the computation.

Likewise, properties of the optimal policy for equipment replacement & maintenance problems can be used to reduce the computation. Whe solvg a vetory repleshmet problem usg a MDP model, kowg that the optmal polcy s of the form (s,s) ca reduce the computatoal burde. That s, f t s optmal to replesh the vetory whe the vetory level s,

More information

Lecture 3 Probability review (cont d)

Lecture 3 Probability review (cont d) STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto

More information

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the

More information

Introduction to local (nonparametric) density estimation. methods

Introduction to local (nonparametric) density estimation. methods Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest

More information

Dependence In Network Reliability

Dependence In Network Reliability Depedece I Network Relablty Nozer D. Sgpurwalla George Washgto Uversty Washgto, DC, U.S.A. ozer@reasearch.crc.gwu.edu Abstract It s farly easy to calculate the relablty of a etwork wth depedet odes va

More information

Lecture 1. (Part II) The number of ways of partitioning n distinct objects into k distinct groups containing n 1,

Lecture 1. (Part II) The number of ways of partitioning n distinct objects into k distinct groups containing n 1, Lecture (Part II) Materals Covered Ths Lecture: Chapter 2 (2.6 --- 2.0) The umber of ways of parttog dstct obects to dstct groups cotag, 2,, obects, respectvely, where each obect appears exactly oe group

More information

6.867 Machine Learning

6.867 Machine Learning 6.867 Mache Learg Problem set Due Frday, September 9, rectato Please address all questos ad commets about ths problem set to 6.867-staff@a.mt.edu. You do ot eed to use MATLAB for ths problem set though

More information

A Study of the Reproducibility of Measurements with HUR Leg Extension/Curl Research Line

A Study of the Reproducibility of Measurements with HUR Leg Extension/Curl Research Line HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) A Study of the Reproducblty of Measuremets wth HUR Leg Eteso/Curl Research Le A mportat property of measuremets s that the results should

More information

Logistic regression (continued)

Logistic regression (continued) STAT562 page 138 Logstc regresso (cotued) Suppose we ow cosder more complex models to descrbe the relatoshp betwee a categorcal respose varable (Y) that takes o two (2) possble outcomes ad a set of p explaatory

More information

The Mathematical Appendix

The Mathematical Appendix The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.

More information

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:

{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution: Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted

More information

VOL. 3, NO. 11, November 2013 ISSN ARPN Journal of Science and Technology All rights reserved.

VOL. 3, NO. 11, November 2013 ISSN ARPN Journal of Science and Technology All rights reserved. VOL., NO., November 0 ISSN 5-77 ARPN Joural of Scece ad Techology 0-0. All rghts reserved. http://www.ejouralofscece.org Usg Square-Root Iverted Gamma Dstrbuto as Pror to Draw Iferece o the Raylegh Dstrbuto

More information

Lecture 4 Sep 9, 2015

Lecture 4 Sep 9, 2015 CS 388R: Radomzed Algorthms Fall 205 Prof. Erc Prce Lecture 4 Sep 9, 205 Scrbe: Xagru Huag & Chad Voegele Overvew I prevous lectures, we troduced some basc probablty, the Cheroff boud, the coupo collector

More information

To use adaptive cluster sampling we must first make some definitions of the sampling universe:

To use adaptive cluster sampling we must first make some definitions of the sampling universe: 8.3 ADAPTIVE SAMPLING Most of the methods dscussed samplg theory are lmted to samplg desgs hch the selecto of the samples ca be doe before the survey, so that oe of the decsos about samplg deped ay ay

More information

CHAPTER 3 POSTERIOR DISTRIBUTIONS

CHAPTER 3 POSTERIOR DISTRIBUTIONS CHAPTER 3 POSTERIOR DISTRIBUTIONS If scece caot measure the degree of probablt volved, so much the worse for scece. The practcal ma wll stck to hs apprecatve methods utl t does, or wll accept the results

More information

MYUNG HWAN NA, MOON JU KIM, LIN MA

MYUNG HWAN NA, MOON JU KIM, LIN MA BAYESIAN APPROACH TO MEAN TIME BETWEEN FAILURE USING THE MODULATED POWER LAW PROCESS MYUNG HWAN NA, MOON JU KIM, LIN MA Abstract. The Reewal process ad the No-homogeeous Posso process (NHPP) process are

More information

Bounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy

Bounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy Bouds o the expected etropy ad KL-dvergece of sampled multomal dstrbutos Brado C. Roy bcroy@meda.mt.edu Orgal: May 18, 2011 Revsed: Jue 6, 2011 Abstract Iformato theoretc quattes calculated from a sampled

More information

CIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights

CIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights CIS 800/002 The Algorthmc Foudatos of Data Prvacy October 13, 2011 Lecturer: Aaro Roth Lecture 9 Scrbe: Aaro Roth Database Update Algorthms: Multplcatve Weghts We ll recall aga) some deftos from last tme:

More information

Availability Equivalence Factors of a General Repairable Parallel-Series System

Availability Equivalence Factors of a General Repairable Parallel-Series System ppled Mathematcs 04 5 73-73 Publshed Ole Jue 04 cres. http://www.scrp.org/joural/am http://dx.do.org/0.436/am.04.564 valablty Equvalece Factors of a Geeral Reparable Parallel-eres ystem bdelfattah Mustafa

More information

22 Nonparametric Methods.

22 Nonparametric Methods. 22 oparametrc Methods. I parametrc models oe assumes apror that the dstrbutos have a specfc form wth oe or more ukow parameters ad oe tres to fd the best or atleast reasoably effcet procedures that aswer

More information

18.413: Error Correcting Codes Lab March 2, Lecture 8

18.413: Error Correcting Codes Lab March 2, Lecture 8 18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse

More information

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5

THE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should

More information

LINEAR REGRESSION ANALYSIS

LINEAR REGRESSION ANALYSIS LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for

More information

CS286.2 Lecture 4: Dinur s Proof of the PCP Theorem

CS286.2 Lecture 4: Dinur s Proof of the PCP Theorem CS86. Lecture 4: Dur s Proof of the PCP Theorem Scrbe: Thom Bohdaowcz Prevously, we have prove a weak verso of the PCP theorem: NP PCP 1,1/ (r = poly, q = O(1)). Wth ths result we have the desred costat

More information

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR

Outline. Point Pattern Analysis Part I. Revisit IRP/CSR Pot Patter Aalyss Part I Outle Revst IRP/CSR, frst- ad secod order effects What s pot patter aalyss (PPA)? Desty-based pot patter measures Dstace-based pot patter measures Revst IRP/CSR Equal probablty:

More information

Statistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura

Statistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura Statstcs Descrptve ad Iferetal Statstcs Istructor: Dasuke Nagakura (agakura@z7.keo.jp) 1 Today s topc Today, I talk about two categores of statstcal aalyses, descrptve statstcs ad feretal statstcs, ad

More information

EP2200 Queueing theory and teletraffic systems. Queueing networks. Viktoria Fodor KTH EES/LCN KTH EES/LCN

EP2200 Queueing theory and teletraffic systems. Queueing networks. Viktoria Fodor KTH EES/LCN KTH EES/LCN EP2200 Queueg theory ad teletraffc systems Queueg etworks Vktora Fodor Ope ad closed queug etworks Queug etwork: etwork of queug systems E.g. data packets traversg the etwork from router to router Ope

More information

CHAPTER 4 RADICAL EXPRESSIONS

CHAPTER 4 RADICAL EXPRESSIONS 6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube

More information

1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i.

1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i. CS 94- Desty Matrces, vo Neuma Etropy 3/7/07 Sprg 007 Lecture 3 I ths lecture, we wll dscuss the bascs of quatum formato theory I partcular, we wll dscuss mxed quatum states, desty matrces, vo Neuma etropy

More information

Continuous Distributions

Continuous Distributions 7//3 Cotuous Dstrbutos Radom Varables of the Cotuous Type Desty Curve Percet Desty fucto, f (x) A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Test scores Desty Curve Percet Probablty Desty Fucto, f

More information

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67. Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please

More information

Optimal Strategy Analysis of an N-policy M/E k /1 Queueing System with Server Breakdowns and Multiple Vacations

Optimal Strategy Analysis of an N-policy M/E k /1 Queueing System with Server Breakdowns and Multiple Vacations Iteratoal Joural of Scetfc ad Research ublcatos, Volume 3, Issue, ovember 3 ISS 5-353 Optmal Strategy Aalyss of a -polcy M/E / Queueg System wth Server Breadows ad Multple Vacatos.Jayachtra*, Dr.A.James

More information

TESTS BASED ON MAXIMUM LIKELIHOOD

TESTS BASED ON MAXIMUM LIKELIHOOD ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal

More information

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA THE ROYAL STATISTICAL SOCIETY 3 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER I STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad

More information

Chapter 4 Multiple Random Variables

Chapter 4 Multiple Random Variables Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:

More information

Multiple Linear Regression Analysis

Multiple Linear Regression Analysis LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple

More information

Lecture 8: Linear Regression

Lecture 8: Linear Regression Lecture 8: Lear egresso May 4, GENOME 56, Sprg Goals Develop basc cocepts of lear regresso from a probablstc framework Estmatg parameters ad hypothess testg wth lear models Lear regresso Su I Lee, CSE

More information

Algorithms Design & Analysis. Hash Tables

Algorithms Design & Analysis. Hash Tables Algorthms Desg & Aalyss Hash Tables Recap Lower boud Order statstcs 2 Today s topcs Drect-accessble table Hash tables Hash fuctos Uversal hashg Perfect Hashg Ope addressg 3 Symbol-table problem Symbol

More information

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several

More information

Homework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015

Homework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015 Fall 05 Homework : Solutos Problem : (Practce wth Asymptotc Notato) A essetal requremet for uderstadg scalg behavor s comfort wth asymptotc (or bg-o ) otato. I ths problem, you wll prove some basc facts

More information

f f... f 1 n n (ii) Median : It is the value of the middle-most observation(s).

f f... f 1 n n (ii) Median : It is the value of the middle-most observation(s). CHAPTER STATISTICS Pots to Remember :. Facts or fgures, collected wth a defte pupose, are called Data.. Statstcs s the area of study dealg wth the collecto, presetato, aalyss ad terpretato of data.. The

More information

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch

More information

(b) By independence, the probability that the string 1011 is received correctly is

(b) By independence, the probability that the string 1011 is received correctly is Soluto to Problem 1.31. (a) Let A be the evet that a 0 s trasmtted. Usg the total probablty theorem, the desred probablty s P(A)(1 ɛ ( 0)+ 1 P(A) ) (1 ɛ 1)=p(1 ɛ 0)+(1 p)(1 ɛ 1). (b) By depedece, the probablty

More information

Part 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))

Part 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971)) art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the

More information

NP!= P. By Liu Ran. Table of Contents. The P vs. NP problem is a major unsolved problem in computer

NP!= P. By Liu Ran. Table of Contents. The P vs. NP problem is a major unsolved problem in computer NP!= P By Lu Ra Table of Cotets. Itroduce 2. Strategy 3. Prelmary theorem 4. Proof 5. Expla 6. Cocluso. Itroduce The P vs. NP problem s a major usolved problem computer scece. Iformally, t asks whether

More information

Qualifying Exam Statistical Theory Problem Solutions August 2005

Qualifying Exam Statistical Theory Problem Solutions August 2005 Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),

More information

Lecture 1 Review of Fundamental Statistical Concepts

Lecture 1 Review of Fundamental Statistical Concepts Lecture Revew of Fudametal Statstcal Cocepts Measures of Cetral Tedecy ad Dsperso A word about otato for ths class: Idvduals a populato are desgated, where the dex rages from to N, ad N s the total umber

More information

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Aalyss of Varace ad Desg of Exermets-I MODULE II LECTURE - GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr Shalabh Deartmet of Mathematcs ad Statstcs Ida Isttute of Techology Kaur Tukey s rocedure

More information

ECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model

ECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model ECON 48 / WH Hog The Smple Regresso Model. Defto of the Smple Regresso Model Smple Regresso Model Expla varable y terms of varable x y = β + β x+ u y : depedet varable, explaed varable, respose varable,

More information

Chapter 11 Systematic Sampling

Chapter 11 Systematic Sampling Chapter stematc amplg The sstematc samplg techue s operatoall more coveet tha the smple radom samplg. It also esures at the same tme that each ut has eual probablt of cluso the sample. I ths method of

More information

3. Basic Concepts: Consequences and Properties

3. Basic Concepts: Consequences and Properties : 3. Basc Cocepts: Cosequeces ad Propertes Markku Jutt Overvew More advaced cosequeces ad propertes of the basc cocepts troduced the prevous lecture are derved. Source The materal s maly based o Sectos.6.8

More information

Applied Mathematics and Materials

Applied Mathematics and Materials Appled Mathematcs ad Materals O a Two-No-Idetcal Stadby Reparable System wth Imperfect Repar MOHAMMED A. HAJEEH, ABDUL-WAHAB ALOTHMAN 2 Techo-Ecoomcs Dvso Kuwat Isttute for Scetfc Research P.O. Box 24885;

More information

9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d

9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d 9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,

More information

Module 1 : The equation of continuity. Lecture 5: Conservation of Mass for each species. & Fick s Law

Module 1 : The equation of continuity. Lecture 5: Conservation of Mass for each species. & Fick s Law Module : The equato of cotuty Lecture 5: Coservato of Mass for each speces & Fck s Law NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg 2 Basc Deftos I Mass Trasfer, we usually

More information

Chapter 13 Student Lecture Notes 13-1

Chapter 13 Student Lecture Notes 13-1 Chapter 3 Studet Lecture Notes 3- Basc Busess Statstcs (9 th Edto) Chapter 3 Smple Lear Regresso 4 Pretce-Hall, Ic. Chap 3- Chapter Topcs Types of Regresso Models Determg the Smple Lear Regresso Equato

More information

BAYESIAN INFERENCES FOR TWO PARAMETER WEIBULL DISTRIBUTION

BAYESIAN INFERENCES FOR TWO PARAMETER WEIBULL DISTRIBUTION Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK (www.eajourals.org BAYESIAN INFERENCES FOR TWO PARAMETER WEIBULL

More information

The Occupancy and Coupon Collector problems

The Occupancy and Coupon Collector problems Chapter 4 The Occupacy ad Coupo Collector problems By Sarel Har-Peled, Jauary 9, 08 4 Prelmares [ Defto 4 Varace ad Stadard Devato For a radom varable X, let V E [ X [ µ X deote the varace of X, where

More information

Bayesian Inferences for Two Parameter Weibull Distribution Kipkoech W. Cheruiyot 1, Abel Ouko 2, Emily Kirimi 3

Bayesian Inferences for Two Parameter Weibull Distribution Kipkoech W. Cheruiyot 1, Abel Ouko 2, Emily Kirimi 3 IOSR Joural of Mathematcs IOSR-JM e-issn: 78-578, p-issn: 9-765X. Volume, Issue Ver. II Ja - Feb. 05, PP 4- www.osrjourals.org Bayesa Ifereces for Two Parameter Webull Dstrbuto Kpkoech W. Cheruyot, Abel

More information

NP!= P. By Liu Ran. Table of Contents. The P versus NP problem is a major unsolved problem in computer

NP!= P. By Liu Ran. Table of Contents. The P versus NP problem is a major unsolved problem in computer NP!= P By Lu Ra Table of Cotets. Itroduce 2. Prelmary theorem 3. Proof 4. Expla 5. Cocluso. Itroduce The P versus NP problem s a major usolved problem computer scece. Iformally, t asks whether a computer

More information

Introduction to Matrices and Matrix Approach to Simple Linear Regression

Introduction to Matrices and Matrix Approach to Simple Linear Regression Itroducto to Matrces ad Matrx Approach to Smple Lear Regresso Matrces Defto: A matrx s a rectagular array of umbers or symbolc elemets I may applcatos, the rows of a matrx wll represet dvduals cases (people,

More information

Dimensionality Reduction and Learning

Dimensionality Reduction and Learning CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that

More information

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty

More information

Sampling Theory MODULE V LECTURE - 14 RATIO AND PRODUCT METHODS OF ESTIMATION

Sampling Theory MODULE V LECTURE - 14 RATIO AND PRODUCT METHODS OF ESTIMATION Samplg Theor MODULE V LECTUE - 4 ATIO AND PODUCT METHODS OF ESTIMATION D. SHALABH DEPATMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOG KANPU A mportat objectve a statstcal estmato procedure

More information

Investigating Cellular Automata

Investigating Cellular Automata Researcher: Taylor Dupuy Advsor: Aaro Wootto Semester: Fall 4 Ivestgatg Cellular Automata A Overvew of Cellular Automata: Cellular Automata are smple computer programs that geerate rows of black ad whte

More information

Lecture 9. Some Useful Discrete Distributions. Some Useful Discrete Distributions. The observations generated by different experiments have

Lecture 9. Some Useful Discrete Distributions. Some Useful Discrete Distributions. The observations generated by different experiments have NM 7 Lecture 9 Some Useful Dscrete Dstrbutos Some Useful Dscrete Dstrbutos The observatos geerated by dfferet eermets have the same geeral tye of behavor. Cosequetly, radom varables assocated wth these

More information

MEASURES OF DISPERSION

MEASURES OF DISPERSION MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda

More information

Real Time Study of a Identical Repairable Elements with Constant Failure and Repair Rates

Real Time Study of a Identical Repairable Elements with Constant Failure and Repair Rates Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 009 Vol II IMECS 009, March 8-0, 009, Hog Kog Real Tme Study of a out of System: Idetcal Reparable Elemets wth Costat Falure ad Repar

More information

BASICS ON DISTRIBUTIONS

BASICS ON DISTRIBUTIONS BASICS ON DISTRIBUTIONS Hstograms Cosder a epermet whch dfferet outcomes are possble (e. Dce tossg). The probablty of all the outcomes ca be represeted a hstogram Dstrbutos Probabltes are descrbed wth

More information

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato

More information