Log-Logistic Software Reliability Growth Model
|
|
- Barbra Mills
- 8 years ago
- Views:
Transcription
1 Log-Logistic Software Reliability Growth Model Swapa S. Gokhale ad Kishor S. Trivedi 2y Bours College of Egg. CACC, Dept. of ECE Uiversity of Califoria Duke Uiversity Riverside, CA 9252 Durham, NC Abstract Fiite failure NHPP models proposed i the literature exhibit either costat, mootoic icreasig or mootoic decreasig failure occurrece rates per fault, ad are iadequate to describe the failure process uderlyig certai failure data sets. I this paper, we propose the loglogistic reliability growth model, which ca capture the icreasig=decreasig ature of the failure occurrece rate per fault. Equatios to estimate the parameters of the existig fiite failure NHPP models, as well as the log-logistic model, based o failure data collected i the form of iterfailure times are developed. We also preset a aalysis of two data sets, where the uderlyig failure process could ot be adequately described by the existig models, which motivated the developmet of the log-logistic model. Itroductio The past 2 years have see the formulatio of several software reliability growth models to predict the reliability ad error cotet of software systems. These models are cocered with forecastig future system operability from the failure data collected durig the testig phase of a software product. A plethora of reliability models have appeared i the literature, however, a extesive validatio of This work was doe whe the author was a graduate studet at Duke Uiversity y This work was supported i part by Bellcore as a core project i the Ceter for Advaced Computig ad Commuicatio, by the Natioal Sciece Foudatio grat umber EEC , ad Charles Stark Draper Laboratory Grat # DL-H these models seems to be lackig. The accuracy of these models whe validated usig the very few available data sets varies sigificatly, ad thus despite the existece of umerous models, oe of them ca be recommeded ureservedly to potetial users. This paper presets a Log-logistic software reliability growth model, the developmet of which was primarily motivated due to the iadequacy of the existig models to describe the ature of failure process uderlyig some of the previously reported as well as ew data sets. The layout of the paper is as follows: Sectio 2 describes the fiite failure NHPP class of software reliability growth models, ad offers a ew decompositio of the mea value fuctio of the fiite failure NHPP models, which eables us to attribute the ature of the failure itesity of the software to the hazard fuctio. Sectio 3 describes some of the limitatios of the existig fiite failure NHPP models. Sectio 4 describes the log-logistic software reliability growth model. Sectio 5 discusses parameter estimatio of the existig fiite failure NHPP models as well as the log-logistic model based o the times betwee failures data. Sectio 6 describes the techiques used for software failure data aalysis. Sectio 7 presets the aalysis of two failure data sets which led us to the developmet of the log-logistic model ad Sectio 8 cocludes the paper. 2 Fiite failure NHPP models This is a class of time-domai [3, 7, 2] software reliability models which assume that software failures display the behavior of a o-homogeeous Poisso process (NHPP).
2 The parameter of the stochastic process, (t), which deotes the failure itesity of the software at time t, is timedepedet. Let N (t) deote the cumulative umber of faults detected by time t, adm(t) deote its expectatio. The m(t) = E[N (t)], ad the failure itesity (t) are Z t m(t) = related as follows: (s)ds () ad, dm(t) = (t) (2) dt N (t) is kow to have a Poisso pmf with parameter m(t), thatis: P fn (t) =g = [m(t)] e,m(t) ; =; ; 2; :::! Various time domai models have appeared i the literature which describe the stochastic failure process by a NHPP. These models differ i their failure itesity fuctio (t), ad hece m(t). The NHPP models ca be further classified ito fiite failure ad ifiite failure categories. Fiite failure NHPP models assume that the expected umber of faults detected give ifiite amout of testig time will be fiite, whereas the ifiite failures models assume that a ifiite umber of faults would be detected i ifiite testig time [3]. Let a deote the expected umber of faults that would be detected give ifiite testig time i case of fiite failure NHPP models. The, the mea value fuctio of the fiite failure NHPP models ca also be writte as: m(t) =af (t) (4) wheref (t) is a distributio fuctio. From Equatio (4), the istataeous failure itesity (t) i case of the fiite failure NHPP models is give by: (t) =af (t) (5) which ca be re-writte as: F (t) (t) =[a, m(t)] =[a, m(t)]h(t) (6), F (t) where h(t) is the failure occurrece rate per fault of the software, or the rate at which the idividual faults maifest themselves as failures durig testig. The quatity [a, m(t)] deotes the expected umber of faults remaiig i the software at time t. Sice [a, m(t)] is a mootoically o-icreasig fuctio of time (actually, [a, m(t)] should decrease as more ad more faults are detected ad (3) removed), the ature of the overall failure itesity, (t), is govered by the ature of failure occurrece rate per fault h(t), from Equatio (6). 3 Limitatios of the NHPP models The failure occurrece rate per fault h(t) ca be a costat, icreasig, decreasig, or icreasig=decreasig. I this sectio, we describe some of the existig fiite failure NHPP models alog with their hazard fuctios. 3. Goel-Okumoto (GO) model The Goel-Okumoto (GO) model [6] has had a strog ifluece o software reliability modelig. The failure occurrece rate per fault is costat i case of the Goel-Okumoto model. Musa-Okumoto model [] is similar to the Goel- Okumoto model, the primary differece beig that it is based o executio time data, whereas the Goel-Okumoto model uses caledar time failure data. Table gives the expressios for m(t), (t), adh(t) for the Goel-Okumoto model. 3.2 Geeralized Goel-Okumoto (GO) model I case of Goel-Okumoto model, the failure occurrece rate per fault is time idepedet; however sice the expected umber of remaiig faults decreases with time, the overall software failure itesity decreases with time. The software quality cotiues to improve as testig progresses. However, i most real-life testig scearios, the software failure itesity icreases iitially ad the decreases. The geeralized Goel-Okumoto(GO) model [5] was proposed to capture this icreasig/decreasig ature of the failure itesity. The ature of the failure occurrece rate per fault is determied by the parameter, ad is icreasig for < ad decreasig for >. Refer to Table for expressios for m(t), (t),adh(t) for the Geeralized Goel-Okumoto model. 3.3 S-shaped model The S-shaped reliability growth model [4] captures the software error removal pheomeo i which there is a time
3 Table. NHPP models - m(t), (t), ad h(t) Coverage Fuctio m(t) (t) h(t) Expoetial a(, e,gt ) age,gt g Weibull a(, e,gt ) age,gt t, gt, S-shaped a[, ( + gt)e,gt ] ag 2 te,gt g2 t +gt Failure occurrece rate per fault Failure occurrece rate per fault vs. time.6 Failure occurrece rate per fault vs. time Failure Occurrece Rate per Fault GO S shaped Ge. GO (gamma < ) Ge. GO (gamma > ) Figure. Hazard for existig NHPP models delay betwee the actual detectio of the fault ad its reportig. The testig process i this case ca be see as cosistig of two phases: fault detectio ad fault isolatio. The S-shaped model has a icreasig failure occurrece rate per fault. The expressios for m(t), (t), adh(t) for the S-shaped model are preseted i Table. The hazard rates for the Goel-Okumoto model, Geeralized Goel-Okumoto model ad S-shaped model are show i Figure. The parameters of the models are set up i a such a way that there are about 26 failures i 25 time uits, for the sake of illustratio. 4 Log-logistic reliability growth model Figure shows that the existig fiite failure NHPP models ca capture costat, mootoically icreasig or mootoically decreasig failure occurrece rates per fault. I some of the failure data sets, the rate at which idividual faults maifest themselves as testig progresses, ca also exhibit a icreasig/decreasig behavior. A icreasig=decreasig tred exhibited by the failure occurrece rate per fault caot be captured by the existig NHPP models. I this sectio, we describe the loglogistic reliability growth model, which ca capture the icreasig=decreasig ature of the hazard fuctio Figure 2. Hazard for Log-logistic model The icreasig/decreasig behavior of the failure occurrece rate per fault ca be captured by the hazard of the loglogistic distributio []. The mea value fuctio, m(t), failure itesity fuctio (t), ad the hazard fuctio, h(t) of the log-logistic software reliability model are give i Equatios (7), (8) ad (9) respectively. m(t) =a (t) +(t) (7) (t) =a (t), [+(t) ] 2 (8) h(t) = c (t), c(t) = (t), +(t) (9) Figure 2 shows the hazard of the log-logistic model. The parameters of the model are set up i such a way that there are 26 failures i 25 time uits, as i case of Figure, merely for the sake of illustratio. Figure 3 shows the coditioal reliability vs. time for the log-logistic model. I Figure 3 failure detectios are assumed to be at fixed itervals. Just after the poit of detectio ad subsequet istataeous repair, reliability jumps up to., ad the decays slowly util the ext detectio. The lowest reliability i each iterval decreases iitially i the log-logistic model, ad the icreases. 5 Estimatio of parameters I this sectio we develop expressios to estimate the parameters of the fiite failure NHPP models preseted i Sectio 2 ad the Log-logistic model based o time betwee failures data. Equatios to estimate the parameters
4 Coditioal reliability Reliability growth for log logistic model The log likelihood i case of Geeralized Goel-Okumoto model ca be writte as: L(a; g; js) =,a(, e,gs )+ log a + log g + log, g P i= s i +(, ) P i= log s i (5) For a fixed value of, Equatio(5), ca be maximized with respect to a ad g to give: Figure 3. Reliability growth of the Goel-Okumoto model have bee developed i the literature [6], ad are preseted here for the sake of completeess. Let ft k ;k = ; 2;:::g deote the sequece of times betwee successive software failures. The t k deotes the time betwee (k, ) st ad k th failure. Let s k deote the time to failure k,sothat: s k = kx i= t k : () The joit desity or the likelihood fuctio of S ;S 2 ;:::;S ca be writte as [3]: f S;S 2;:::;S (s ;s 2 ;:::;s )=e,m(s) Y i= (s i ): () For a give sequece of software failure times s = (s ;s 2 ;:::;s ), that are realizatios of the radom variables S ;S 2 ;:::;S, the parameters of the software reliability growth models are estimated usig the maximum likelihood method. The log likelihood i case of Goel-Okumoto model is giveby[6]: L(a; gjs) = log a + log g, a(, e,gs ),g P i= s i (2) Maximizig Equatio (2) with respect to a ad g, we have: ad g = a =, e,gs (3) X i= s i + as e,gs (4) Solvig these two simultaeous o-liear equatios, we obtai the poit estimates of a ad g. ad g = =, e,gs a X i= (6) s i + as e,gs (7) Simultaeously solvig the above two equatios for a fixed value of, gives the poit estimates of a ad g. Differet values of give a family of Geeralized Goel-Okumoto models, ad we choose the best amog these. Similarly, the log likelihood i case of S-shaped model ca be writte as: L(a; gjs) =,a(, ( + gs )e,gs )+2 log g + log a + P i= log s i, g P i= s i (8) Maximizig Equatio (8) with respect to a ad g, we have: ad a =, ( + gs )e,gs (9) 2 g = ags 2 e,gs + X i= s i (2) Solvig the above two coupled o-liear equatios, we obtai the poit estimates of parameters a ad g. The log likelihood i case of log-logistic model is: L(a; ; js) =,a (s) +(s ) + log a + log + log +(, ) P i= log s i, 2 P i= log ( + (s i) ) (2) Maximizig Equatio (2) with respect to a,,ad gives: a = (+(s ) ) (s ) (22)
5 s = 2( + s X (23) s i ) +s i= i P log s k =, [+(t) ] 2 log, i= log s P i (24) (s +2 i) k log(s i) i= +(s i) Solvig the above three equatios simultaeously, yields the poit estimates of the parameters a, ad. 6 Data aalysis techiques I this sectio we describe the techiques used for the aalysis of the software failure data sets. 6. Tred aalysis Software reliability studies are usually based o the applicatio of reliability growth models to obtai various measures of iterest. I order to determie which reliability growth model to use, tred tests are very useful. I this sectio we discuss the most frequetly used tred tests as precursors to reliability growth modelig for failure data collected i the form of iterfailure times. The two tred tests that are commoly carried out are [8]: Arithmetic Mea Test: This test cosists of computig the arithmetic mea (i) of the observed iterfailure times t j ;j =; 2;:::;i: (i) = i ix j= t j (25) A icreasig sequece of (i) idicates reliability growth ad a decreasig sequece idicates reliability decay. Laplace Test: Laplace test is superior from a optimality poit of view ad is recommeded for use whe the NHPP assumptio is made [4]. Let N (t) deote the cumulative umber of faults detected over the period (;t). The failure itesity (t) determies reliability growth=reliability decay=stable reliability. A decreasig (t) implies reliability growth, icreasig (t) implies reliability decay, ad a costat (t) implies stable reliability. The test procedure is to compute the Table 2. Tred ad correspodig models Tred Coverage Fuctio Reliability growth Go ad Geeralized GO ( <) Reliability decay followed by growth Log-logistic / S-shaped Stable reliability Homogeeous Poisso process Laplace factor l(t) giveby[2]: l(t) = N(t) N(t) X X = j= t q 2N(t) t j, t 2 (26) The Laplace factor is evaluated step by step, after every failure occurrece. Here t is the equal to the time of occurrece of the i th failure, ad the failure at time t is excluded. Equatio (26) ca the be modified as: ix t Xi, X j j= t i, j, 2 = j= l(t) = q t 2(i,) Laplace factor ca be iterpreted as follows: (27) Negative values idicate a decreasig failure itesity, ad thus reliability growth. Positive values idicate a icreasig failure itesity, ad thus a decrease i the reliability. Values betwee,2 ad +2 idicate stable reliability. Tred aalysis ca sigificatly help i choosig the appropriate model for a give sequece of iterfailure times, so that they ca be applied to data displayig treds i accordace with their assumptios rather tha blidly. Usig a model for the aalysis of a failure data set, without takig ito cosideratio the tred displayed by the data ca lead to urealistic results, whe the tred displayed is differet tha that assumed by the model [9]. The classificatio of the failure data accordig to the tred ad the correspodig model are summarized i Table 2.
6 6.2 Goodess of fit Arithemtic mea vs. i The ability of a model to reproduce the observed failure behavior of the software, also kow as its retrodictive capability [8], is determied by the goodess-of-fit test. The observed failure data is used to estimate the parameters of the model. The estimated mea value fuctio is computed ad plotted alog with the observed mea value fuctio. The error sum of squares is the calculated to evaluate the goodess-of-fit. The lower the error sum of squares the better is the fit. Arithmetic Mea i Figure 4. Arithmetic mea test - DS I 6.3 Model bias 2 Laplace factor vs. i Model bias determies whether the predictios are o a average close to the true distributios. The estimated probability of failure for each failure iterval is used to determie the extet to which a model itroduces bias ito its predictios. If the model is optimistically biased, the estimates of time to ext failure are higher tha what is actually observed, whereas i case of a pessimistic bias the estimate of times to ext failure are lower tha the observed oes []. Model bias is determied by u-plot. 6.4 Model bias tred Model bias tred measures the o-statioarity i the predictios, i other words whether model bias chages over time. The aalysis is similar to that of model bias, except that the estimated failure probabilities are trasformed i a maer that preserves temporal iformatio []. Model bias tred is determied by y-plot. 7 Software failure data aalysis I this sectio we preset the aalysis of the two data sets which led us to the developmet of the log-logistic software reliability model. Data Set I (referred to hereafter as DS) is from the U. S. Navy Feet Computer Programmig Ceter, ad cosists of errors i the developmet of software for the real-time, multicomputer complex which forms the core of the Naval Tactical Data System (NTDS). The time (days) betwee software failures are reported for oe of the larger modules Laplace Factor i Figure 5. Laplace test - DS I comprisig the NTDS software [6]. The data set cosists of 26 failures i 25 days. Data Set II (referred to hereafter as DS2) is collected durig the test phase of a software developmet project at the Charles Stark Draper Laboratories. DS II cosists of 6 failures i approximately 5 days. Figures 4 ad 5 show the arithmetic mea ad Laplace tred tests for DS II respectively. Tred tests idicate reliability decay followed by reliability growth which suggests the use of either S-shaped or log-logistic model. Loglogistic model provides the best retrodictive capability as see i the Figure 6. Bias ad bias tred tests are ot carried out i case of DS I due to a small umber of failures i the data set. Figures 7 ad 8 show the arithmetic mea ad Laplace tred tests for DS II respectively. As see from the figures, Tred tests idicate reliability decay followed by reliability growth which suggests the use of either S-shaped or loglogistic model. Log-logistic model provides the best retrodictive capability as see i the Figure 9, ad the best predictive capability as see i the Figure. S-shaped as well
7 3 25 Expected umber of faults durig testig Field Data GO Ge. GO (.2) 8 Laplace factor vs. i Expected Number of Faults 2 5 Laplace Factor Figure 6. Goodess of fit test - DS I i Figure 8. Laplace test - DS II 2 Arithmetic mea vs. i 8 6 Expected Number of Faults vs. Field data Ge. GO (.2) 8 4 Arithmetic Mea Expected Number of Faults i Figure 7. Arithmetic mea test - DS II as log-logistic models exhibits the least o-statioarity i the predictios. Kolomogorov-Smirov distace for the goodess of fit test i case of DS I ad DS II, ad Kolmogorov distace for bias ad bias tred i case of DS II are summarized i Tables 3 ad 4 respectively. 8 Coclusios I this paper, we have preseted the log-logistic software reliability growth model which was motivated by the fact that the existig fiite failure NHPP models were i Figure 9. Goodess of fit test - DS II u plot S shaped Weibull (.5) Figure. Bias - DS II Table 3. KS distace - Goodess of fit Model DS I DS II GO 29: S-shaped 7: Log-logistic Ge. GO ( =:95) 4: Ge. GO ( =:5) 2:9 648 Ge. GO ( =:) : Table 4. Kolmogorov distace - DS II Model Bias Bias tred GO :377 :2645 S-shaped Log-logistic.377 :2673 Ge. GO ( =:95) :377 :258
8 y plot S shaped Weibull (.5) [4] O. Gauodi. Optimal Properties of the Laplace Tred Test for Software-Reliability Models. IEEE Tras. o Reliability, 4(4): , Dec [5] A. L. Goel. Software Reliability Models: Assumptios, Limitatios ad Applicability. IEEE Tras. o Software Egieerig, SE-(2):4 423, Dec Figure. Bias tred - DS II adequate to describe the failure process uderlyig some of the data sets. We offer a ew decompositio of the mea value fuctio of fiite failure NHPP models which eables us to attribute the ature of the failure itesity to the failure occurrece rate per fault. The existig fiite failure NHPP models ca capture costat, mootoic icreasig, ad mootoic decreasig failure occurrece rate per fault, whereas the log-logistic model proposed here ca capture a icreasig=decreasig ature of the failure occurrece rate per fault. Equatios to obtai the maximum likelihood estimates of the parameters of the existig fiite failure NHPP models, as well as the log-logistic model based o times betwee failures data are developed. Aalysis of two failure data sets which led us to the log-logistic model, usig arithmetic ad Laplace tred tests, goodess-of-fit test, bias ad bias tred tests is preseted. Refereces [] A. A. Abdel-Ghally, P. Y. Cha, ad B. Littlewood. Evaluatio of Competig Software Reliability Predictios. IEEE Tras. o Software Egieerig,SE- 2(9): , Sept [2] D.R.CoxadP.A.W.Lewis.The Statistical Aalysis of a Series of Evets. Lodo: Chapma ad Hall, Lodo, 978. [3] W. Farr. Hadbook of Software Reliability Egieerig, M. R. Lyu, Editor, chapter Software Reliability Modelig Survey, pages 7 7. McGraw-Hill, New York, NY, 996. [6] A. L. Goel ad K. Okumoto. -Depedet Error- Detectio Rate Models for Software Reliability ad Other Performace Measures. IEEE Tras. o Reliability, R-28(3):26 2, Aug [7] S. Gokhale, P. N. Marios, ad K. S. Trivedi. Importat Milestoes i Software Reliability Modelig. I Proc. 8th Itl. Coferece o Software Egieerig ad Kowledge Egieerig (SEKE 96), pages , Lake Tahoe, Jue 996. [8] K. Kaou ad J. C. Laprie. Software Reliability Tred Aalysis from Theoretical to Practical Cosideratios. IEEE Tras. o Software Egieerig, 2(9):74 747, Sept [9] K. Kaou ad J. C. Laprie. Hadbook of Software Reliability Egieerig, M. R. Lyu, Editor, chapter Tred Aalysis, pages McGraw-Hill, New York, NY, 996. [] L. M. Leemis. Reliability - Probalistic Models ad Statistical Methods. Pretice-Hall, Eglewood Cliffs, New Jersey, 995. [] J. D. Musa. A Theory of Software Reliability ad its Applicatio. IEEE Tras. o Software Egieerig, SE-():32 327, Sept [2] M. Ohba. Software Reliability Aalysis Models. IBM Joural Res. Develop., 28(4): , July 984. [3] K. S. Trivedi. Probability ad Statistics with Reliability, Queuig ad Computer Sciece Applicatios. Pretice-Hall, Eglewood Cliffs, New Jersey, 982. [4] S. Yamada, M. Ohba, ad S. Osaki. S-Shaped Reliability Growth Modelig for Software Error Detectio. IEEE Tras. o Reliability, R-32(5): , Dec. 983.
Reliability Analysis in HPC clusters
Reliability Aalysis i HPC clusters Narasimha Raju, Gottumukkala, Yuda Liu, Chokchai Box Leagsuksu 1, Raja Nassar, Stephe Scott 2 College of Egieerig & Sciece, Louisiaa ech Uiversity Oak Ridge Natioal Lab
More informationNon-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 informationSPC on Ungrouped Data: Power Law Process Model
Iteratioal Joural of Software Egieerig. ISSN 0974-3162 Volume 5, 1 (2014), pp. 7-16 Iteratioal Research Publicatio House http://www.irphouse.com SPC o Ugrouped Data: Power Law Process Model DR. R. Satya
More informationSPC 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 informationA 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 informationIn 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 informationThe 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 informationModified 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 information1 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 informationData 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 informationInstallment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate
Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate Nia-Nia JI a,*, Yue LI, Dog-Hui WNG College of Sciece, Harbi
More informationChapter 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 informationChapter 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 informationI. 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 informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationMaximum 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 informationThe 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 informationConfidence 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 informationVladimir 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 informationLECTURE 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 informationHow To Improve Software Reliability
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.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationAutomatic 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 informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC
TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies
More informationUniversity 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 informationPSYCHOLOGICAL 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 informationChapter 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 informationChapter 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 informationTaking 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 informationVolatility 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 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 informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationSubject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More information1. 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 informationCONTROL 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 informationPROCEEDINGS 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 informationTheorems 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 informationCase 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 informationZ-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 informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationD I S C U S S I O N P A P E R
I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S ( I S B A UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 2012/27 Worst-case
More informationHypothesis 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 informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationOutput 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 informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationChapter 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, 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 informationTradigms of Astundithi and Toyota
Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationCHAPTER 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 informationCantilever Beam Experiment
Mechaical Egieerig Departmet Uiversity of Massachusetts Lowell Catilever Beam Experimet Backgroud A disk drive maufacturer is redesigig several disk drive armature mechaisms. This is the result of evaluatio
More informationReview: Classification Outline
Data Miig CS 341, Sprig 2007 Decisio Trees Neural etworks Review: Lecture 6: Classificatio issues, regressio, bayesia classificatio Pretice Hall 2 Data Miig Core Techiques Classificatio Clusterig Associatio
More informationForecasting. Forecasting Application. Practical Forecasting. Chapter 7 OVERVIEW KEY CONCEPTS. Chapter 7. Chapter 7
Forecastig Chapter 7 Chapter 7 OVERVIEW Forecastig Applicatios Qualitative Aalysis Tred Aalysis ad Projectio Busiess Cycle Expoetial Smoothig Ecoometric Forecastig Judgig Forecast Reliability Choosig the
More informationInference 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 informationNEW 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 informationInverse Gaussian Distribution
5 Kauhisa Matsuda All rights reserved. Iverse Gaussia Distributio Abstract Kauhisa Matsuda Departmet of Ecoomics The Graduate Ceter The City Uiversity of New York 65 Fifth Aveue New York NY 6-49 Email:
More informationSoving 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 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 informationDepartment 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 informationW. Sandmann, O. Bober University of Bamberg, Germany
STOCHASTIC MODELS FOR INTERMITTENT DEMANDS FORECASTING AND STOCK CONTROL W. Sadma, O. Bober Uiversity of Bamberg, Germay Correspodig author: W. Sadma Uiversity of Bamberg, Dep. Iformatio Systems ad Applied
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success
More informationEntropy of bi-capacities
Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics
More informationSpam Detection. A Bayesian approach to filtering spam
Spam Detectio A Bayesia approach to filterig spam Kual Mehrotra Shailedra Watave Abstract The ever icreasig meace of spam is brigig dow productivity. More tha 70% of the email messages are spam, ad it
More informationConfidence 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 informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationActuarial Models for Valuation of Critical Illness Insurance Products
INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 9, 015 Actuarial Models for Valuatio of Critical Illess Isurace Products P. Jidrová, V. Pacáková Abstract Critical illess
More informationA 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 informationUM USER SATISFACTION SURVEY 2011. Final Report. September 2, 2011. Prepared by. ers e-research & Solutions (Macau)
UM USER SATISFACTION SURVEY 2011 Fial Report September 2, 2011 Prepared by ers e-research & Solutios (Macau) 1 UM User Satisfactio Survey 2011 A Collaboratio Work by Project Cosultat Dr. Agus Cheog ers
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationAn Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function
A Efficiet Polyomial Approximatio of the Normal Distributio Fuctio & Its Iverse Fuctio Wisto A. Richards, 1 Robi Atoie, * 1 Asho Sahai, ad 3 M. Raghuadh Acharya 1 Departmet of Mathematics & Computer Sciece;
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/
More informationA RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY
J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON
More informationEvaluating 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 informationA Fuzzy Model of Software Project Effort Estimation
TJFS: Turkish Joural of Fuzzy Systems (eissn: 309 90) A Official Joural of Turkish Fuzzy Systems Associatio Vol.4, No.2, pp. 68-76, 203 A Fuzzy Model of Software Project Effort Estimatio Oumout Chouseioglou
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationChapter XIV: Fundamentals of Probability and Statistics *
Objectives Chapter XIV: Fudametals o Probability ad Statistics * Preset udametal cocepts o probability ad statistics Review measures o cetral tedecy ad dispersio Aalyze methods ad applicatios o descriptive
More informationCHAPTER 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 informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationA Mathematical Perspective on Gambling
A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal
More informationPlug-in martingales for testing exchangeability on-line
Plug-i martigales for testig exchageability o-lie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,vovk}@cs.rhul.ac.uk
More informationBasic Measurement Issues. Sampling Theory and Analog-to-Digital Conversion
Theory ad Aalog-to-Digital Coversio Itroductio/Defiitios Aalog-to-digital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica Pós-Graduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationNr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996
Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationChair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics
Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to
More information0.7 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 informationPartial Di erential Equations
Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio
More informationODBC. Getting Started With Sage Timberline Office ODBC
ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationStudy 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 informationUC 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 information1 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