Reliability Measures of a Series System with Weibull Failure Laws
|
|
- Alban Pope
- 5 years ago
- Views:
Transcription
1 Iteratioal Joural of Statistics ad Systems ISSN Volume, Number 2 (26), pp Research Idia Publicatios Reliability Measures of a Series System with Weibull Failure Laws S.K. Chauha ad S.C. Malik Departmet of Statistics, M.D. Uiversity, Rohtak 24 (Haryaa) statskumar@gmail.com & sc_malik@rediffmail.com Abstract The Weibull distributio is widely used i reliability ad life data aalysis due to its versatility. Ad, this distributio has bee cosidered as a popular life time distributio which describes modelig pheomea with mootoic failure rates of compoets. Depedig o the values of the parameters it ca be used to model a variety of life behaviors. I this paper, reliability measures such as reliability ad mea time to system failure (MTSF) of a series system of idetical compoets by cosiderig Weibull failure laws are obtaied. The results for these measures are also evaluated for the special case of Weibull distributio i.e. by assumig Rayleigh failure laws. The behavior of MTSF ad reliability has bee observed graphically for arbitrary values of the parameters related to umber of compoets, failure rates ad operatig time. Keywords: Series System, Reliability, MTSF, Weibull Failure Laws. INTRODUCTION It has commoly kow that performace of operatig systems depeds etirely o the cofiguratios of their compoets. The system may have simple or complex structure of the compoets. Ad, accordigly several cofiguratios of the compoets have bee evolved as a result of research i the field of reliability egieerig. The series systems are oe of them beig used i may systems like wheat harvestig system where a tractor, wago ad combie are coected i series. I a series system, the compoets are arraged i such a way that the successive operatio of the system depeds o the proper operatio of all the compoets. Therefore, reliability of such systems has become a matter of cocer for the egieers ad researchers i order to idetify the factors which ca be used to improve their performace. There are several systems i which compoets have
2 74 S.K. Chauha ad S.C. Malik mootoic failure rates. For example, the hazard rate of rotatig shafts, valves ad cams are of o liear ature due to agig ad workig stress. I such systems, the compoet s life time distributed by cumulative damage ad thus they have icreasig failure rate with passage of time. Balagurusamy (984) ad Sriath(985) determie reliability measures of a series system for Expoetial distributio. Elsayed(22) developed reliability measure of some system cofiguratios usig Expoetial, Rayleigh ad Weibull distributios. Navarro ad Spizzichio(2) made a Compariso of series ad parallel systems with compoets sharig the same copula. Recetly, Nadal et al. (25) evaluated the Reliability ad Mea time to System Failure (MTSF) of a Series System with expoetial failure laws. The Weibull distributio is widely used i reliability ad life data aalysis due to its versatility. Ad, this distributio has bee cosidered as a popular life time distributio which describes modelig pheomea with mootoic failure rates of compoets. Depedig o the values of the parameters it ca be used to model a variety of life behaviors. I this paper, reliability measures such as reliability ad mea time to system failure (MTSF) of a series system of idetical compoets by cosiderig Weibull failure laws are obtaied. The results for these measures are also evaluated for the special case of Weibull distributio i.e. by assumig Rayleigh failure laws. The behavior of MTSF ad reliability has bee observed graphically for arbitrary values of the parameters related to umber of compoets, failure rates ad operatig time. 2. NOTATIONS R(t) = Reliability of the system, R i (t) = Reliability of the i th compoet h(t)= Istataeous failure rate of the system, h i (t) = Istataeous failure rate of i th compoet, λ = Costat failure rate T = Life time of the system, T i = Life time of the i th compoet. 3. SYSTEM DESCRIPTION Here, a series system of compoets is cosidered which ca fail at the failure of ay oe of the compoets. The state trasitio diagram is show i Fig. Fig: A series system of compoets. The reliability of the system is give by R(t) = Pr[T>t] = Pr[mi(T, T 2,.., T )>t] = Pr[T >t, T 2 >t,.,t >t] = i= Pr [Ti > t] = R i (t) i= ()
3 Reliability Measures of a Series System with Weibull Failure Laws 75 The mea time to system failure is give by MTSF= R i (t) dt i= (2) 4. RELIABILITY MEASURES OF A SERIES SYSTEM WITH WEIBULL DISTRIBUTION: Suppose failure rate of all compoets are govered by the Weibull failure law i.e. h i (t) = λ i t β i The, the compoets reliability is give by R i (t) = e t h i (u)du = e t λ iu β idu = e λ i t Therefore, the system reliability is give by R s (t) = R i (t) = e λ t i Ad, i= MTSF = e i= λ i t β i + β i + β i + β i + β i + β i + i= = e λ t β i + i= i β i + dt = For idetical compoets we ca have h i (t) = λt β Г β i + i= [λ i (β i +) β i] β i + The the system reliability is give by λtβ+ R s (t) = e β+ ad MTSF= e λtβ+ β+ dt = Г β+ [λ(β+) β ] β+ Illustratios. For a sigle compoet, the system reliability is give by R s (t) = i= R i (t) = e λ i t β i + β i + i= = e λ t β + For idetical compoets, we ca have h i (t) = λt β The the system reliability is give by R s (t) = e λtβ+ β+ ad MTSF= Г β+ [λ(β+) β ] β+ β+ ad MTSF= Г β + [λ (β +) β ] β+ 2. Suppose system has two compoets, the the system reliability is give by R s (t) = e λ t β i + i β i + = e 2 λ tβ i + i= t β + 2 i β i + = β+ +λ 2 tβ 2+ β2+ ] i= e [λ
4 76 S.K. Chauha ad S.C. Malik MTSF= Г β + Г β 2 + [λ (β +) β ] β+ [λ 2 (β 2 +) β 2] β2+ For idetical compoets, we ca have h i (t) = λt β The the system reliability is give by 2λtβ+ R s (t) = e β+ ad MTSF= R s (t)dt = e 2λtβ+ β+ dt = Г β+ [2λ(β+) β ] β+ I a similar way we ca obtai reliability ad MTSF of a system havig three or more compoets coected i series. 5. RELIABILITY MEASURES FOR ARBITRARY VALUES OF THE PARAMETERS Reliability ad mea time to system failure (MTSF) of the system has bee obtaied for arbitrary values of the parameters associated with umber of compoets(), failure rate (λ), operatig time of the compoet (t) ad shape parameter (β). The results are show umerically ad graphically as: No. of Compoets λ=., t=, β=. Table : Reliability Vs No. of Compoets () Reliability λ=.2, t=, β=. λ=.3, t=, β=. λ=.4, t=, β=. λ=.5, t=, β=
5 Reliability Reliability Measures of a Series System with Weibull Failure Laws No. of Compoets λ=. λ=.2 λ=.3 λ=.4 λ=.5 Fig.2: Reliability Vs No. of Compoets () Table 2: MTSF Vs No. of Compoets () No. of MTSF Compo ets λ=., t=, β=. λ=.2, t=, β=. λ=.3, t=, β=. λ=.4, t=, β=. λ=.5, t=, β=
6 MTSF 78 S.K. Chauha ad S.C. Malik No. of Compoets λ=. λ=.2 λ=.3 λ=.4 λ=.5 Fig.3: MTSF Vs No. of Compoets () Table 3: Reliability Vs No. of Compoets () No. of Reliability Compoets β=., λ=.,t= β=.2, λ=.,t= β=.3, λ=.,t= β=.4, λ=.,t= β=.5, λ=.,t=
7 Reliability Reliability Measures of a Series System with Weibull Failure Laws No. of Compoets β=. β=.2 β=.3 β=.4 β=.5 Fig.4: Reliability Vs No. of Compoets () Table 4: MTSF Vs No. of Compoets () No. of Compo ets MTSF β=.5, λ=.,t= β=., λ=.,t= β=.2, λ=.,t= β=.3, λ=.,t= β=.4, λ=.,t=
8 MTSF 8 S.K. Chauha ad S.C. Malik No. of Compoets β=. β=.2 β=.3 β=.4 β=.5 Fig.5: MTSF Vs No. of Compoets () Table 5: Reliability Vs No. of Compoets () No. of Reliability Compoe ts t=5, λ=., β=. t=, λ=., β=. t=5, λ=., β=. t=2, λ=., β=. t=25, λ=., β=
9 Reliability Reliability Measures of a Series System with Weibull Failure Laws No. of Compoets t=5 t= t=5 t=2 t=25 Fig.6: Reliability Vs No. of Compoets ad Time 6. RELIABILITY MEASURES FOR A SPECIAL CASE (RAYLEIGH DISTRIBUTION) OF WEIBULL DISTRIBUTION: The Rayleigh distributio has extesively bee used i life testig experimets, reliability aalysis, commuicatio egieerig, cliical studies ad applied statistics. This distributio is a special case of Weibull distributio with the shape parameter β=. Whe compoets are govered by Rayleigh failure laws, the compoet reliability is give by R i (t) = e t h i (u)du = e t λ iudu = e λ i t2 2, where h i (t) = λ i t Therefore, the system reliability is give by R s (t) = R i (t) = e λ it 2 2 = e λ it 2 i= 2 i= Ad, MTSF = t= R(t)dt i= = e λ i t2 i= t= For idetical compoets we ca have λ i t = λt The system reliability is give by 2 dt = Π 2 i= λ i R s (t) = e λt2 2 ad MTSF= R(t)dt = e λt2 2 dt t= t= = Π 2λ
10 82 S.K. Chauha ad S.C. Malik Illustratios:. For a sigle compoet the system reliability is give by R s (t) = Ad MTSF= t= e λ i t2 λ it2 2 i= = e i= 2 R(t)dt = e λ i t2 i= t= For idetical compoets, we ca have λ i t = λt The the system reliability is give by 2 dt R s (t) = e λt2 2 ad MTSF= e λt2 2 dt = Π t= 2λ = Π 2 i= λ i = Π 2λ 2. Suppose system has two compoets, the the system reliability is give by 2 R s (t) = i= R i (t) = e Ad MTSF= t= R(t)dt λ 2 it2 i= 2 = e 2 λ i t2 i= t= For idetical compoet, we ca have λ i t = λt The the system reliability is give by R s (t) = e λt2 ad MTSF= e λt2 dt t= 2 dt = 2 Π λ = Π 2 2 i= λ i = Π 2(λ +λ 2 ) I a similar way we ca obtai reliability ad MTSF of a system havig three or more compoets coected i series. 7. RELIABILITY MEASURES FOR ARBITRARY VALUES OF THE PARAMETERS Reliability ad mea time to system failure (MTSF) of the system has bee obtaied for arbitrary values of the parameters associated with umber of compoets(), failure rate (λ) ad operatig time of the compoet (t) The results are show umerically ad graphically as:
11 Reliability Reliability Measures of a Series System with Weibull Failure Laws 83 Number of Compoets Table 6: Reliability Vs No. of Compoets () Reliability λ=.,t= λ=.2,t= λ=.3,t= λ=.4, t= λ=.5, t= λ=. λ=.2 λ=.3 λ=.4 λ= No. of Compoets Fig.7: Reliability Vs No. of Compoets ()
12 MTSF 84 S.K. Chauha ad S.C. Malik Table 7: MTSF Vs No. of Compoets () No. of MTSF Compoets λ=.,t= λ=.2,t= λ=.3,t= λ=.4,t= λ=.5,t= λ=. λ=.2 λ=.3 λ=.4 λ= No. of Compoets Fig.8: MTSF Vs No. of Compoets ()
13 Reliability Reliability Measures of a Series System with Weibull Failure Laws 85 Table 8: Reliability Vs No. of Compoets () No. of Compoets t=5, λ=. t=, λ=. Reliability t=5, λ=. t=2, λ=. t=25, λ= No. of Compoets t=5 t= t=5 t=2 t=25 Fig.9: Reliability Vs No. of Compoets () 8. DISCUSSION OF THE RESULTS The results obtaied for arbitrary values of the parameters idicate that reliability ad mea time to system failure of a series system of idetical compoets keep o decreasig with the icrease of the umber of compoets ad their failure rates. However, the effect of umber of compoets ad their failure rates o reliability of the system is much more i case compoets govered by Rayleigh failure laws the
14 86 S.K. Chauha ad S.C. Malik that of Weibull failure laws. I case of mea time to system failure, the effect is much more whe compoets follow Weibull failure laws rather tha Rayleigh failure laws. The reliability of the system goes o decreasig with the icrease of operatig time irrespective of distributios govered by failure time of the compoets. The effect of operatig time o reliability is much more i case compoets follow Rayleigh failure laws as compare to Weibull failure laws. However, there is o effect of operatig time o mea time to system failure (MTSF). The results obtaied for some more particular values of the shape parameter β (.,.2,.3,.4 ad.5) idicate that reliability ad mea time to system failure of a series system of idetical compoets declie with the icrease of the value of β. The results are show umerically ad graphically i respective tables ad figures. CONCLUSION I preset study, we coclude that the reliability ad MTSF keep o decreasig with the icrease the umber of compoets, failure rates ad operatig time of the compoet. It is suggested that least umber of compoet should be used i a series system for better performace. However, the performace of such systems ca be improved by utilizig compoets which follow Weibull failure laws. REFERENCES [] Balagurusamy, E. (984): Reliability Egieerig, Tata McGraw Hill Publishig Co. Ltd., Idia. [2] Sriath, L.S. (985): Cocept i Reliability Egieerig, Affiliated East-West Press (P) Ltd. [3] Rausad, M. ad Hsylad, A. (24): System Reliability Theory, Joh Wiley & Sos, Ic., Hoboke, New Jersey, [4] Navarro, J. ad Spizzichio, F. (2): Comparisos of series ad parallel systems with compoets sharig the same copula, Applied Stochastic Models I Busiess ad Idustry, Vol. 26(6), pp [5] Elsayed, A. (22): Reliability Egieerig, Wiley Series i Systems Egieerig ad Maagemet. [6] Nadal, J., Chauha, S.K. & Malik, S.C. (25): Reliability ad MTSF of a Series ad Parallel systems, Iteratioal Joural of Statistics ad Reliability Egieerig, Vol. 2(), pp [7] Chauha, S.K. ad Malik, S.C. (26): Reliability Evaluatio of a Series ad Parallel systems for Arbitrary Values of the Parameters, Iteratioal Joural of Statistics ad Reliability Egieerig, Vol. 3(), pp.-9,
Bayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function
Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume 12, Number 4 (217), pp. 791-796 Research Idia Publicatios http://www.ripublicatio.com Bayesia ad E- Bayesia Method of Estimatio of Parameter
More informationMathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution
America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 (http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.05040. ISSN: 6-8999 (Prit; ISSN: 6-9006 (Olie Mathematical
More informationEstimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable
Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya
More informationDepartment of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment HW5 Solution
Departmet of Civil Egieerig-I.I.T. Delhi CEL 899: Evirometal Risk Assessmet HW5 Solutio Note: Assume missig data (if ay) ad metio the same. Q. Suppose X has a ormal distributio defied as N (mea=5, variace=
More informationConfidence interval for the two-parameter exponentiated Gumbel distribution based on record values
Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationThere is no straightforward approach for choosing the warmup period l.
B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.
More informationControl chart for number of customers in the system of M [X] / M / 1 Queueing system
Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Cotrol chart for umber of customers i the system of M [X] / M / Queueig system T.Poogodi, Dr.
More informationEstimating the Change Point of Bivariate Binomial Processes Experiencing Step Changes in Their Mean
Proceedigs of the 202 Iteratioal Coferece o Idustrial Egieerig ad Operatios Maagemet Istabul, Turey, July 3 6, 202 Estimatig the Chage Poit of Bivariate Biomial Processes Experiecig Step Chages i Their
More informationANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY
ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa,
More informationPOWER AKASH DISTRIBUTION AND ITS APPLICATION
POWER AKASH DISTRIBUTION AND ITS APPLICATION Rama SHANKER PhD, Uiversity Professor, Departmet of Statistics, College of Sciece, Eritrea Istitute of Techology, Asmara, Eritrea E-mail: shakerrama009@gmail.com
More informationComparison of Methods for Estimation of Sample Sizes under the Weibull Distribution
Iteratioal Joural of Applied Egieerig Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 14273-14278 Research Idia Publicatios. http://www.ripublicatio.com Compariso of Methods for Estimatio of Sample
More informationMechanical Efficiency of Planetary Gear Trains: An Estimate
Mechaical Efficiecy of Plaetary Gear Trais: A Estimate Dr. A. Sriath Professor, Dept. of Mechaical Egieerig K L Uiversity, A.P, Idia E-mail: sriath_me@klce.ac.i G. Yedukodalu Assistat Professor, Dept.
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationInternational Journal of Mathematical Archive-5(7), 2014, Available online through ISSN
Iteratioal Joural of Mathematical Archive-5(7), 214, 11-117 Available olie through www.ijma.ifo ISSN 2229 546 USING SQUARED-LOG ERROR LOSS FUNCTION TO ESTIMATE THE SHAPE PARAMETER AND THE RELIABILITY FUNCTION
More informationExtreme Value Charts and Analysis of Means (ANOM) Based on the Log Logistic Distribution
Joural of Moder Applied Statistical Methods Volume 11 Issue Article 0 11-1-01 Extreme Value Charts ad Aalysis of Meas (ANOM) Based o the Log Logistic istributio B. Sriivasa Rao R.V.R & J.C. College of
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationReliability and Queueing
Copyright 999 Uiversity of Califoria Reliability ad Queueig by David G. Messerschmitt Supplemetary sectio for Uderstadig Networked Applicatios: A First Course, Morga Kaufma, 999. Copyright otice: Permissio
More informationBayesian inference for Parameter and Reliability function of Inverse Rayleigh Distribution Under Modified Squared Error Loss Function
Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN:99-878 EISSN: 239-844 Joural home page: www.ajbasweb.com Bayesia iferece
More informationSample Size Estimation in the Proportional Hazards Model for K-sample or Regression Settings Scott S. Emerson, M.D., Ph.D.
ample ie Estimatio i the Proportioal Haards Model for K-sample or Regressio ettigs cott. Emerso, M.D., Ph.D. ample ie Formula for a Normally Distributed tatistic uppose a statistic is kow to be ormally
More information2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2
Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:
More informationEstimation of the Population Mean in Presence of Non-Response
Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, 537 548 DOI: 0.535/CKSS.0.8.4.537 Estimatio of the Populatio Mea i Presece of No-Respose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,
More informationModeling and Estimation of a Bivariate Pareto Distribution using the Principle of Maximum Entropy
Sri Laka Joural of Applied Statistics, Vol (5-3) Modelig ad Estimatio of a Bivariate Pareto Distributio usig the Priciple of Maximum Etropy Jagathath Krisha K.M. * Ecoomics Research Divisio, CSIR-Cetral
More informationModified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis
America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry
More informationReliability and Availablity
Reliability ad Availablity This set of otes is a combiatio of material from Prof. Doug Carmichael's otes for 13.21 ad Chapter 8 of Egieerig Statistics Hadbook. NIST/SEMATECH e-hadbook of Statistical Methods,
More informationSome Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation
; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 097-70)]; ISSN 0975-556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial Ratio-Product
More informationMA131 - Analysis 1. Workbook 9 Series III
MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationTHE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE ABSTRACT
Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN 56-586 THE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE Atif Nazir, Tahir Mahmood ad
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
More informationB. Maddah ENMG 622 ENMG /27/07
B. Maddah ENMG 622 ENMG 5 3/27/7 Queueig Theory () What is a queueig system? A queueig system cosists of servers (resources) that provide service to customers (etities). A Customer requestig service will
More informationMOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationBootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests
Joural of Moder Applied Statistical Methods Volume 5 Issue Article --5 Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed Al-Ha Ebrahem
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationControl Charts for Mean for Non-Normally Correlated Data
Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article 5 5-1-017 Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies
More informationMaximum likelihood estimation from record-breaking data for the generalized Pareto distribution
METRON - Iteratioal Joural of Statistics 004, vol. LXII,. 3, pp. 377-389 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN Maximum likelihood estimatio from record-breakig data for the geeralized Pareto distributio
More informationNew Correlation for Calculating Critical Pressure of Petroleum Fractions
IARJSET ISSN (Olie) 2393-8021 ISSN (Prit) 2394-1588 Iteratioal Advaced Research Joural i Sciece, Egieerig ad Techology New Correlatio for Calculatig Critical Pressure of Petroleum Fractios Sayed Gomaa,
More informationNumerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
More informationResearch Article Control of Traffic Intensity in Hyperexponential and Mixed Erlang Queueing Systems with a Method Based on SPRT
Mathematical Problems i Egieerig Volume 23, Article ID 2424, 9 pages http://dx.doi.org/.55/23/2424 Research Article Cotrol of Traffic Itesity i Hyperexpoetial ad Mixed Erlag Queueig Systems with a Method
More informationA Block Cipher Using Linear Congruences
Joural of Computer Sciece 3 (7): 556-560, 2007 ISSN 1549-3636 2007 Sciece Publicatios A Block Cipher Usig Liear Cogrueces 1 V.U.K. Sastry ad 2 V. Jaaki 1 Academic Affairs, Sreeidhi Istitute of Sciece &
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationResearch Article A Unified Weight Formula for Calculating the Sample Variance from Weighted Successive Differences
Discrete Dyamics i Nature ad Society Article ID 210761 4 pages http://dxdoiorg/101155/2014/210761 Research Article A Uified Weight Formula for Calculatig the Sample Variace from Weighted Successive Differeces
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationMinimax Estimation of the Parameter of Maxwell Distribution Under Different Loss Functions
America Joural of heoretical ad Applied Statistics 6; 5(4): -7 http://www.sciecepublishiggroup.com/j/ajtas doi:.648/j.ajtas.654.6 ISSN: 6-8999 (Prit); ISSN: 6-96 (Olie) Miimax Estimatio of the Parameter
More informationOptimal Design of Accelerated Life Tests with Multiple Stresses
Optimal Desig of Accelerated Life Tests with Multiple Stresses Y. Zhu ad E. A. Elsayed Departmet of Idustrial ad Systems Egieerig Rutgers Uiversity 2009 Quality & Productivity Research Coferece IBM T.
More informationResearch on Dependable level in Network Computing System Yongxia Li 1, a, Guangxia Xu 2,b and Shuangyan Liu 3,c
Applied Mechaics ad Materials Olie: 04-0-06 ISSN: 66-748, Vols. 53-57, pp 05-08 doi:0.408/www.scietific.et/amm.53-57.05 04 Tras Tech Publicatios, Switzerlad Research o Depedable level i Network Computig
More informationDERIVING THE 12-LEAD ECG FROM EASI ELECTRODES VIA NONLINEAR REGRESSION
DERIVING THE 1-LEAD ECG FROM EASI ELECTRODES VIA NONLINEAR REGRESSION 1 PIROON KAEWFOONGRUNGSI, DARANEE HORMDEE 1, Departmet of Computer Egieerig, Faculty of Egieerig, Kho Kae Uiversity 4, Thailad E-mail:
More information-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION
NEW NEWTON-TYPE METHOD WITH k -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds West Yorkshire, LS7 JS, ENGLAND ABSTRACT The objective
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationLinear Regression Models
Liear Regressio Models Dr. Joh Mellor-Crummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
More informationA Fixed Point Result Using a Function of 5-Variables
Joural of Physical Scieces, Vol., 2007, 57-6 Fixed Poit Result Usig a Fuctio of 5-Variables P. N. Dutta ad Biayak S. Choudhury Departmet of Mathematics Begal Egieerig ad Sciece Uiversity, Shibpur P.O.:
More informationHigher-order iterative methods by using Householder's method for solving certain nonlinear equations
Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem
More informationContinuous Functions
Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More information6340(Print), ISSN (Online) Volume 4, Issue 2, March - April (2013) IAEME AND TECHNOLOGY (IJMET)
INTERNATIONAL Iteratioal Joural of Mechaical JOURNAL Egieerig OF MECHANICAL ad Techology (IJMET), ENGINEERING ISSN 0976 AND TECHNOLOGY (IJMET) ISSN 0976 6340 (Prit) ISSN 0976 6359 (Olie) Volume 4, Issue
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationxn = x n 1 α f(xn 1 + β n) f(xn 1 β n)
Proceedigs of the 005 Witer Simulatio Coferece M E Kuhl, N M Steiger, F B Armstrog, ad J A Joies, eds BALANCING BIAS AND VARIANCE IN THE OPTIMIZATION OF SIMULATION MODELS Christie SM Currie School of Mathematics
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationAdjacent vertex distinguishing total coloring of tensor product of graphs
America Iteratioal Joural of Available olie at http://wwwiasiret Research i Sciece Techology Egieerig & Mathematics ISSN Prit): 38-3491 ISSN Olie): 38-3580 ISSN CD-ROM): 38-369 AIJRSTEM is a refereed idexed
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationELEC1200: A System View of Communications: from Signals to Packets Lecture 3
ELEC2: A System View of Commuicatios: from Sigals to Packets Lecture 3 Commuicatio chaels Discrete time Chael Modelig the chael Liear Time Ivariat Systems Step Respose Respose to sigle bit Respose to geeral
More informationADVANCED SOFTWARE ENGINEERING
ADVANCED SOFTWARE ENGINEERING COMP 3705 Exercise Usage-based Testig ad Reliability Versio 1.0-040406 Departmet of Computer Ssciece Sada Narayaappa, Aeliese Adrews Versio 1.1-050405 Departmet of Commuicatio
More informationA New Multivariate Markov Chain Model with Applications to Sales Demand Forecasting
Iteratioal Coferece o Idustrial Egieerig ad Systems Maagemet IESM 2007 May 30 - Jue 2 BEIJING - CHINA A New Multivariate Markov Chai Model with Applicatios to Sales Demad Forecastig Wai-Ki CHING a, Li-Mi
More informationAs metioed earlier, directly forecastig o idividual product demads usually result i a far-off forecast that ot oly impairs the quality of subsequet ma
Semicoductor Product-mix Estimate with Dyamic Weightig Scheme Argo Che, Ziv Hsia ad Kyle Yag Graduate Istitute of Idustrial Egieerig, Natioal Taiwa Uiversity Roosevelt Rd. Sec. 4, Taipei, Taiwa, 6 ache@tu.edu.tw
More informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
More informationPower Comparison of Some Goodness-of-fit Tests
Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School 7-6-2016 Power Compariso of Some Goodess-of-fit Tests Tiayi Liu tliu019@fiu.edu DOI: 10.25148/etd.FIDC000750
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationFirst Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationDEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES
DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES Icreasig ad Decreasig Auities ad Time Reversal by Jim Farmer Jim.Farmer@mq.edu.au Research Paper No. 2000/02 November 2000 Divisio of Ecoomic ad Fiacial
More informationBecause it tests for differences between multiple pairs of means in one test, it is called an omnibus test.
Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal
More informationQueueing theory and Replacement model
Queueig theory ad Replacemet model. Trucks at a sigle platform weigh-bridge arrive accordig to Poisso probability distributio. The time required to weigh the truck follows a expoetial probability distributio.
More informationMath 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
More informationAssessment of extreme discharges of the Vltava River in Prague
Flood Recovery, Iovatio ad Respose I 05 Assessmet of extreme discharges of the Vltava River i Prague M. Holický, K. Jug & M. Sýkora Kloker Istitute, Czech Techical Uiversity i Prague, Czech Republic Abstract
More informationPAijpam.eu IRREGULAR SET COLORINGS OF GRAPHS
Iteratioal Joural of Pure ad Applied Mathematics Volume 109 No. 7 016, 143-150 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://www.ijpam.eu doi: 10.173/ijpam.v109i7.18 PAijpam.eu
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationPresenting A Framework To Study Linkages Among Tqm Practices, Supply Chain Management Practices, And Performance Using Dematel Technique
Australia Joural of Basic ad Applied Scieces, 5(9): 885-890, 20 ISSN 99-878 Presetig A Framework To Study Likages Amog Tqm Practices, Supply Chai Maagemet Practices, Ad Performace Usig Dematel Techique
More informationVARDHAMAN COLLEGE OF ENGINEERING
VARDHAMAN COLLEGE OF ENGINEERING (Autoomous) Shamshabad, Hyderabad -50 28 S. No UNITS BASIC PROBABILITY THEORY: Rules for combiig probability, Probability Distri butios, Radom variables, desity ad distributio
More informationA STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD
IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: 39-63 pissn: 3-7308 A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD Satish
More informationAlternative Ratio Estimator of Population Mean in Simple Random Sampling
Joural of Mathematics Research; Vol. 6, No. 3; 014 ISSN 1916-9795 E-ISSN 1916-9809 Published by Caadia Ceter of Sciece ad Educatio Alterative Ratio Estimator of Populatio Mea i Simple Radom Samplig Ekaette
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More information( θ. sup θ Θ f X (x θ) = L. sup Pr (Λ (X) < c) = α. x : Λ (x) = sup θ H 0. sup θ Θ f X (x θ) = ) < c. NH : θ 1 = θ 2 against AH : θ 1 θ 2
82 CHAPTER 4. MAXIMUM IKEIHOOD ESTIMATION Defiitio: et X be a radom sample with joit p.m/d.f. f X x θ. The geeralised likelihood ratio test g.l.r.t. of the NH : θ H 0 agaist the alterative AH : θ H 1,
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationThe Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution
Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 1263-1277 HIKARI Ltd, www.m-hikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters
More informationProbability of error for LDPC OC with one co-channel Interferer over i.i.d Rayleigh Fading
IOSR Joural of Electroics ad Commuicatio Egieerig (IOSR-JECE) e-issn: 2278-2834,p- ISSN: 2278-8735.Volume 9, Issue 4, Ver. III (Jul - Aug. 24), PP 59-63 Probability of error for LDPC OC with oe co-chael
More informationFLUID LIMIT FOR CUMULATIVE IDLE TIME IN MULTIPHASE QUEUES. Akademijos 4, LT-08663, Vilnius, LITHUANIA 1,2 Vilnius University
Iteratioal Joural of Pure ad Applied Mathematics Volume 95 No. 2 2014, 123-129 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v95i2.1
More informationSimple Linear Regression
Chapter 2 Simple Liear Regressio 2.1 Simple liear model The simple liear regressio model shows how oe kow depedet variable is determied by a sigle explaatory variable (regressor). Is is writte as: Y i
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationJambulingam Subramani 1, Gnanasegaran Kumarapandiyan 2 and Saminathan Balamurali 3
ISSN 1684-8403 Joural of Statistics Volume, 015. pp. 84-305 Abstract A Class of Modified Liear Regressio Type Ratio Estimators for Estimatio of Populatio Mea usig Coefficiet of Variatio ad Quartiles of
More informationPerformance of Stirling Engines * (Arranging Method of Experimental Results and Performance Prediction) Abstract. 1. Introduction
1 Performace of Stirlig Egies (Arragig Method of Experimetal Results ad Performace Predictio) Shoichi IWAMOTO Koichi HIRATA ad Fujio TODA Key Words: Stirlig Egie, Heat Exchager, Frictio, Egie Desig, Mechaical
More informationx = Pr ( X (n) βx ) =
Exercise 93 / page 45 The desity of a variable X i i 1 is fx α α a For α kow let say equal to α α > fx α α x α Pr X i x < x < Usig a Pivotal Quatity: x α 1 < x < α > x α 1 ad We solve i a similar way as
More informationSolution of Differential Equation from the Transform Technique
Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from
More informationScheduling under Uncertainty using MILP Sensitivity Analysis
Schedulig uder Ucertaity usig MILP Sesitivity Aalysis M. Ierapetritou ad Zheya Jia Departmet of Chemical & Biochemical Egieerig Rutgers, the State Uiversity of New Jersey Piscataway, NJ Abstract The aim
More informationSimulation of Discrete Event Systems
Simulatio of Discrete Evet Systems Uit 9 Queueig Models Fall Witer 2014/2015 Uiv.-Prof. Dr.-Ig. Dipl.-Wirt.-Ig. Christopher M. Schlick Chair ad Istitute of Idustrial Egieerig ad Ergoomics RWTH Aache Uiversity
More informationPreponderantly increasing/decreasing data in regression analysis
Croatia Operatioal Research Review 269 CRORR 7(2016), 269 276 Prepoderatly icreasig/decreasig data i regressio aalysis Darija Marković 1, 1 Departmet of Mathematics, J. J. Strossmayer Uiversity of Osijek,
More information[412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION
[412] A TEST FOR HOMOGENEITY OF THE MARGINAL DISTRIBUTIONS IN A TWO-WAY CLASSIFICATION BY ALAN STUART Divisio of Research Techiques, Lodo School of Ecoomics 1. INTRODUCTION There are several circumstaces
More information