TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN


 Belinda Howard
 10 months ago
 Views:
Transcription
1 HARDMEKO 004 Hardess Measuremets Theory ad Applicatio i Laboratories ad Idustries  November, 004, Washigto, D.C., USA TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN Koichiro HATTORI, Satoshi TAKAGI, Hajime ISHIDA ad Takashi USUDA Natioal Metrology Istitute of Japa, Natioal Istitute of Advaced Idustrial Sciece ad Techology, Tsukuba, Japa Abstract I the early of 003, the traceability system of the Rockwell hardess stadard have bee developed i Japa. The Natioal Metrology Istitute of Japa (NMIJ started to provide the atioal primary hardess stadards, oe of which is the hardess referece blocks, ad the other oe is the accreditatio of the Rockwell hardess tester. I this paper, we preset the detail of ucertaity trasfer betwee the NMIJ ad the secodary stadard laboratories. Keywords: Rockwell hardess, traceability, ucertaity. CALIBRATION SERVICE AND ACCREDITATION PROGRAM IN JAPAN NMIJ started calibratio service of secodary stadard Rockwell testig machie (jcss i 00, ad the Natioal Istitute of Techology ad Evaluatio (NITE also started two accreditatio program of traceability of Rockwell hardess, oe is the traceability of calibratio of testig machies ad the other is calibratio of stadard blocks. The ame of the service is Japa calibratio service system (JCSS. The JCSS accredits compay, supplyig the calibratio service to a customer with the traceable Rockwell hardess values. Because of the traceability system operated uder the measuremetlaw of Japa, jcss ad JCSS have some differece. The NMIJ oly calibrates the secodary stadard Rockwell hardess machie o the jcss calibratio system, ad there are o differece of machies betwee calibratio of stadard blocks ad calibratio of testig machie at this poit. The calibrated laboratories declare their calibratio services (calibratio of testig machie/ stadard block ad their ucertaity, after jcss calibratio. The ucertaity sources of Rockwell hardess block issued by secodary laboratory are show i the table that is calculated for the stadard blocks provided by the secodary laboratory. The terms,, 3 ad 4 is determied as the result of jcss calibratio, certified by NMIJ. The ucertaity of stadard ideter is calculated of combiatio of the accidetal error ad ucertaity of atioal primary ideters whe the ideter is compared with the primary ideter. Fially the declared ucertaity of the Rockwell hardess value is tested uder the accreditatio program of the JCSS traceability system coducted by NITE. The NMIJ certified stadard hardess blocks, which are used i this proficiecy test (Fig.. The details of the ucertaity calculatios ad evaluatio of declared ucertaity are preseted as followig sectios. jcss calibratio NMIJ Primary stadard trasfer stadard Blocks commo ideter with calibratio of. test forces. depth 3. compariso of hardess Secodary stadard Fig. schematic of secodary stadard o the jcss calibratio system. JCSS accreditatio Primary stadard Proefficiecy test to evaluate the declared ucertaity usig E value NMIJ stadard blocks Secodary stadard Fig. evaluatio of declared ucertaity through the proficiecy test o the JCSS accreditatio program. Table. The ucertaity sources of the JCSS certified block. Source of ucertaity Prelimiary test force u F0 Total test force u F 3 Depth measurig device u h 4 Hardess compariso with commo ideter u comp 5 Stadard ideter u Id 6 Hardess variatio of the block u block Sources 4 are evaluated by NMIJ 5,6 are evaluated by secodary laboratory.
2 . UNCERTAINTY OF FORCE To Estimate the ucertaity of the force, we measured the force at three differet heights ad evaluate the ucertaity as u f 3 3 ( fh, mesured h h fstadard, (. where the h is the umber of measured height ad is the umber of repeat at the same height. f stadard is the ad 47 N for the prelimiary test force ad the total test force, respectively. The total stadard ucertaity of force is calculated by the combiatio of ucertaity of force measurig device, u f, as u. (. f u f u f I this method, the ucertaity of the force is estimated as a total stadard deviatio from the stadard values. The advatage of this method is that it ca be all measured deviatio ca be cosidered, ad the bias compoet ca be also cosidered. The coverage factor is useful whe the ucertaity is give oly by the accidetal error i the measuremet. I additio the treatmet of the effect of ocorrected bias compoet is ot easy. We believe that this method is oe of the best solutios of the ucertaity evaluatio icludig the ocorrected bias compoet. 3. UNCERTAINTY OF THE DEPTHMEASUREING DEVICE Depth measurig device is evaluated as follows: determiatio of the zero poit as a set positio, where the equivalet of 00HRC. After the depth measurig device is oce moved to the dowward over the 00 m (0 HRC ad after moved to measuremet positio (ex. 00 m. The the measuremet was carried out at each positio that iterval is the 0 m (~0HRC. The corrected depth measurig data was evaluated as, uh 3 ( hij i j hstd i (3. where the i is the positio ad the j is the umber of repeats, ad the is the total umber of measuremet, respectively. The total ucertaity of the depthmeasurig device is give by the combiatio of the stadard ucertaity of the depthcalibratio device, u h, ad the ucertaity due to the resolutio of the depthmeasurig device (or hardess idicator, u hres ad u h, h uhstd uh uhres u. (4. 4. UNCERTAINTY OF THE INDENTER The ideter characteristic is the most sigificat parameter for measuremet value, eve if the ideter is suitable for the ISO requiremets []. 4. Stadard ideters It is recommeded that the each secodary laboratory should have oe or more stadard ideters. The defiitio of the stadard ideter i the JCSS system is: the ideters a are suitable for the ISO requiremet [] ad b have the correctio values for each hardess levels. The hardess values of the JCSS certified blocks expressed as a corrected hardess values. To determie the correctio values the compariso of the hardess values betwee the atioal primary ideter are carried out usig stable testig machie ad made a sufficiet umber of idetatios to the same blocks []. The correctio values are determied as a bias from the atioal primary ideters. The correctio values are determied 0HRC iterval, from 0 to 65 HRC, typically. The atioal primary ideters also have correctio values. This is based o the followig assumptios that the hardess variatio of each ideter comes from the variatio of the cotact area ad this variatio ca be comparable if the stable testig machie is used to the compariso. 4. Ucertaity of atioal primary stadard ideters The NMIJ has about 30 atioal primary stadard ideters. These ideters have the correctio values, which value is defied as a hardess deviatio from the ideal stadard ideter [3]. The detail of the determiatio of the correctio values is give i the other presetatio[4]. We itroduce it briefly: There are six characteristic parameters for each ideter, the five differet average curvatures (for five differet average agle for the sphere part of the ideter ad oe parameter for the coe agle. May actual shape ideters are prepared ad the hardess measuremet was carried out for each ideter. The multiple regressio aalysis was carried out usig followig model, H i a5 ( p5i a ( pi a6 ( p6i a ( pi 0 i 00..., (5. where H i ad are the hardess values of measured ad ideal for the block, respectively. The parameters p i is the measured characteristic parameters ad a is the coefficiets of the regressio. The error term icludes accidetal error ad the error due to the hardess variatio of the positio. The ideal hardess is estimated, as that is the value of the regressio curve at characteristic parameters set to that of ideal shape ideter, i.e. 00 m for curvature ad 0 for coe agle. The ucertaity of the ideter is determied by the remaied error of the regressio aalysis. The stadard ucertaity of correctio values of the atioal primary stadard ideters is about 0.3 HRC at 60HRC level, ad it depeds o the hardess levels due to the differece of ouiformity of the blocks. 5. COMPARISON OF HARDNESS VALUES USING COMMON INDENTER
3 To determie the total ucertaity of testig machie of secodary laboratory, we compare the measuremet values of same blocks usig commo ideter. The total deviatio from the primary hardess tester icludig test cycle effect is evaluated through the compariso. 5. Hardess compariso usig 4d method The NMIJ adopted the followig method for the hardess compariso, the method had developed by the oe of the authors, ad we call it 4d method. Geerally, the hardess varies getly i the block. Therefore the differece of measured values will be very small with decreasig the iterval of two idetatios. Because of the hardess chage due to the preperformed idetatio, the limit of actual distace betwee the two idetatios will be 4 times of diameter of idetatio. This iterval betwee the idets is the origi of the ame of this method. I the 4d method, the hardess differece betwee two idetatios ca be regarded as a accidetal error. The theoretical of this compariso ca be calculated as follows: The hardess values of secod idetatio, H i ad correspodig referece idetatio, H i is give by, Hi H i m m pij ij ik (6. where is the ideal hardess value of the block, p i is the hardess variatio depedig o the idet positio i, m is the hardess variatio due to the differeces of machies, i is the accidetal error ad, deote the referece ad secod idetatios, respectively. The hardess differece of the correspodig poit is give by, H i H i Hi m m pij ik ij, (7. ad the mea value of the hardess differece is, H m m. (8. Here, we eglect terms of the mea values of the p i ad i, usig after th measuremets usig, pi i 0 ad i i 0. (9. The stadard deviatio of the compariso usig 4d method will be expressed as ( H i H ( i i pij ik ij. (0. Cosider the idepedece betwee p i ad i, ij ad ik.. It ca be assumed that the amplitude of the accidetal error will be the same. We obtai, i ( pij i pij (. O the other had, the compariso of the hardess values is carried out usig the mea hardess values of the block. We also calculate the deviatio of the mea hardess compariso. The mea hardess values are give by, H i H i m m (. Here the mea terms of the p i ad i are eglected. The differece betwee the mea values m ad m is give by the same as that obtaied by the 4d method, H m m. (3. The stadard deviatio of the mea hardess value for the referece H i is give by,. ( Hi H i ( pij ij ( pij i i i (4. Here we use the result of above discussio, the cross terms ca be eglected ad the amplitude of stadard deviatio of the accidetal error is set to the. The total stadard deviatio of the mea hardess compariso is give by the addig the two idepedet deviatios of two measuremets, pi i i. (5. The stadard deviatio of the 4d method is decreased whe the correlatio betwee the p ij ad p ik becomes strog. The result of 4d method is correspodig to that of mea hardess compariso if the ocorrelatio is observed, it is the worst deviatio limit of the 4d method. We also check the validity of the method, experimetally. 5. experimetal evaluatio of validity of 4d method The atioal primary hardess tester SHT3 Rockwell hardess tester was used i the experimet. SHT3 is the leveramplified dead weight type ad the holographic gages used to depth measurig device. The omial hardess levels are 64, 60, 40, 30 ad 0 HRC. The hardess measuremet was carried out four times; the itervals of the measuremets are about oeweek for the d measuremet, about fourmoths for the d3rd ad oeweek for the 3rd 4th measuremet. Table shows the typical example of measuremet obtaied the 40HRC block. The d4th idetatio data are obtaied by the measuremet usig 4d method aroud the st idet. The hardess variatio of the each measuremet poit is show i the Fig. 3. The coectig lie shows a measured hardess observed i the same day. The hardess variatio depedig o the measuremet positio shows the similar tedecy. That meas the correlatios betwee the idetatios may be strog.
4 Table. The hardess observed usig 4d method for 40HRC block. Idet positio st 0 week d week 3rd 6 weeks 4th 7 weeks Average Idet positio Fig. 3. The hardess observed usig 4d method. st d Table. Oeway aalysis of variace for 40HRC block. Idet positio factor ss dof variace p value betwee groups E05 withi groups rd Measured day factor ss dof variace p value betwee groups E0 withi groups The result of oeway aalysis of variace is show i table. We regardig that the lie data ad the row data as the same group i the aalysis of idet positio ad measured day i the table, respectively. The 4d method ad the mea hardess compariso are correspodig to the idet positio ad the measured day i the table, respectively. The aalysis idicates that the effect of testig machie istability depedig measuremet day is much smaller tha that effect of the hardess variatio of the block. The calculated stadard deviatios of these comparisos are 0.06 HRC for 4d method ad 0.5 HRC for the mea hardess compariso ad that is directly reflected the ucertaity of the calibratio. The ucertaity of the trasfer block usig hardess compariso is estimated as about 0.07 HRC. As the largest ucertaity source, the ideter is eglected because the ideter is commoly used i the compariso. The result of oeway aalysis obtaied other hardess levels are show i table 3. The p values of the compariso 4th is small exceptig 60HRC block that is because the 60HRC block is very good so that the measured rage of hardess variatio was very small (0.HRC. The the effect of 4d method looks slightly small. Table 3. The oeway aalysis of variace for 4d method. 64HRC factor ss dof variace p value betwee groups E0 withi groups HRC factor ss dof variace p value betwee groups E0 withi groups HRC factor ss dof variace p value betwee groups E03 withi groups HRC factor ss dof variace p value betwee groups E0 withi groups Fially the ucertaity of hardess compariso usig 4d method is give by, ucomp _ ( H i i Hi. (6. where, the hardess value, H i is compared with correspodig refereceidetatio, H i ad is the umber of idets. The the ucertaity of compariso is determied after combied with the ucertaity of trasfer stadard block, u comp_. comp ucomp _ ucomp _ u (7. The u comp_ is cosidered a lower limit of ucertaity, if the measuremet values equal to the stadard oe ad is about 0.07 HRC that is determied by the direct calibratio result of NMIJ without ideter ad discussed stadard deviatio of the block usig 4d method. The ucertaity of compariso will be expected about 0.09 HRC usig u comp_ =0.07 HRC ad u comp_ = 0.06 HRC whe the same tester is used i the compariso. 6 PROFICIENCY TEST 6. Proficiecy test to evaluate the declared ucertaity As metioed at sectio, the declared ucertaity is evaluated by the blid test coducted by the NITE. The stadard block certified by the NMIJ is used i the test. Geerally the a ideter is selected from the atioal primary stadard ideters, arbitrary, to determiatio of the
5 hardess of block, ad that meas the hardess values of the block is dispersed artificially. The hardess of the block is determied as a mea value of the six idetatios; we perform oe idetatio for each sectio. The secodary laboratory also perform six idetatios for each sectio, the idetatio positio is selected arbitrary iside the sectio. The sectio of the block is show i the fig M Fig. 4 The sectio of the block. 6. The evaluatio of the JCSS ucertaity I this subsectio, we show the result of the simulative hardess blid test, performed at 0, 40 ad 60HRC levels. The hardess levels of compariso are selected to cover the Rockwell hardess rages. The calculated JCSSequivalet ucertaity is evaluated experimetally usig E value, H E (8. U NMIJ U JCSS Here, the U NMIJ ad U JCSS are the expaded combied ucertaities usig coverage factor (k=. The U NMIJ is our declared expaded ucertaity of the Rockwell hardess block calibratio. The U JCSS is the calculated expadig ucertaity followig the metioed method. The sesitivity coefficiets of the ref.[5] is used i the ucertaity calculatio. Two differet testers are used i the evaluatio, oe of which is the (A NMIJ primary hardess machie, SHT3, ad the other oe is the (B commercially available tester, which oe of the participated machie for the roud robi test of the JCSS program. The features of that is the leveramplified dead weight type ad the prelimiary test force is applied by the sprig. The result of the tester A is show i table 5. The ucertaity equivalet to the jcss certificate is determied usig metioed ucertaity estimatio. The uit of ucertaity i the tables are HRC. The ucertaity of the hardess compariso is determied experimetally usig 4d method ad commo ideter. The determied ucertaity of compariso i the table 5(a is about HRC, which value is almost same expected i the 5.. The total ucertaity of tester A idicated the table 5(b, ad is equivalet to the ucertaity of JCSS calibratio of block. I the jcss is ot ivolvig the ucertaity of ideter, the the ucertaity of the ideter is combied at here. Fially the result of the JCSS ucertaity is tested by the E value. The differece of hardess betwee the stadard hardess blocks ad the hardess of tested machie is about 0., which is caused by the differece of ideter i this case. The E values are less tha 0.3. I the table 6, we show the result obtaied by the commercial tester usig the metioed calibratio scheme. The expected ucertaity, show i the table 6(b, is equivalet to that of JCSS calibratio of blocks. The machie used i this evaluatio is ot suitable for the ISO part 3[], however, the all E values show less tha, ad that is earby the E = 0.5. These results show the metioed scheme may estimate the ucertaity of Rockwell hardess correctly. Table 5. The simulative estimatio of total ucertaity for NMIJ primary hardess tester (A, uit of ucertaity is HRC. (a The ucertaity equivalet to the jcss Source of ucertaity 0HRC 40HRC 60HRC Prelimiary test force, u F Total test force, u F Depth measurig device, u h Hardess compariso with commo ideter, u comp Comb. std. Ucertaity Exp. std. Ucertaity (k= (b Total ucertaity of simulative d machie (JCSS Source of ucertaity 0HRC 40HRC 60HRC Machie (jcss for A Stadard ideter, u std ouiformity of blocks, u b Comb. std. Ucertaity Exp. std. Ucertaity (k= (c E Source of ucertaity 0HRC 40HRC 60HRC Differece of Hardess Exp. ucertaity of d Lab Exp. ucertaity of NMIJ E value Table 6. The simulative estimatio of total ucertaity for commercial type hardess tester (B, uit of ucertaity is HRC. (a Expected ucertaity of commercial tester (B Source of ucertaity 0HRC 40HRC 60HRC Machie (jcss for B Stadard ideter, u std ouiformity of blocks, u b Comb. std. Ucertaity Exp. std. Ucertaity (k= (b E Source of ucertaity 0HRC 40HRC 60HRC Differece of Hardess Exp. ucertaity of d Lab Exp. ucertaity of NMIJ E value
6 7. CONCLUDING REMARKS The traceability of Rockwell hardess ad ucertaity trasfer system i Japa is preseted. The ucertaity calculatio method is verified two differet simulative experimets. We also verify the validity of this ucertaity trasfer method through the actual JCSS accreditatio program. The Rockwell hardess traceability system i Japa is just started. The JCSS accredited compaies are rapidly icreased ad may compaies are waitig jcss calibratio. Because of the jcss allows oly the direct calibratio of machie by the NMIJ. We are tryig to develop aother ucertaity estimatio system usig their certified calibratio devices ad certified hardess blocks. REFERENCES [] ISO6508,3,"Metallic materials Rockwell hardess test ". [] H. Ishida et al., "The characters of the Rockwell diamod ideters ad tatalizatio method" J. Mater. Test. Res., No.3, pp.96, 978. [3] H. Yao et al., "Characterizatio of stadard Rockwell diamod ideters ad method of establishig stadard ideters", proceedigs of the roudtable discussio o hardess testig 7th, IMEKO, Lodo, pp. 650, 996. [4] S. Takagi et al., Direct Verificatio ad Calibratio of Rockwell Diamod Coe Ideters, proceedigs of Hardmeko 004, submitted to Hardmeko 004. [5] Europea cooperatio for Accreditatio, EA0/6, "EA Guidelies o the Estimatio of Ucertaity i Hardess Measuremets", 00 [6] The detail of JCSS system please refer: Authors: Dr. Koichiro Hattori, NMIJ, AIST Address: AIST Tsukuba Cet.3, Umezoo  Phoe: , FAX: Mr. Satoshi TAKAGI, NMIJ, AIST, Address: AIST Tsukuba Cet.3, Umezoo  Phoe: , FAX: Mr. Hajime ISHIDA, NMIJ, AIST Address: AIST Tsukuba Cet.3, Umezoo  Phoe: , FAX: Dr. Takashi USUDA, NMIJ AIST Address: AIST Tsukuba Cet.3, Umezoo  Phoe: , FAX:
OBJECTIVES. Chapter 1 INTRODUCTION TO INSTRUMENTATION FUNCTION AND ADVANTAGES INTRODUCTION. At the end of this chapter, students should be able to:
OBJECTIVES Chapter 1 INTRODUCTION TO INSTRUMENTATION At the ed of this chapter, studets should be able to: 1. Explai the static ad dyamic characteristics of a istrumet. 2. Calculate ad aalyze the measuremet
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
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 information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationAssessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions
Assessmet ad Modelig of Forests FR 48 Sprig Assigmet Solutios. The first part of the questio asked that you calculate the average, stadard deviatio, coefficiet of variatio, ad 9% cofidece iterval of the
More informationSome Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation
Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 705041010,
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 informationActivity 3: Length Measurements with the FourSided Meter Stick
Activity 3: Legth Measuremets with the FourSided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a foursided meter
More information71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationStat 200 Testing Summary Page 1
Stat 00 Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece
More information(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
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 informationAPPENDIX A EARLY MODELS OF OXIDE CMP
APPENDIX A EALY MODELS OF OXIDE CMP Over the past decade ad a half several process models have bee proposed to elucidate the mechaism ad material removal rate i CMP. Each model addresses a specific aspect
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationExample 3.3: Rainfall reported at a group of five stations (see Fig. 3.7) is as follows. Kundla. Sabli
3.4.4 Spatial Cosistecy Check Raifall data exhibit some spatial cosistecy ad this forms the basis of ivestigatig the observed raifall values. A estimate of the iterpolated raifall value at a statio is
More informationRademacher Complexity
EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for
More informationPaired Data and Linear Correlation
Paired Data ad Liear Correlatio Example. A group of calculus studets has take two quizzes. These are their scores: Studet st Quiz Score ( data) d Quiz Score ( data) 7 5 5 0 3 0 3 4 0 5 5 5 5 6 0 8 7 0
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationDISTRIBUTION LAW Okunev I.V.
1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated
More informationMedian and IQR The median is the value which divides the ordered data values in half.
STA 666 Fall 2007 Webbased Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5umber summary mea ad stadard deviatio Media
More informationElementary Statistics
Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform largesample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationUsing An Accelerating Method With The Trapezoidal And MidPoint Rules To Evaluate The Double Integrals With Continuous Integrands Numerically
ISSN 50 (Paper) ISSN 505 (Olie) Vol.7, No., 017 Usig A Acceleratig Method With The Trapezoidal Ad MidPoit Rules To Evaluate The Double Itegrals With Cotiuous Itegrads Numerically Azal Taha Abdul Wahab
More informationA goodnessoffit test based on the empirical characteristic function and a comparison of tests for normality
A goodessoffit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
More information6.867 Machine learning, lecture 7 (Jaakkola) 1
6.867 Machie learig, lecture 7 (Jaakkola) 1 Lecture topics: Kerel form of liear regressio Kerels, examples, costructio, properties Liear regressio ad kerels Cosider a slightly simpler model where we omit
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationBHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13
BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the
More information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationMOMENTMETHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMETMETHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationDefinitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.
Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,
More informationMeasures of Spread: Variance and Standard Deviation
Lesso 16 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the zaxis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationA LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!
A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited
More informationMeasuring Scales. Measuring Scales
Measurig Scales To measure a legth, a metre scale is geerally used, which is graduated to cetimeter ad millimeter, ad is oe metre i legth. For the measuremet of a legth with a metre scale we adopt the
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationDiscrete probability distributions
Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
More informationOFFSET CORRECTION IN A DIGITAL INTEGRATOR FOR ROTATING COIL MEASUREMENTS
XX IMEKO World Cogress Metrology for Gree Growth September 9 4, 0, Busa, Republic of Korea OFFSET CORRECTION IN A DIGITAL INTEGRATOR FOR ROTATING COIL MEASUREMENTS P. Arpaia,, P. Cimmio,, L. De Vito, ad
More informationSALES AND MARKETING Department MATHEMATICS. 2nd Semester. Bivariate statistics LESSONS
SALES AND MARKETING Departmet MATHEMATICS d Semester Bivariate statistics LESSONS Olie documet: http://jffduttc.weebly.com sectio DUT Maths S. IUT de SaitEtiee Départemet TC J.F.Ferraris Math S StatVar
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationSimple Linear Regression
Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio
More informationy ij = µ + α i + ɛ ij,
STAT 4 ANOVA Cotrasts ad Multiple Comparisos /3/04 Plaed comparisos vs uplaed comparisos Cotrasts Cofidece Itervals Multiple Comparisos: HSD Remark Alterate form of Model I y ij = µ + α i + ɛ ij, a i
More informationInformativeness Improvement of Hardness Test Methods for Metal Product Assessment
IOP Coferece Series: Materials Sciece ad Egieerig PAPER OPEN ACCESS Iformativeess Improvemet of ardess Test Methods for Metal Product Assessmet To cite this article: S Osipov et al 016 IOP Cof. Ser.: Mater.
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3337 HIKARI Ltd, www.mhikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationVERTICAL MOVEMENTS FROM LEVELLING, GRAVITY AND GPS MEASUREMENTS
rd IAG / 2th FIG Symposium, Bade, May 2224, 26 VERTICAL MOVEMENTS FROM LEVELLING, GRAVITY AND GPS MEASUREMENTS N. Hatjidakis, D. Rossikopoulos Departmet of Geodesy ad Surveyig, Faculty of Egieerig Aristotle
More informationEDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR
Joural of Statistical Research 26, Vol. 37, No. 2, pp. 4355 Bagladesh ISSN 256422 X EDGEORTH SIZE CORRECTED, AND TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Zahirul Hoque Departmet of Statistics
More informationESTIMATION AND PREDICTION BASED ON KRECORD VALUES FROM NORMAL DISTRIBUTION
STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON KRECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum 695581, Kerala, Idia M. Shy
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationTesting Statistical Hypotheses with Fuzzy Data
Iteratioal Joural of Statistics ad Systems ISS 973675 Volume 6, umber 4 (), pp. 44449 Research Idia Publicatios http://www.ripublicatio.com/ijss.htm Testig Statistical Hypotheses with Fuzzy Data E. Baloui
More informationA Simplified Binet Formula for kgeneralized Fibonacci Numbers
A Simplified Biet Formula for kgeeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More informationSEQUENCES AND SERIES
9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
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 I1 Part I  203A A radom variable X is distributed with the margial desity: >
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Twoparameter 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 informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationDecoupling Zeros of Positive DiscreteTime Linear Systems*
Circuits ad Systems,,, 448 doi:.436/cs..7 Published Olie October (http://www.scirp.org/oural/cs) Decouplig Zeros of Positive DiscreteTime Liear Systems* bstract Tadeusz Kaczorek Faculty of Electrical
More informationA RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider ksample ad chage poit problems for idepedet data i a
More informationBounds of HerfindahlHirschman index of banks in the European Union
MPRA Muich Persoal RePEc Archive Bouds of HerfidahlHirschma idex of bas i the Europea Uio József Tóth Kig Sigismud Busiess School, Hugary 4 February 2016 Olie at https://mpra.ub.uimueche.de/72922/ MPRA
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry DaaPicard Departmet of Applied Mathematics Jerusalem College of Techology
More informationCTL.SC0x Supply Chain Analytics
CTL.SC0x Supply Chai Aalytics Key Cocepts Documet V1.1 This documet cotais the Key Cocepts documets for week 6, lessos 1 ad 2 withi the SC0x course. These are meat to complemet, ot replace, the lesso videos
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 informationTo the use of Sellmeier formula
To the use of Sellmeier formula by Volkmar Brücker Seior Experte Service (SES) Bo ad HfT Leipzig, Germay Abstract Based o dispersio of pure silica we proposed a geeral Sellmeier formula for various dopats
More informationNotes on the use of the GoldfeldQuandt test for heteroscedasticity in environment research
Biometrical Letters Vol. 45(008), No., 4355 Notes o the use of the GoldfeldQuadt test for heteroscedasticity i eviromet research Aa Budka, Dauta Kachlicka, Maria Kozłowska Departmet of Mathematical ad
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol DiscreteEvent System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol DiscreteEvet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationII. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation
II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso productmomet correlatio
More informationChapter Objectives. Bivariate Data. Terminology. Lurking Variable. Types of Relations. Chapter 3 Linear Regression and Correlation
Chapter Objectives Chapter 3 Liear Regressio ad Correlatio Descriptive Aalysis & Presetatio of Two Quatitative Data To be able to preset twovariables data i tabular ad graphic form Display the relatioship
More informationHOMEWORK #10 SOLUTIONS
Math 33  Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More informationUCLA STAT 110B Applied Statistics for Engineering and the Sciences
UCLA SA 0B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology eachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More information(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)
Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationCS / MCS 401 Homework 3 grader solutions
CS / MCS 401 Homework 3 grader solutios assigmet due July 6, 016 writte by Jāis Lazovskis maximum poits: 33 Some questios from CLRS. Questios marked with a asterisk were ot graded. 1 Use the defiitio of
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationVaranasi , India. Corresponding author
A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationStatistical Test for Multidimensional Uniformity
Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School 020 Statistical Test for Multidimesioal Uiformity Tieyog Hu Florida Iteratioal Uiversity,
More informationTable 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab
Sectio 12 Tests of idepedece ad homogeeity I this lecture we will cosider a situatio whe our observatios are classified by two differet features ad we would like to test if these features are idepedet
More informationLecture 9: Independent Groups & Repeated Measures ttest
Brittay s ote 4/6/207 Lecture 9: Idepedet s & Repeated Measures ttest Review: Sigle Sample ztest Populatio (otreatmet) Sample (treatmet) Need to kow mea ad stadard deviatio Problem with this? Sigle
More informationROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND
PacificAsia Joural of Mathematics, Volume 5, No., JauaryJue 20 ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND SHAKEEL JAVAID, Z. H. BAKHSHI & M. M. KHALID ABSTRACT: I this paper, the roll cuttig problem
More informationTrue Nature of Potential Energy of a Hydrogen Atom
True Nature of Potetial Eergy of a Hydroge Atom Koshu Suto Key words: Bohr Radius, Potetial Eergy, Rest Mass Eergy, Classical Electro Radius PACS codes: 365Sq, 365w, 33+p Abstract I cosiderig the potetial
More informationMECHANICAL INTEGRITY DESIGN FOR FLARE SYSTEM ON FLOW INDUCED VIBRATION
MECHANICAL INTEGRITY DESIGN FOR FLARE SYSTEM ON FLOW INDUCED VIBRATION Hisao Izuchi, Pricipal Egieerig Cosultat, Egieerig Solutio Uit, ChAS Project Operatios Masato Nishiguchi, Egieerig Solutio Uit, ChAS
More informationEstimation of the Population Mean in Presence of NonResponse
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 NoRespose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,
More informationVerification of continuous predictands
barbara.casati@ec.gc.ca Verificatio of cotiuous predictads Barbara Casati 9 Ja 007 Exploratory ethods: joit distributio Scatterplot: plot of observatio versus forecast values Perfect forecast obs, poits
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationCENTRIFUGAL PUMP SPECIFIC SPEED PRIMER AND THE AFFINITY LAWS Jacques Chaurette p. eng., Fluide Design Inc. November 2004
CENTRIFUGAL PUMP SPECIFIC SPEE PRIMER AN THE AFFINITY LAWS Jacques Chaurette p. eg., Fluide esig Ic. November 004 www.fluidedesig.com There is a umber called the specific speed of a pump whose value tells
More informationCorrelation and Covariance
Correlatio ad Covariace Tom Ilveto FREC 9 What is Next? Correlatio ad Regressio Regressio We specify a depedet variable as a liear fuctio of oe or more idepedet variables, based o covariace Regressio
More informationNonparametric Tests for Two Factor Designs
Uiversity of Wollogog Research Olie Applied Statistics Educatio ad Research Collaboratio (ASEARC)  Coferece apers Faculty of Egieerig ad Iformatio Scieces 011 Noparametric Tests for Two Factor Desigs
More informationLainiotis filter implementation. via Chandrasekhar type algorithm
Joural of Computatios & Modellig, vol.1, o.1, 2011, 115130 ISSN: 17927625 prit, 17928850 olie Iteratioal Scietific Press, 2011 Laiiotis filter implemetatio via Chadrasehar type algorithm Nicholas Assimais
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationGini Index and Polynomial Pen s Parade
Gii Idex ad Polyomial Pe s Parade Jules Sadefo Kamdem To cite this versio: Jules Sadefo Kamdem. Gii Idex ad Polyomial Pe s Parade. 2011. HAL Id: hal00582625 https://hal.archivesouvertes.fr/hal00582625
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More information