Output Analysis (2, Chapters 10 &11 Law)


 Monica Hodges
 1 years ago
 Views:
Transcription
1 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 be doe o the basis of a statistical aalysis. For example, whe comparig two systems, oe system ca be better o some replicatios ad worse o others. Tellig which system is better requires ca be oly doe approximately based o statistical aalysis. Example 1 Suppose a bak is cosiderig two possible ATM systems. o Zippy: M/M/1 queue with arrival rate 1 ad oe fast server with mea service time 0.9 miutes. o Kluky: M/M/ queue with arrival rate 1 ad two slow servers with mea service time 1.8 miutes each. 1
2 The performace measure of iterest is the mea delay of the first 100 customers, d(100) (system starts empty). The true measures (evaluated based o exact aalysis) are d Z (100) = 4.13 ad d K (100) = So Kluky is better. Simulatig these two systems ad comparig the output of oe ru at a time i 100 replicatios, gives the followig. I 53 Out of 100 replicatios, d Z (100) < d K (100). That is, P{wrog aswer} = This meas that a aalysis based o a sigle ru is likely to lead to wrog aswers. To improve the compariso oe could simulate each system for replicatio ad the base the decisio o comparig the averages of the replicatios.
3 Doig 100 comparisos this way (with a total of 100 replicatios) gave the followig results for differet s. A dot plot illustrates what s happeig. This example highlights the eed for methods to assess the ucertaity ad give statistical bouds or guaratees for coclusios ad decisios. 3
4 Cofidece Itervals for the Differece of Systems Cosider two alterative systems with performace measures μ i, i = 1, (the mea of somethig). Suppose that i replicatios are made for system i. Let X ij be the observatio from system i o replicatio j. The idea is to use the X ij s to build a cofidece iterval for ζ = μ 1 μ. If the iterval misses 0, we coclude there is a statistical differece betwee the two systems. We ca also have a idea of how sigificat is the differece betwee the two systems. There are two approaches for buildig cofidece itervals for ζ: Pairedt ad twosamplet. Pairedt cofidece Iterval Assume replicatios of each system are performed. Let Z j = X 1j X j. The, the Z j s are IID. The pairedt approach works by buildig a cofidece iterval based o the sample mea ad variace for Z, Z ( Z Z( )) j j= 1 j= 1 Z Z ( ) =, S( ) = j. 1 4
5 Assume that the Z j s are ormally distributed (which may be justified by the cetral limit theorem). The, the followig is a approximate 100(1 α) percet cofidece iterval for ζ SZ ( ) Z ( ) ± t 1,1 α /. Oe importat fact about the pairedt cofidece iterval is that X 1j ad X j eed ot to be idepedet. This could be useful whe utilizig a variace reductio techique. Example Cosider comparig two alterative orderig policies i a (s, S) system ((0, 40) ad (0, 80)). The measure of performace of iterest is the average mothly cost i a 10 moth horizo. The simulatio results of five replicatios of the two systems are as follows. 5
6 The, Z (5) = 4.98 ad S Z (5) = 1.19, ad usig a 10% sigificace level, t 4, 0.95 =.13, A approximate 90% tpaired cofidece iterval is ±.13 = 4.98 ± 3.33 ζ (1.65,8.31). 5 Sice the iterval does ot cotai 0, this implies that the two (s, S) policies are sesibly differet. I additio, the secod policy seems to be better (as it has lower cost). A twosample cofidece iterval This method allows samples of uequal sizes (i.e. 1 ) from both systems. However, it requires that X 1j ad X j be idepedet. This ca be doe by usig differet radom umber streams for obtaiig X 1j ad X j. The method works by estimatig the mea ad variace for each sample separately. That is, oe first estimates 1 1 X ( X X ( )) X ( ), S ( ), 1j 1j 1 1 j= 1 j= = 1 1 = X ( X X ( )) X ( ), S ( ). j j j= 1 j= 1 = = 1 6
7 A approximate 100(1 α) % cofidece iterval for ζ is S1 ( 1) S( ) X1( 1) X( ) ± tfˆ +,1 α /, 1 where S1 ( 1)/ 1+ S( )/ fˆ =. S1 ( 1)/ 1 /( 1 1) + S( )/ /( 1) Example 3 Applyig the twosample cofidece iterval approach for the ivetory system i Example, gives X (5) = 15.57, X (5) = 10.59, S (5) = 4, S (5) = The, f ˆ = , ad t ˆ,0.95 = 1.86, ad approximate 90% cofidece iterval is f 4.98 ± 1.86(1.46) = 4.98 ±.3 ζ (.66, 7.3). This also idicates that the two policies are sesibly differet with the secod policy beig better. Pairedt versus twosample cofidece iterval 7
8 Comparig steadystate measures of two system The approaches for buildig cofidece itervals require iid observatios X 11, X 1,, X 11, ad X1, X,, X each system. These ca be obtaied easily for termiatig simulatio. For otermiatig steadystate simulatios such observatios ca be obtaied by the methods discussed before such as replicatio/deletio ad batch meas. Cofidece Itervals for the Differece of > Systems If more that two systems are to be compared, the, from cofidece itervals for the differece of the performace measure betwee differet pairs of the systems are required. A key issue here is the validity of idividual cofidece itervals for multiple comparisos. Specifically, if c cofidece itervals are to be developed with a overall cofidece level of 1 α, the each iterval must have a cofidece level 1 α/c. (This is called the Boferroi iequality.) This implies that makig multiple comparisos require a large umber of replicatios to achieve the high cofidece level 1 α/c. 8
9 Comparig with a stadard Here, oe of the systems (say system 1) is the stadard (e.g., the existig cofiguratio). We eed to compare each of the other k systems with the stadard, ad obtai cofidece itervals o μ μ 1, μ 3 μ 1,..., μ k μ 1. The, to obtai a overall cofidece of 1 α, each iterval should have a cofidece level 1 α/(k 1). Example 4 The simulatio of five differet (s, S) policies i a ivetory system gave the followig results i 5 replicatios. The performace measure of iterest is the average mothly cost over a horizo o 10 moths. The first policy is cosidered the stadard system. 9
10 The desired overall cofidece level is 90%. So, itervals for μ k μ 1, k =,3,4,5, are developed with a cofidece level (1 0.1/4)% = 97.5%. The cofidece itervals are as follows. Here we see that system 4 ad 5 are obviously worse tha the stadard, while system ad 3 are ot ecessarily better. All pairwise compariso Here all systems are compared pairwise. This would be doe if all alteratives receive equal cosideratio. Cofidece itervals for μi μ 1 i, i 1 = 1,, k, i = 1,, k, i 1 i. The total umber of comparisos to be made is k(k 1)/. So to obtai a overall cofidece of 1 α, the cofidece level for every iterval should be 1 α/[k(k 1)/]. 10
11 Selectig the best of k systems Suppose we wat to select oe of the k alteratives as the best (assume smaller mea is better). We propose a method here that requires specifyig a correctselectio probability P* ad a idifferece zoe d*. That is, the probability that the selected system has a mea that does exceed the best oe by at more tha d* is P*. E.g., d* = 0.01 ad P* = 0.95, implies that there is a 95% chace that the selected mea will ot exceed the best mea by more tha The method work i two stages. First simulate 0 replicatios of each of the k system ( 0 should be large, > 0), ad compute the sample mea ad variace for each system, X ( ) ad (1) i 0 S ( ). i 0 The, compute the sample size N i for each system, as where h 1 are give i Table fuctio of P*, k, ad 0. The, make (N i 0 ) additioal replicatio for each system ad estimate the sample meas X ( N ). () i i 0 The, for each system i compute the weight 11
12 The, compute the weighted sample mea as X ( N ) = W X ( ) + W X ( N ), (1) () i i i1 i 0 i i i 0 where W i = 1 W i1. Pick the systems with the smallest weighted sample mea as best system. Variace Reductio Variace reductio techiques (VRT) are methods to reduce the variace (i.e. icrease precisio) of simulatio output without doig more rus. They are based o settig up the simulatio i smart ways that beefit from correlatios betwee differet radom variables to reduce variability. VRTs are ot free though. E.g., some additioal programmig effort may be eeded. The basic relatio used i a VRT is var( ax + by ) = a var( X ) + b var( Y ) + ab cov( X, Y ). If the last term is egative tha the variace of ax + by could be reduced. Recall that the sample covariace based o observatios of X ad Y is cov( XY, ) = i= 1 ( X X)( Y Y) i 1 i 1
13 Variace reductio usig commo radom umbers (CRN) Thus is used whe comparig the simulatio outputs from two systems. The idea is to use the same stream of radom umbers to compare both systems. Ituitively, this is equivalet to comparig the two systems uder similar coditios. Suppose that iid observatios are available from the output of each system, X 11, X 1,, X 1, ad X 1, X,, X. Cosider the paired differece Z j = X 1j X j, j = 1,,. The, var( Z ) = var( X ) + var( X ) cov( X, X ) j 1j j 1j j This implies var(z j ) is reduced if X 1j ad X j are positively correlated (i.e. cov(x 1j, X 1j ) > 0). This ca be achieved by usig the same stream of radom umber to estimate X 1j ad X 1j. Oe critical poit is sychroizatio. That is, usig radom umbers for the same purpose i both simulatios. Oe way to achieve this is to use dedicated streams for each source of radomess (e.g. oe for arrival times ad oe for service times.) 13
14 I additio, usig a radom variates geeratio method which uses 1 U(0,1) to get 1 variate X ad which uses a mootoe trasformatio U X helps i sychroizatio. The iverse trasform method is highly desired here. CRM works well if the two systems uder compariso react i a similar way to chages i the uderlyig radom umber streams. If ot, we could get cov(x 1, X ) < 0 ad the method bacfires. Example 5 CRN was applied to the comparig the two ATM cofiguratios i Example 1 based o 100 replicatios. CRN was applied i differet ways by sychroizig arrival times oly (A), service times oly (S), ad both arrival ad service times. The results were as follows. (I meas o sychroizatio e.g. usig oe radom umber stream for all purposes.) 14
15 Variace reductio usig atithetic variates (AV) This method is used for variace reductio of a sigle system (i order to get a more precise output). Like CRN, AV works by recyclig radom umbers. The idea is to ru replicatio j based o pairs of two replicatios. Oe replicatio, uses radom umbers U j1, U j,, U jm to estimate a measure of performace X (1) j, ad the other ru uses radom umbers 1 U j1, 1 U j,, 1 U jm, to estimate a similar measure X () j. The, use X j = (X (1) j + X () j )/ as the output of replicatio j. The use of U ad (1 U) is sought to iduce egative correlatio betwee X (1) j ad X () j. The, the variace of X j is less tha the variace of X (1) j ad X () j sice var(x j ) = [var(x (1) j ) + var(x () j ) + cov(x (1) j, X () j )] / 4. The ituitio behid AV is that couterbalacig a large observatio with a small oe leads somewhere close to the mea. Note that AV requires sychroizatio like CRN so that U ad 1 U for the same purpose. 15
16 Variace reductio usig cotrol variates (CV) This method works by relatig the radom variable of iterest, X, (for which we wish to estimate the mea) to aother radom variable with kow mea, Y. E.g, i a queueig simulatio, X could be the delay time ad Y the service time. The idea is to use our kowledge of E[Y] = v to cotrol X whe X ad Y are correlated. Specifically, defie X C = X a(y v), The, E[X C ] = E[X]. The choice of a ca be made i a way that miimizes var[x C ]. Note that var[x C ] = var[x] + a var[y] acov(x,y). This is a secod degree polyomial i a whose miimum is a* = cov(x, Y)/var[Y]. The issue ow is how to estimate cov(x, Y) or eve var[y]. This ca ofte be oly doe through sample data, which teds to bias the estimatio of E[X] through E[X C ]. 16
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 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 informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
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 KolmogorovSmirov 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 informationOnesample test of proportions
Oesample 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 informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo 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 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 informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
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 information5: 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 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 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 information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationCHAPTER 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 informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
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 information, a Wishart distribution with n 1 degrees of freedom and scale matrix.
UMEÅ UNIVERSITET Matematiskstatistiska istitutioe Multivariat dataaalys D MSTD79 PA TENTAMEN 00409 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multivariat dataaalys D, 5 poäg.. Assume that
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 : pvalue
More informationCenter, 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 informationChapter 10. Hypothesis Tests Regarding a Parameter. 10.1 The Language of Hypothesis Testing
Chapter 10 Hypothesis Tests Regardig a Parameter A secod type of statistical iferece is hypothesis testig. Here, rather tha use either a poit (or iterval) estimate from a simple radom sample to approximate
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationTIEE Teaching Issues and Experiments in Ecology  Volume 1, January 2004
TIEE Teachig Issues ad Experimets i Ecology  Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
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 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 informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
More informationMeasures 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 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 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 informationPractice 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 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 informationPage 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 informationNormal 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 information1 Hypothesis testing for a single mean
BST 140.65 Hypothesis Testig Review otes 1 Hypothesis testig for a sigle mea 1. The ull, or status quo, hypothesis is labeled H 0, the alterative H a or H 1 or H.... A type I error occurs whe we falsely
More informationx : X bar Mean (i.e. Average) of a sample
A quick referece for symbols ad formulas covered i COGS14: MEAN OF SAMPLE: x = x i x : X bar Mea (i.e. Average) of a sample x i : X sub i This stads for each idividual value you have i your sample. For
More informationNow 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 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 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 informationAnalyzing 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 informationConfidence 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 informationOverview. 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 informationNotes on Hypothesis Testing
Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter
More informationCOMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
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 information9.8: THE POWER OF A TEST
9.8: The Power of a Test CD91 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based
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 informationIncremental 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 information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationRobust and Resistant Regression
Chapter 13 Robust ad Resistat Regressio Whe the errors are ormal, least squares regressio is clearly best but whe the errors are oormal, other methods may be cosidered. A particular cocer is logtailed
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 informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife 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 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 informationStandard Errors and Confidence Intervals
Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5yearold boys. If we assume
More information15.075 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 highdefiitio
More informationChapter 10 Student Lecture Notes 101
Chapter 0 tudet Lecture Notes 0 Basic Busiess tatistics (9 th Editio) Chapter 0 Twoample Tests with Numerical Data 004 PreticeHall, Ic. Chap 0 Chapter Topics Comparig Two Idepedet amples Z test 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 informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationLesson 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 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 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 informationLearning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.
Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the
More informationBASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.
BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts
More informationQuadrat 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 informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More informationJoint Probability Distributions and Random Samples
STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margial
More informationConfidence intervals and hypothesis tests
Chapter 2 Cofidece itervals ad hypothesis tests This chapter focuses o how to draw coclusios about populatios from sample data. We ll start by lookig at biary data (e.g., pollig), ad lear how to estimate
More informationDetermining 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 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 Chisquare (χ ) distributio.
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
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 informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70450) 18004186789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, 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 informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationMannWhitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)
NoParametric ivariate Statistics: WilcoxoMaWhitey 2 Sample Test 1 MaWhitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo) MaWhitey (WMW) test is the oparametric equivalet of a pooled
More informationUnit 20 Hypotheses Testing
Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect
More informationProblem Set 1 Oligopoly, market shares and concentration indexes
Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet
More informationSampling 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 informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationSolutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork
Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the
More informationMEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)
MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:
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 informationConfidence Interval Estimation of the Shape. Parameter of Pareto Distribution. Using Extreme Order Statistics
Applied Mathematical Scieces, Vol 6, 0, o 93, 4674640 Cofidece Iterval Estimatio of the Shape Parameter of Pareto Distributio Usig Extreme Order Statistics Aissa Omar, Kamarulzama Ibrahim ad Ahmad Mahir
More information*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 informationQuantitative Computer Architecture
Performace Measuremet ad Aalysis i Computer Quatitative Computer Measuremet Model Iovatio Proposed How to measure, aalyze, ad specify computer system performace or My computer is faster tha your computer!
More informationLearning objectives. Duc K. Nguyen  Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the timevalue
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS 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 BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
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 informationSample size for clinical trials
Outcome variables for trials British Stadards Istitutio Study Day Sample size for cliical trials Marti Blad Prof. of Health Statistics Uiversity of York http://martiblad.co.uk A outcome variable is oe
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 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 informationThis is arithmetic average of the x values and is usually referred to simply as the mean.
prepared by Dr. Adre Lehre, Dept. of Geology, Humboldt State Uiversity http://www.humboldt.edu/~geodept/geology51/51_hadouts/statistical_aalysis.pdf STATISTICAL ANALYSIS OF HYDROLOGIC DATA This hadout
More informationSpss Lab 7: Ttests Section 1
Spss Lab 7: Ttests Sectio I this lab, we will be usig everythig we have leared i our text ad applyig that iformatio to uderstad ttests for parametric ad oparametric data. THERE WILL BE TWO SECTIONS FOR
More informationOMG! Excessive Texting Tied to Risky Teen Behaviors
BUSIESS WEEK: EXECUTIVE EALT ovember 09, 2010 OMG! Excessive Textig Tied to Risky Tee Behaviors Kids who sed more tha 120 a day more likely to try drugs, alcohol ad sex, researchers fid TUESDAY, ov. 9
More informationCorrelation. example 2
Correlatio Iitially developed by Sir Fracis Galto (888) ad Karl Pearso (8) Sir Fracis Galto 8 correlatio is a much abused word/term correlatio is a term which implies that there is a associatio betwee
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 information