Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation"

Transcription

1 Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio

2 Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace. V( ˆ ) Almost always the case whe {,, } is a sequece of output observatios from withi a sigle replicatio (autocorrelated sequece, time-series). Suppose the poit estimator q is the sample mea i i / Variace of is almost impossible to estimate. For system with steady state, produce a output process that is approximately covariace statioary (after passig the trasiet phase). The covariace betwee two radom variables i the time series depeds oly o the lag (the # of observatios betwee them).

3 3 Error Estimatio For a covariace statioary time series, {,, }: Lag-k autocovariace is: Lag-k autocorrelatio is: If a time series is covariace statioary, the the variace of is: The expected value of the variace estimator is: ), cov( ), cov( k i i k k k k ) ( k k k V / where ), ( c B BV S E c

4 Error Estimatio The expected value of the variace estimator is: E S BV ( ), where B / c ad V ( ) is the variace of If i are idepedet, the S / is a ubiased estimator of V ( ) If the autocorrelatio k are primarily positive, the S / is biased low as a estimator of V ( ). If the autocorrelatio k are primarily egative, the S / is biased high as a estimator of V ( ). 4

5 C.I. with Specified Precisio [Termiatig Simulatios] The half-legth H of a 00( a)% cofidece iterval for a mea q, based o the t distributio, is give by: S H t a /, R R R is the # of replicatios S is the sample variace Suppose that a error criterio e is specified with probability - a, a sufficietly large sample size should satisfy: P q e a.. 5

6 C.I. with Specified Precisio [Termiatig Simulatios] Assume that a iitial sample of size R 0 (idepedet) replicatios has bee observed. Obtai a iitial estimate S 0 of the populatio variace. The, choose sample size R such that R R 0 : Sice t a/, R- z a/, a iitial estimate of R: za / S0 R, za / is the stadard ormal distributio. e t /, 0 R is the smallest iteger satisfyig R R 0 ad a R S R e Collect R - R 0 additioal observatios. The 00(-a)% C.I. for q:.. t a /, R S R 6

7 C.I. with Specified Precisio [Termiatig Simulatios] Call Ceter Example: estimate the aget s utilizatio over the first hours of the workday. Iitial sample of size R 0 = 4 is take ad a iitial estimate of the populatio variace is S 0 = (0.07) = The error criterio is e= 0.04 ad cofidece coefficiet is -a = 0.95, hece, the fial sample size must be at least: 0 z.05s0.96 * e 0.04 For the fial sample size:.4 R t 0.05, R ta /, RS0 /e R = 5 is the smallest iteger satisfyig the error criterio, so R - R 0 = additioal replicatios are eeded. After obtaiig additioal outputs, half-width should be checked. 7

8 Output Aalysis for Steady-State Simulatio Cosider a sigle ru of a simulatio model to estimate a steady-state or log-ru characteristics of the system. The sigle ru produces observatios,,... (geerally the samples of a autocorrelated time series). Performace measure: q lim T i lim 0 T E E T, E for discrete measure Idepedet of the iitial coditios. i ( t) dt, for cotiuous measure (with probability ) (with probability ) 8

9 Output Aalysis for Steady-State Simulatio The sample size is a desig choice, with several cosideratios i mid: Ay bias i the poit estimator that is due to artificial or arbitrary iitial coditios (bias ca be severe if ru legth is too short). Desired precisio of the poit estimator. Budget costraits o computer resources. Notatio: the estimatio of q from a discrete-time output process. Oe replicatio (or ru), the output data:,, 3, With several replicatios, the output data for replicatio r: r, r, r3, 9

10 Iitializatio Bias Methods to reduce the poit-estimator bias caused by usig artificial ad urealistic iitial coditios: Itelliget iitializatio. Divide simulatio ito a iitializatio phase ad data-collectio phase. Itelliget iitializatio Iitialize the simulatio i a state that is more represetative of log-ru coditios. If the system exists, collect data o it ad use these data to specify more early typical iitial coditios. If the system ca be simplified eough to make it mathematically solvable, e.g. queueig models, solve the simplified model to fid log-ru expected or most likely coditios, use that to iitialize the simulatio. 0

11 Iitializatio Bias Divide each simulatio ito two phases: A iitializatio phase, from time 0 to time T 0. A data-collectio phase, from T 0 to the stoppig time T 0 +T E. The choice of T 0 is importat: After T 0, system should be more early represetative of steady-state behavior. System has reached steady state: the probability distributio of the system state is close to the steady-state probability distributio (bias of respose variable is egligible).

12 Iitializatio Bias M/G/ queueig example: A total of 0 idepedet replicatios were made. Each replicatio begiig i the empty ad idle state. Simulatio ru legth o each replicatio was T 0 +T E = 5,000 miutes. Respose variable: queue legth, L Q (t,r) (at time t of the rth replicatio). Batchig itervals of,000 miutes, batch meas Esemble averages: To idetify tred i the data due to iitializatio bias The average correspodig batch meas across replicatios:. j rj R r The preferred method to determie deletio poit. R R replicatios

13 Iitializatio Bias 3

14 Iitializatio Bias 4

15 Iitializatio Bias A plot of the esemble averages,..(, d), versus 000j, for j =,,,5. Illustrates the dowward bias of the iitial observatios. 5

16 Iitializatio Bias Cumulative average sample mea (after deletig d observatios):.. (, d). j d jd Not recommeded to determie the iitializatio phase. It is apparet that dowward bias is preset ad this bias ca be reduced by deletio of oe or more observatios. 6

17 Iitializatio Bias No widely accepted, objective ad prove techique to guide how much data to delete to reduce iitializatio bias to a egligible level. Plots ca, at times, be misleadig but they are still recommeded. Cumulative average becomes less variable as more data are averaged. The more correlatio preset, the loger it takes for approach steady state.. j to Differet performace measures could approach steady state at differet rates. 7

It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.

It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable. Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig

More information

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { } UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to: STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Probability and statistics: basic terms

Probability and statistics: basic terms Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample

More information

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1. Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

More information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced

More information

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.

The variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2. SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample

More information

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give

More information

(all terms are scalars).the minimization is clearer in sum notation:

(all terms are scalars).the minimization is clearer in sum notation: 7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1

More information

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 13, Part A Analysis of Variance and Experimental Design Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of

More information

Statisticians use the word population to refer the total number of (potential) observations under consideration

Statisticians use the word population to refer the total number of (potential) observations under consideration 6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space

More information

Sampling Distributions, Z-Tests, Power

Sampling Distributions, Z-Tests, Power Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace

More information

ARIMA Models. Dan Saunders. y t = φy t 1 + ɛ t

ARIMA Models. Dan Saunders. y t = φy t 1 + ɛ t ARIMA Models Da Sauders I will discuss models with a depedet variable y t, a potetially edogeous error term ɛ t, ad a exogeous error term η t, each with a subscript t deotig time. With just these three

More information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals 7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses

More information

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N. 3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear

More information

Probability, Expectation Value and Uncertainty

Probability, Expectation Value and Uncertainty Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

More information

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10 DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set

More information

y ij = µ + α i + ɛ ij,

y ij = µ + α i + ɛ ij, STAT 4 ANOVA -Cotrasts ad Multiple Comparisos /3/04 Plaed comparisos vs uplaed comparisos Cotrasts Cofidece Itervals Multiple Comparisos: HSD Remark Alterate form of Model I y ij = µ + α i + ɛ ij, a i

More information

Stat 200 -Testing Summary Page 1

Stat 200 -Testing Summary Page 1 Stat 00 -Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece

More information

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes. Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (

More information

HOMEWORK 2 SOLUTIONS

HOMEWORK 2 SOLUTIONS HOMEWORK SOLUTIONS CSE 55 RANDOMIZED AND APPROXIMATION ALGORITHMS 1. Questio 1. a) The larger the value of k is, the smaller the expected umber of days util we get all the coupos we eed. I fact if = k

More information

11. Simulation. Contents. Alternative to what? What is simulation?

11. Simulation. Contents. Alternative to what? What is simulation? Cotets Itroductio Geeratio of traffic process realizatios Geeratio of radom variable realizatios Collectio of data Statistical aalsis Lect.ppt S-38.45 - Itroductio to Teletraffic Theor Sprig 005 What is

More information

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1 PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured

More information

Discrete probability distributions

Discrete probability distributions Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop

More information

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

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

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13 BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the

More information

Chapter 6 Principles of Data Reduction

Chapter 6 Principles of Data Reduction Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a

More information

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios

More information

Output Analysis and Run-Length Control

Output Analysis and Run-Length Control IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%

More information

IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 2006 SECTION A IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

V. Nollau Institute of Mathematical Stochastics, Technical University of Dresden, Germany

V. Nollau Institute of Mathematical Stochastics, Technical University of Dresden, Germany PROBABILITY AND STATISTICS Vol. III - Correlatio Aalysis - V. Nollau CORRELATION ANALYSIS V. Nollau Istitute of Mathematical Stochastics, Techical Uiversity of Dresde, Germay Keywords: Radom vector, multivariate

More information

Statistics 20: Final Exam Solutions Summer Session 2007

Statistics 20: Final Exam Solutions Summer Session 2007 1. 20 poits Testig for Diabetes. Statistics 20: Fial Exam Solutios Summer Sessio 2007 (a) 3 poits Give estimates for the sesitivity of Test I ad of Test II. Solutio: 156 patiets out of total 223 patiets

More information

Binomial Distribution

Binomial Distribution 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible

More information

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >

More information

6. Uniform distribution mod 1

6. Uniform distribution mod 1 6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which

More information

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m? MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle

More information

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ. 2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For

More information

Two-Stage Controlled Fractional Factorial Screening for Simulation Experiments

Two-Stage Controlled Fractional Factorial Screening for Simulation Experiments Two-Stage Cotrolled Fractioal Factorial Screeig for Simulatio Experimets Hog Wa Assistat Professor School of Idustrial Egieerig Purdue Uiversity 35 N. Grat Street West Lafayette, IN 4797-223 hwa@purdue.edu

More information

The Sample Variance Formula: A Detailed Study of an Old Controversy

The Sample Variance Formula: A Detailed Study of an Old Controversy The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace

More information

Sampling, Sampling Distribution and Normality

Sampling, Sampling Distribution and Normality 4/17/11 Tools of Busiess Statistics Samplig, Samplig Distributio ad ormality Preseted by: Mahedra Adhi ugroho, M.Sc Descriptive statistics Collectig, presetig, ad describig data Iferetial statistics Drawig

More information

ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION

ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy

More information

Lecture 4. Random variable and distribution of probability

Lecture 4. Random variable and distribution of probability Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za

More information

Simple Linear Regression

Simple Linear Regression Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio

More information

Review Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn

Review Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom

More information

Math 2784 (or 2794W) University of Connecticut

Math 2784 (or 2794W) University of Connecticut ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties

More information

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig

More information

Varanasi , India. Corresponding author

Varanasi , India. Corresponding author A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet

More information

UCLA STAT 110B Applied Statistics for Engineering and the Sciences

UCLA STAT 110B Applied Statistics for Engineering and the Sciences UCLA STAT 110B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles,

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak

More information

Jacob Hays Amit Pillay James DeFelice 4.1, 4.2, 4.3

Jacob Hays Amit Pillay James DeFelice 4.1, 4.2, 4.3 No-Parametric Techiques Jacob Hays Amit Pillay James DeFelice 4.1, 4.2, 4.3 Parametric vs. No-Parametric Parametric Based o Fuctios (e.g Normal Distributio) Uimodal Oly oe peak Ulikely real data cofies

More information

Confidence Level We want to estimate the true mean of a random variable X economically and with confidence.

Confidence Level We want to estimate the true mean of a random variable X economically and with confidence. Cofidece Iterval 700 Samples Sample Mea 03 Cofidece Level 095 Margi of Error 0037 We wat to estimate the true mea of a radom variable X ecoomically ad with cofidece True Mea μ from the Etire Populatio

More information

NCSS Statistical Software. Tolerance Intervals

NCSS Statistical Software. Tolerance Intervals Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided

More information

Confidence Intervals for the Population Proportion p

Confidence Intervals for the Population Proportion p Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:

More information

Probability theory and mathematical statistics:

Probability theory and mathematical statistics: N.I. Lobachevsky State Uiversity of Nizhi Novgorod Probability theory ad mathematical statistics: Law of Total Probability. Associate Professor A.V. Zorie Law of Total Probability. 1 / 14 Theorem Let H

More information

LECTURE 11 LINEAR PROCESSES III: ASYMPTOTIC RESULTS

LECTURE 11 LINEAR PROCESSES III: ASYMPTOTIC RESULTS PRIL 7, 9 where LECTURE LINER PROCESSES III: SYMPTOTIC RESULTS (Phillips ad Solo (99) ad Phillips Lecture Notes o Statioary ad Nostatioary Time Series) I this lecture, we discuss the LLN ad CLT for a liear

More information

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters? CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter

More information

Lecture 1 Probability and Statistics

Lecture 1 Probability and Statistics Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark

More information

Section 14. Simple linear regression.

Section 14. Simple linear regression. Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo

More information

Median and IQR The median is the value which divides the ordered data values in half.

Median and IQR The median is the value which divides the ordered data values in half. STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media

More information

DISTRIBUTION LAW Okunev I.V.

DISTRIBUTION LAW Okunev I.V. 1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated

More information

INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS

More information

Advanced Engineering Mathematics Exercises on Module 4: Probability and Statistics

Advanced Engineering Mathematics Exercises on Module 4: Probability and Statistics Advaced Egieerig Mathematics Eercises o Module 4: Probability ad Statistics. A survey of people i give regio showed that 5% drak regularly. The probability of death due to liver disease, give that a perso

More information

Fig. 2. Block Diagram of a DCS

Fig. 2. Block Diagram of a DCS Iformatio source Optioal Essetial From other sources Spread code ge. Format A/D Source ecode Ecrypt Auth. Chael ecode Pulse modu. Multiplex Badpass modu. Spread spectrum modu. X M m i Digital iput Digital

More information

Grouping 2: Spectral and Agglomerative Clustering. CS 510 Lecture #16 April 2 nd, 2014

Grouping 2: Spectral and Agglomerative Clustering. CS 510 Lecture #16 April 2 nd, 2014 Groupig 2: Spectral ad Agglomerative Clusterig CS 510 Lecture #16 April 2 d, 2014 Groupig (review) Goal: Detect local image features (SIFT) Describe image patches aroud features SIFT, SURF, HoG, LBP, Group

More information

Probability and Statistics

Probability and Statistics ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,

More information

HOMEWORK #10 SOLUTIONS

HOMEWORK #10 SOLUTIONS Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous

More information

o <Xln <X2n <... <X n < o (1.1)

o <Xln <X2n <... <X n < o (1.1) Metrika, Volume 28, 1981, page 257-262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow

More information

Element sampling: Part 2

Element sampling: Part 2 Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig

More information

STATISTICAL INFERENCE

STATISTICAL INFERENCE STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample

More information

SRC Technical Note June 17, Tight Thresholds for The Pure Literal Rule. Michael Mitzenmacher. d i g i t a l

SRC Technical Note June 17, Tight Thresholds for The Pure Literal Rule. Michael Mitzenmacher. d i g i t a l SRC Techical Note 1997-011 Jue 17, 1997 Tight Thresholds for The Pure Literal Rule Michael Mitzemacher d i g i t a l Systems Research Ceter 130 Lytto Aveue Palo Alto, Califoria 94301 http://www.research.digital.com/src/

More information

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

More information

R. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State

R. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

More information

4.1 Sigma Notation and Riemann Sums

4.1 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 information

A LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!

A LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!! A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited

More information

B Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets

B Supplemental Notes 2 Hypergeometric, Binomial, Poisson and Multinomial Random Variables and Borel Sets B671-672 Supplemetal otes 2 Hypergeometric, Biomial, Poisso ad Multiomial Radom Variables ad Borel Sets 1 Biomial Approximatio to the Hypergeometric Recall that the Hypergeometric istributio is fx = x

More information

Matrix Representation of Data in Experiment

Matrix Representation of Data in Experiment Matrix Represetatio of Data i Experimet Cosider a very simple model for resposes y ij : y ij i ij, i 1,; j 1,,..., (ote that for simplicity we are assumig the two () groups are of equal sample size ) Y

More information

Correlation and Covariance

Correlation and Covariance Correlatio ad Covariace Tom Ilveto FREC 9 What is Next? Correlatio ad Regressio Regressio We specify a depedet variable as a liear fuctio of oe or more idepedet variables, based o co-variace Regressio

More information

1. By using truth tables prove that, for all statements P and Q, the statement

1. By using truth tables prove that, for all statements P and Q, the statement Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3

More information

Eksamen 2006 H Utsatt SENSORVEILEDNING. Problem 1. Settet består av 9 delspørsmål som alle anbefales å telle likt. Svar er gitt i <<.. >>.

Eksamen 2006 H Utsatt SENSORVEILEDNING. Problem 1. Settet består av 9 delspørsmål som alle anbefales å telle likt. Svar er gitt i <<.. >>. Eco 43 Eksame 6 H Utsatt SENSORVEILEDNING Settet består av 9 delspørsmål som alle abefales å telle likt. Svar er gitt i . Problem a. Let the radom variable (rv.) X be expoetially distributed with

More information

Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Length Measurements with the Four-Sided Meter Stick Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

More information

Lecture 10 October Minimaxity and least favorable prior sequences

Lecture 10 October Minimaxity and least favorable prior sequences STATS 300A: Theory of Statistics Fall 205 Lecture 0 October 22 Lecturer: Lester Mackey Scribe: Brya He, Rahul Makhijai Warig: These otes may cotai factual ad/or typographic errors. 0. Miimaxity ad least

More information

Question 1: Exercise 8.2

Question 1: Exercise 8.2 Questio 1: Exercise 8. (a) Accordig to the regressio results i colum (1), the house price is expected to icrease by 1% ( 100% 0.0004 500 ) with a additioal 500 square feet ad other factors held costat.

More information

Modified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis

Modified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry

More information

Some Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables

Some Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables Some Basic Probability Cocepts 2. Experimets, Outcomes ad Radom Variables A radom variable is a variable whose value is ukow util it is observed. The value of a radom variable results from a experimet;

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

Research Article Health Monitoring for a Structure Using Its Nonstationary Vibration

Research Article Health Monitoring for a Structure Using Its Nonstationary Vibration Advaces i Acoustics ad Vibratio Volume 2, Article ID 69652, 5 pages doi:.55/2/69652 Research Article Health Moitorig for a Structure Usig Its Nostatioary Vibratio Yoshimutsu Hirata, Mikio Tohyama, Mitsuo

More information

Solutions to Final Exam Review Problems

Solutions to Final Exam Review Problems . Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

More information

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4. 4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad

More information

Analytic Continuation

Analytic Continuation Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for

More information

Lecture 3 The Lebesgue Integral

Lecture 3 The Lebesgue Integral Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified

More information

STATISTICAL method is one branch of mathematical

STATISTICAL method is one branch of mathematical 40 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 Optimizig Forest Samplig by usig Lagrage Multipliers Suhud Wahyudi, Farida Agustii Widjajati ad Dea Oktaviati

More information

A note on self-normalized Dickey-Fuller test for unit root in autoregressive time series with GARCH errors

A note on self-normalized Dickey-Fuller test for unit root in autoregressive time series with GARCH errors Appl. Math. J. Chiese Uiv. 008, 3(): 97-0 A ote o self-ormalized Dickey-Fuller test for uit root i autoregressive time series with GARCH errors YANG Xiao-rog ZHANG Li-xi Abstract. I this article, the uit

More information

DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES FREE BOOKLET CSIR-UGC/NET MATHEMATICAL SCIENCES

DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES FREE BOOKLET CSIR-UGC/NET MATHEMATICAL SCIENCES DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES FREE BOOKLET DEEPAK SERIES CSIR-UGC/NET MATHEMATICAL SCIENCES SOLVED PAPER DEC- DEEPAK SERIES DEEPAK SERIES DEEPAK SERIES Note : This material is issued as complimetary

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

More information