COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS

Size: px
Start display at page:

Download "COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS"

Transcription

1 COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat Uiversity, Bagkok, Thailad Puamee Sachakamol Iteratioal Graduated Program for Idustrial Egieerig (IGP), Departmet of Idustrial Egieerig, Faculty of Egieerig, Kasetsart Uiversity, Bagkok, Thailad Abstract: Statistical Process Cotrol (SPC) is a method of quality cotrol which uses statistical methods to moitorig ad cotrollig the process to esure that the maufacturig lie operates at its full potetial. At its full potetial, the process ca be applied to make as much coformig product as possible with miimum waste. However, the quality of the process cotrol is determied by samples chose ad Cotrol Chart tools whereas this research aims to determie ad compariso the efficiecy of S-Cotrol Chart ad EWMA-S Chart. The determiatio factors are fluctuatio i variaces, differet sizes of samples, ad variatios of samplig method. From, iteratio ru i R Versio.. with Media-Set Samplig, variaces rage from. to 3., the EWMA-S Cotrol Chart yield higher efficiecy i every possible sceario tested. Keywords: cotrol chart, quality cotrol, productio, statistical process cotrol, radom samplig, stadard deviatio 63

2 . THE ORIGIN AND SIGNIFICANCE OF THE PROBLEM Geeral maufacturig process teds to always vary. This variatio will cause the quality of the product varies as well. The variatio sometimes is ot so much. Maufacturer allows this to happe because it does ot affect the overall quality of the product. However, if the variatio is severe ad is ot acceptable because it may make the product becomes a waste or caot be used eve though there have bee attempts to reduce or cotrol the variatio i each factor, but it still has the variatio evetually i the maufacturig process. Therefore, the variatio should be cotrolled, which is the quality cotrol to esure that the quality of products meets the required stadards or is i withi the acceptable limit. Statistical Process Cotrol or SPC is the implemetatio of statistical process to cotrol the maufacturig process so that the properties of products i the maufacturig process is cosistet as well as raisig maufacturig capacity or reducig the variatio of the maufacturig process util the properties of products meet or close to the stadards, which SPC ow is very popular for aalyzig the performace of the maufacturig process. (Shewhart, 939) has developed a cotrol chart, which is cosidered as the mai tool of SPC by relyig o statistical methods. The mai fuctio of cotrol chart is to detect variatios i the maufacturig process ad to be a tool to idicate how the maufacturig process operates, whether variatio occurs out of cotrol or is egligible, which is cosidered i cotrol. This cotrol chart, if the maufacturer uses to cotrol the maufacturig process regularly, will prevet quality problems well sice the cotrol chart will detect chages i the maufacturig process i a timely maer allowig maufacturers to rapidly kow maufacturig coditios. I the evet of a problem, it ca stop the maufacturig process ad improve it. I this study, 4 samplig plas will be used icludig Simple Radom Samplig (SRS), Systematic Samplig (SYS), Raked-Set Samplig (RSS) ad Media-Set Samplig (MSS) by usig R software package for simulatig radom samplig of each pla as a represetative for moitorig ad create a cotrol chart. From the study of the chart above, the researcher was iterested to study the process with little chage i the stadard deviatio with stadard deviatio cotrol chart ad expoetially weighted movig average cotrol chart for examiig the variatio of the process fiacial. The study foud that there was o compariso of the two cotrol charts uder the circumstaces of the maufacturig process i the stadard deviatio with egligible chage at the same time ad the researcher will study differet sample sizes i order to compare the performace of all cotrol charts with the Average Ru Legth() at each samplig pla.. RELATED THEORIES AND RESEARCHES I this study, the researcher was iterested i several types of radom samplig to create S cotrol charts ad expoetially weighted movig average cotrol chart for measurig the variace of the process (EWMA-S Cotrol Chart) uder data with a ormal distributio by comparig cotrol charts obtaied from various samplig methods. The related theories ad researches are as follows. (Maravelakis & Castagliola, 9) foud that EWMA cotrol chart is more effective tha other charts if is low for ay costat i chages ( ) τ whe are equal i all charts i which low λ value is suited to detect small to medium chages ad high λ value is suited for large-scale chage τ.8. ( ) More researches are also compared the stregth betwee the expoetially weighted Shewhart average cotrol chart ad sythetic by usig Average Ru Legth () that is out of cotrol as a basis o makig a decisio from the coclusios. The distributios made the expoetially weighted average cotrol chart the most effective. 64

3 (Tawatira & Mayureesawa, ) studied the cotrol charts by Raked-set Samplig ad compared the value from data simulatio of 5 charts which were x chart, x RSS Chart, MRSS Chart, x MSS Chart, ad MMSS Chart ad showed the results at some m values adσ at all k levels ad at specified δ value i the experimet pla. The results that the average of the maufacturig process has ot chaged (δ = ) was foud that at all levels of k, m, σ the MRSS-Chart had the best ability to moitor the maufacturig process sice it gave the value closest to the target value of 37 while x RSS, Chart, x MSS Chart, ad MMSS Chart และ x Chart has subsequetly lower ability to moitor the maufacturig process, havig the value more differet from 37, respectively. 3. RESEARCH OPERATION I this research the researcher wated to study the performace of S-cotrol charts ad expoetially weighted movig average cotrol chart for measurig the variace of a process (EWMA-S Cotrol Chart) for a process with a chage i the stadard deviatio of various samplig methods by simulatig data with differet samplig methods. The the average ru legth () was calculated usig R software to compare the performace of both charts. The followigs are the procedure to coduct the research. 3.. Procedure The procedure is divided as follows: Data simulatio: Data simulatio was performed usig R software package versio 3.. to simulate data with ormal distributio, average equals to, variace rages from. to 3. ad icreases at. icremet(.,.,.4,.6,.8,.,.,.4,.6,.8,.,.,.4,.6,.8, 3.) for samples (m = ). Sampligs were doe usig the followig methods:. Simple Radom Samplig (SRS). Systematic Samplig (SYS) 3. Raked-Set Samplig (RSS) 4. Media-Set Samplig (MSS) The sample size was desigated as 5,, 5, 3, respectively ( = 5,, 5, 3). Specificatio of Cotrol Rage. Cotrol Chart (S Cotrol Chart): The cotrol rage for S cotrol chart was LCL = B3 S UCL = B4 S i which B 3 ad B 4 ca be opeed from the appedix of the Factors for Costructig Variables Cotrol Charts.. Expoetially Weighted Movig Average Cotrol Chart for Moitorig the Process Variace (EWMA-S Cotrol Chart): The cotrol rage for EWMA-S Cotrol Chart was LCL = UCL = σ + c λ λ i which c is the critical value ad k is the degree of freedom ad k = -. Average Ru Legth () is calculated from M = RL i M i= Where M is the umber of experimet i each situatio i order to fid, RLi is the umber of samples moitored util the maufacturig process is out of cotrol at experimet. σ k 65

4 To fid, Mote Carlo simulatio techique was used with the followig procedure. Cotrol Chart(S Cotrol Chart). Simulate data ~ N( σ ) X, j, i with variace from. to3. at. icremet at = 5,, 5, 3 for samples.. Specify the lower cotrol limit LCL = B3 S ad upper cotrol limitucl = B4 S 3. Calculate statistical values from the chart from s i = s i = j= j= ( x x) i, j x i, j or ( xi, j ) j= ( ) 4. Cosider statistical value LCL < s i < UCL (the statistical value is uder cotrol) whe t = util LCL < s i ad s i > UCL (the statistical value is out of cotrol) ad cout the ru legth (RL). 5. Repeat steps, 3, ad 4 util M =,. 6. Calculate. Expoetially Weighted Movig Average Cotrol Chart for Moitorig the Process Variace (EWMA- S Cotrol Chart). Simulate data X i, j ~ N(, σ ) with variace from. to3.at. icremet at = 5,, 5, 3 for samples.. Specify the upper cotrol limit UCL σ + c λ λ σ k = where c = 3 ad k = - Specify the lower cotrol limit LCL = 3. Calculate Expoetially Weighted Movig Average Cotrol Chart for Moitorig the Process Variace (EWMA) zi = ( λ) zi + λvi, i where λ =. ad v i = l s i. 4. Cosider statistical value LCL < zi < UCL (the statistical value is uder cotrol) whe t = util LCL < ad z i > UCL (the statistical value is out of cotrol) ad cout the ru legth zi (RL). 5. Repeat steps, 3 ad4util M =,. 6. Calculate. 3.. Compariso of the chart efficiecy: I this study, was used to compare the performace of S Cotrol Chart ad expoetially weighted movig average cotrol chart (EWMA-S Cotrol Chart) for measurig the process variace for a process that has a chage i the stadard deviatio for various sample methods, where is the average umber of samples to be moitored util the discovery of the out of cotrol ad ca be determied as follows: If is high the the chart has low efficiecy. If is low the the chart has high efficiecy. 4. RESULTS This research aims to study ad compare the performace of S Cotrol Chart ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace i differet situatio such as whe the data has a chage i the stadard deviatio at differet levels, the data has differet sample sizes, ad the data has differet samplig methods i 66

5 order to draw coclusios about the performace of S Cotrol Charts ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace usig the average ru legth () as a value to compare the performace of both charts, which was preseted i the form of tables ad graphs showig the average ru legth (). From Tables -4, it was foud that EWMA-S Cotrol Chart had the average ru legth () less tha S Cotrol Chart at all levelsof stadard deviatio ad all samplig plas except RSS ad MSS samplig methodidicatig that EWMA-S Cotrol Chartis more efficiettha S Cotrol Chart whe the sample size is 3. Table : from cotrol chart of differet samplig sizes =5 Sig ma S Cotrol Chart SRS SYS RSS MSS EWM A-S S Cotrol Chart EWM A-S S Cotrol Chart EWM A-S S Cotrol Chart EWMA -S = = Figure : i the SRS ad SYS EWMA-S (SRS) SRS =5 SRS = SRS =5 SRS =3 EWMA-S (SYS) SYS =5 SYS = SYS =5 SYS =

6 From figure, it was foud that i the Simple Radom Samplig (SRS) ad Systematic Samplig (SYS) will icrease whe the stadard deviatio rises ad be costat whe the stadard deviatio is betwee. to.6 ad decrease whe the stadard deviatio is greater tha.6. From Figure, Raked-Set Samplig (RSS) ad MSS give higher whe the stadard deviatio icreases. Figure : i the RSS ad MSS EWMA-S (RSS) EWMA-S (MSS) RSS =5 RSS = RSS =5 RSS =3 RSS =5 RSS = RSS =5 RSS = Figure 3: S Cotrol Chart i the SRS, SYS, ad MSS 6 S Cotrol Chart (SRS) SRS =5 SRS = SRS =5 SRS =3 68

7 S Cotrol Chart (SYS) SYS =5 SYS = SYS =5 SYS = S Cotrol Chart (MSS) MSS =5 MSS = MSS =5 MSS =3 From figure 3, it was foud that the sample size does ot affect of S-Chart, meaig the sample size does ot affect the performace of S-Cotrol Chart. Figure 4: S Cotrol Chart i the RSS, S Cotrol Chart (RSS),5,, RSS =5 RSS = RSS =5 RSS =3 From figure 4, it was foud that Raked-Set Samplig (RSS) gives = at all sample sizes. This is because Raked-Set Samplig (RSS) is a samplig method that sorts the value from low to high ad uses the lowest umber as the first sample ad repeat util all samples are used. Therefore, sample umber is the sample that has the lowest observatio. The lowest observatio will be out of the cotrol rage whe plotted o a graph resultig i =. 69

8 The other figure are plots but ot listed i this paper. The coclusios ca be draw as below:. It was foud that SRS ad SYS are more efficiet tha other samplig methods whe the stadard deviatio is high. If the stadard deviatio is lower, MSS will be more efficiet.. It was foud that samplig method did ot affect the performace of both charts because they gave the same for all samplig methods. 5. CONCLUSIONS AND RECOMMENDATIONS The fidigs i this research are to study ad compare the performace of S Cotrol Chart ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace i differet cases as follows. Data chaged the stadard deviatio at differet levels. Data had with differet sample sizes. Samplig method was differet for the data. The coclusios about the performace of S Cotrol Chart ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace usig the average ru legth () as a value to compare the performace of two charts were that whe cosiderig the appropriate samplig method it was foud that i the case of a mior variace (σ.) Media-Set Samplig (MSS) was more efficiet tha other samplig methods, whe applied to a large variace (σ >.) Simple Rasom Samplig (SRS) ad Systematic Samplig (SYS) were more efficiet tha other samplig methods, ad for whe the sample size was larger, the cotrol charts were also efficiet cotrol chart as well. I all cases, EWMA-S Cotrol Chart was more efficiet tha S Cotrol Chart. Table : The coclusio of the efficiecy of the cotrol charts at various sample sizes. Sample Size() Most Efficiet Cotrol Chart 5 EWMA-S Cotrol Chart EWMA-S Cotrol Chart 5 EWMA-S Cotrol Chart 3 EWMA-S Cotrol Chart Table 3: The coclusio of the efficiecy of the cotrol charts whe the samplig method was differet. Samplig Method SRS SYS RSS MSS Most Efficiet Cotrol Chart EWMA-S Cotrol Chart EWMA-S Cotrol Chart EWMA-S Cotrol Chart EWMA-S Cotrol Chart I this study, ormal distributio was used to simulate the data for the compariso of S Cotrol Chart ad Expoetially Weighted Movig Average Cotrol Chart (EWMA-S Cotrol Chart) for measurig the process variace. If ayoe is iterested i coductig further study, other kids of distributios may be used such as U-Shaped distributio or J-Shaped distributio, etc. REFERENCE LIST. Crowder, S., & Hamilto, M. (99). A EWMA for moitorig stadard deviatio. Joural of Quality Techology, 4, -.. Maravelakis, P., & Castagliola, P. (9). A EWMA chart for moitorig the process stadard deviatio whe parameters are estimated. omputatioal Statistics ad Data Aalysis, 53, Shewhart, W. A. (939). Statistical method from the viewpoit of quality cotrol. (W. Edwards Demig). Graduate School, the Departmet of Agriculture, Tawatira, N., & Mayureesawa, T. (). The cotrol chart by arraged order samplig. Joural of KigMogkut's Uiveristy Techology North Bagkok, 3,

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION

CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION www.arpapress.com/volumes/vol8issue2/ijrras_8_2_04.pdf CONTROL CHART BASED ON A MULTIPLICATIVE-BINOMIAL DISTRIBUTION Elsayed A. E. Habib Departmet of Statistics ad Mathematics, Faculty of Commerce, Beha

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.

This document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC. SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol

More information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number. GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN

Analyzing Longitudinal Data from Complex Surveys Using SUDAAN Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval

Inference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

Quadrat Sampling in Population Ecology

Quadrat Sampling in Population Ecology Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may

More information

Data Analysis and Statistical Behaviors of Stock Market Fluctuations

Data Analysis and Statistical Behaviors of Stock Market Fluctuations 44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Email:

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature. Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown

Z-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about

More information

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series

Automatic Tuning for FOREX Trading System Using Fuzzy Time Series utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which

More information

Statistical inference: example 1. Inferential Statistics

Statistical 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 information

Convention Paper 6764

Convention Paper 6764 Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or

More information

A Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as:

A Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as: A Test of Normality Textbook Referece: Chapter. (eighth editio, pages 59 ; seveth editio, pages 6 6). The calculatio of p values for hypothesis testig typically is based o the assumptio that the populatio

More information

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights

Center, Spread, and Shape in Inference: Claims, Caveats, and Insights Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the

More information

SPC for Software Reliability: Imperfect Software Debugging Model

SPC for Software Reliability: Imperfect Software Debugging Model IJCSI Iteratioal Joural of Computer Sciece Issues, Vol. 8, Issue 3, o., May 0 ISS (Olie: 694-084 www.ijcsi.org 9 SPC for Software Reliability: Imperfect Software Debuggig Model Dr. Satya Prasad Ravi,.Supriya

More information

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Chapter 7: Confidence Interval and Sample Size

Chapter 7: Confidence Interval and Sample Size Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

More information

Confidence Intervals

Confidence Intervals Cofidece Itervals Cofidece Itervals are a extesio of the cocept of Margi of Error which we met earlier i this course. Remember we saw: The sample proportio will differ from the populatio proportio by more

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

Research Method (I) --Knowledge on Sampling (Simple Random Sampling)

Research Method (I) --Knowledge on Sampling (Simple Random Sampling) Research Method (I) --Kowledge o Samplig (Simple Radom Samplig) 1. Itroductio to samplig 1.1 Defiitio of samplig Samplig ca be defied as selectig part of the elemets i a populatio. It results i the fact

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

Simulation-based Analysis of Service Levels in Stable Production- Inventory Systems

Simulation-based Analysis of Service Levels in Stable Production- Inventory Systems Simulatio-based Aalysis of Service Levels i Stable Productio- Ivetory Systems Jayedra Vekateswara, Kaushik Margabadu#, D. Bijulal*, N. Hemachadra, Idustrial Egieerig ad Operatios Research, Idia Istitute

More information

One-sample test of proportions

One-sample test of proportions Oe-sample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:

More information

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

More information

Volatility of rates of return on the example of wheat futures. Sławomir Juszczyk. Rafał Balina

Volatility of rates of return on the example of wheat futures. Sławomir Juszczyk. Rafał Balina Overcomig the Crisis: Ecoomic ad Fiacial Developmets i Asia ad Europe Edited by Štefa Bojec, Josef C. Brada, ad Masaaki Kuboiwa http://www.hippocampus.si/isbn/978-961-6832-32-8/cotets.pdf Volatility of

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011 15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes high-defiitio

More information

PSYCHOLOGICAL STATISTICS

PSYCHOLOGICAL STATISTICS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

The Forgotten Middle. research readiness results. Executive Summary

The Forgotten Middle. research readiness results. Executive Summary The Forgotte Middle Esurig that All Studets Are o Target for College ad Career Readiess before High School Executive Summary Today, college readiess also meas career readiess. While ot every high school

More information

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

More information

Realtime Metrics for Process Performance By focusing on realtime measures, plant operators can improve bottom-line results

Realtime Metrics for Process Performance By focusing on realtime measures, plant operators can improve bottom-line results Feature Egieerig Report Practice Realtime Metrics for Process Performace By focusig o realtime measures, plat operators ca improve bottom-lie results George Buckbee ExperTue, Ic. Process performace, quality

More information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

Evaluating Model for B2C E- commerce Enterprise Development Based on DEA

Evaluating Model for B2C E- commerce Enterprise Development Based on DEA , pp.180-184 http://dx.doi.org/10.14257/astl.2014.53.39 Evaluatig Model for B2C E- commerce Eterprise Developmet Based o DEA Weli Geg, Jig Ta Computer ad iformatio egieerig Istitute, Harbi Uiversity of

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

Page 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville

Page 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville Real Optios for Egieerig Systems J: Real Optios for Egieerig Systems By (MIT) Stefa Scholtes (CU) Course website: http://msl.mit.edu/cmi/ardet_2002 Stefa Scholtes Judge Istitute of Maagemet, CU Slide What

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

A GENERAL FRAMEWORK OF IMPORTANCE SAMPLING FOR VALUE-AT-RISK AND CONDITIONAL VALUE-AT-RISK. Lihua Sun L. Jeff Hong

A GENERAL FRAMEWORK OF IMPORTANCE SAMPLING FOR VALUE-AT-RISK AND CONDITIONAL VALUE-AT-RISK. Lihua Sun L. Jeff Hong Proceedigs of the 2009 Witer Simulatio Coferece M. D. Rossetti, R. R. Hill, B. Johasso, A. Duki, ad R. G. Igalls, eds. A GENERAL FRAMEWORK OF IMPORTANCE SAMPLING FOR VALUE-AT-RISK AND CONDITIONAL VALUE-AT-RISK

More information

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu

Multi-server Optimal Bandwidth Monitoring for QoS based Multimedia Delivery Anup Basu, Irene Cheng and Yinzhe Yu Multi-server Optimal Badwidth Moitorig for QoS based Multimedia Delivery Aup Basu, Iree Cheg ad Yizhe Yu Departmet of Computig Sciece U. of Alberta Architecture Applicatio Layer Request receptio -coectio

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Normal Distribution.

Normal Distribution. Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued

More information

BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)

BASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1) BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet

More information

On Formula to Compute Primes. and the n th Prime

On Formula to Compute Primes. and the n th Prime Applied Mathematical cieces, Vol., 0, o., 35-35 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih Abdul-Nabi

More information

THE ROLE OF EXPORTS IN ECONOMIC GROWTH WITH REFERENCE TO ETHIOPIAN COUNTRY

THE ROLE OF EXPORTS IN ECONOMIC GROWTH WITH REFERENCE TO ETHIOPIAN COUNTRY - THE ROLE OF EXPORTS IN ECONOMIC GROWTH WITH REFERENCE TO ETHIOPIAN COUNTRY BY: FAYE ENSERMU CHEMEDA Ethio-Italia Cooperatio Arsi-Bale Rural developmet Project Paper Prepared for the Coferece o Aual Meetig

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

Using Four Types Of Notches For Comparison Between Chezy s Constant(C) And Manning s Constant (N)

Using Four Types Of Notches For Comparison Between Chezy s Constant(C) And Manning s Constant (N) INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH OLUME, ISSUE, OCTOBER ISSN - Usig Four Types Of Notches For Compariso Betwee Chezy s Costat(C) Ad Maig s Costat (N) Joyce Edwi Bategeleza, Deepak

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

DAME - Microsoft Excel add-in for solving multicriteria decision problems with scenarios Radomir Perzina 1, Jaroslav Ramik 2

DAME - Microsoft Excel add-in for solving multicriteria decision problems with scenarios Radomir Perzina 1, Jaroslav Ramik 2 Itroductio DAME - Microsoft Excel add-i for solvig multicriteria decisio problems with scearios Radomir Perzia, Jaroslav Ramik 2 Abstract. The mai goal of every ecoomic aget is to make a good decisio,

More information

Optimal Adaptive Bandwidth Monitoring for QoS Based Retrieval

Optimal Adaptive Bandwidth Monitoring for QoS Based Retrieval 1 Optimal Adaptive Badwidth Moitorig for QoS Based Retrieval Yizhe Yu, Iree Cheg ad Aup Basu (Seior Member) Departmet of Computig Sciece Uiversity of Alberta Edmoto, AB, T6G E8, CAADA {yizhe, aup, li}@cs.ualberta.ca

More information

TIGHT BOUNDS ON EXPECTED ORDER STATISTICS

TIGHT BOUNDS ON EXPECTED ORDER STATISTICS Probability i the Egieerig ad Iformatioal Scieces, 20, 2006, 667 686+ Prited i the U+S+A+ TIGHT BOUNDS ON EXPECTED ORDER STATISTICS DIMITRIS BERTSIMAS Sloa School of Maagemet ad Operatios Research Ceter

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

Rainbow options. A rainbow is an option on a basket that pays in its most common form, a nonequally

Rainbow options. A rainbow is an option on a basket that pays in its most common form, a nonequally Raibow optios INRODUCION A raibow is a optio o a basket that pays i its most commo form, a oequally weighted average of the assets of the basket accordig to their performace. he umber of assets is called

More information

Practice Problems for Test 3

Practice Problems for Test 3 Practice Problems for Test 3 Note: these problems oly cover CIs ad hypothesis testig You are also resposible for kowig the samplig distributio of the sample meas, ad the Cetral Limit Theorem Review all

More information

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal Advaced Sciece ad Techology Letters, pp.31-35 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phase-locked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College

More information

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

More information

Iran. J. Chem. Chem. Eng. Vol. 26, No.1, 2007. Sensitivity Analysis of Water Flooding Optimization by Dynamic Optimization

Iran. J. Chem. Chem. Eng. Vol. 26, No.1, 2007. Sensitivity Analysis of Water Flooding Optimization by Dynamic Optimization Ira. J. Chem. Chem. Eg. Vol. 6, No., 007 Sesitivity Aalysis of Water Floodig Optimizatio by Dyamic Optimizatio Gharesheiklou, Ali Asghar* + ; Mousavi-Dehghai, Sayed Ali Research Istitute of Petroleum Idustry

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

Now here is the important step

Now here is the important step LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

More information

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

Systems Design Project: Indoor Location of Wireless Devices

Systems Design Project: Indoor Location of Wireless Devices Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised

More information

France caters to innovative companies and offers the best research tax credit in Europe

France caters to innovative companies and offers the best research tax credit in Europe 1/5 The Frech Govermet has three objectives : > improve Frace s fiscal competitiveess > cosolidate R&D activities > make Frace a attractive coutry for iovatio Tax icetives have become a key elemet of public

More information

Universal coding for classes of sources

Universal coding for classes of sources Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric

More information

Measuring Magneto Energy Output and Inductance Revision 1

Measuring Magneto Energy Output and Inductance Revision 1 Measurig Mageto Eergy Output ad Iductace evisio Itroductio A mageto is fudametally a iductor that is mechaically charged with a iitial curret value. That iitial curret is produced by movemet of the rotor

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 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 information

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

More information

A Software Reliability Growth Model for Three-Tier Client Server System

A Software Reliability Growth Model for Three-Tier Client Server System 2 Iteratioal Joural of Computer Applicatios (975 8887) A Software Reliability Growth Model for Three-Tier Cliet Server System Pradeep Kumar Iformatio Techology Departmet ABES Egieerig College, Ghaziabad

More information

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,

NEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff, NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical

More information

Research Article Sign Data Derivative Recovery

Research Article Sign Data Derivative Recovery Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

More information

, a Wishart distribution with n -1 degrees of freedom and scale matrix.

, a Wishart distribution with n -1 degrees of freedom and scale matrix. UMEÅ UNIVERSITET Matematisk-statistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 004-0-9 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information