The standard deviation of the mean

 Janel Cooper
 5 months ago
 Views:
Transcription
1 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 a radom variable x ad the correspodig probability distributio p(x). For coveiece, we cosider the case of a discrete radom variable, although the geeralizatio to cotiuous radom variables is straightforward. Give p(x), oe ca easily compute the expectatio value ad the variace, E(x) µ = x xp(x), () Var(x) = x (x µ) 2 p(x) = E(x 2 ) [E(x)] 2. (2) The stadard deviatio of x is deoted by σ Var(x). I the real world, p(x) is usually ukow, i which case µ ad σ are ukow. However, oe ca perform experimets to measure x. Suppose measuremets are made, ad the values x, x 2,...x are obtaied. Ideally, we would like to recostruct the probability distributio p(x) from the data, but here we are iterested i determiig the expectatio value µ ad the stadard deviatio σ from the experimetal results. We ca regard x, x 2,...x as idepedet ad idetically distributed radom variables (ofte abbreviated as iid or IID radom variables). These are idepedet, sice separate measuremets of x are idepedet of each other. These are idetically distributed, sice the experimet is measurig the same radom variable x each time (although, of course, the outcome of each measuremet will ot be the same). This meas that E(x i ) = µ ad Var(x i ) =, for i =,2,3,...,. Of course, the above iformatio is ot very practical, sice a priori we do ot kow the values of µ ad σ. Havig made idepedet measuremets, we would like to ascertai the best possible estimates for µ ad σ. I class, we defied the sample average x ad the sample variace Σ 2 by x x i, (3) Σ 2 (x i x) 2. (4)
2 These quatities are easily computed from the data. We ow assert that the sample average x provides a best estimate for the actual mea µ ad the sample variace Σ 2 provides a best estimate for the actual variace. I the mathematical statistics literature, there is some debate as to the meaig of the word best. I the preset cotext, the word best simply meas that the estimates are ubiased, that is E(x) = µ ad E(Σ 2 ) =, (5) where the expectatio values are computed assumig for a momet that we do kow the uderlyig probability distributio p(x). Let us verify eq. (5) explicitly. First, recallig that E(cx) = ce(x) ad E(x+y) = E(x)+E(y), we have E(x) = E(x i ) = µ = µ = µ. I the Appedix, we demostrate that E(Σ 2 ) =. To reiterate, x provides a best estimate of the ukow µ, which is the expectatio value of the radom variable x. Similarly, Σ 2 provides a best estimate of the ukow, which is the expectatio value of (x µ) The stadard deviatio of the mea Although x provides a best estimate of the ukow µ, its determiatio does ot tell us howlikely it is that themeasured valuexis close toµ. After all, if Iperformadditioal measuremets of x, I would expect the value of the average x to chage (although the chage is expected to be small oce is large eough). Thus, what we would really like to kow is the probability distributio of the radom variable x. Of course, sice we do ot kow i geeral the expectatio value ad variace of x, we also do ot kow i geeral the expectatio value ad variace of x. Ideed, we have already see that E(x) = µ, which we do ot kow. Likewise, we ca compute Var(x) as follows: Var(x) = Var ( ) x i = Var(x 2 i ) = 2 = 2 σ2 = σ2, (6) which depeds o the ukow. However, we do have a best estimate for based o our data, amely Σ 2 defied i eq. (4). Hece, we shall defie the stadard deviatio of the mea (also called the stadard error) to be σ m, where m Σ2 = () (x i x) 2. (7) Theexperimetalistowcocludesaftertakigdataadobtaiigthevaluesx,x 2,...,x after measuremets, that the best estimate of the mea is x±σ m. 2
3 If there is a theoretical value of µ to compare this to, the experimetalist ca ow make statemets ivolvig cofidece itervals (e.g., the probability that the data is cosistet with the theoretical expectatio), as discussed i Boas. It is very importat to distiguish σ m, which is obtaied from data ad σ which is the ukow stadard deviatio of the radom variable x. We have = Var(x), which is determied by the probability distributio p(x) ad does ot deped o the umber measuremets performed by the experimetalist. The experimetalist ca make a estimate for, amely Σ 2 give by eq. (4). It may look like Σ 2 depeds o, but the depedece is pretty weak (if is large). After all, x also depeds weakly o (if is large), which provides the best estimate for µ. However, σm 2 = Σ2 / depeds strogly o. The more measuremets that are made, the smaller σm 2 is. This is ot surprisig, sice oe expects that the larger is, the better x is as a estimate for µ. As emphasized above, σm 2 is a estimate of the variace of x, which is obviously ot the same as the variace of x [they differ by a factor of as show i eq. (6)]. Equivaletly, Σ is a estimate of the ucertaity i a sigle measuremet of the radom variable x, whereas σ m is a estimate o the ucertaity of the mea value of the radom variable x as determied by measuremets. A simple example illustrates the above discussio. Suppose that p(x) is the biomial distributio with probability p that a tossed coi will lad o heads. Defie the radom variable, { x =, the coi lads o heads, x = x = 0, the coi lads o tails. Give this coi, the experimetalist is asked to determie the mea µ = p ad the variace = p( p) by flippig the coi times. After flips, the experimetalist obtais a data set, x, x 2,...,x, which is a series of s ad 0s. From this data, the experimetalist computes x which is equal to the umber of heads divided by. The experimetalist also computes Σ usig eq. (4) ad σ m usig eq. (7). The experimetalist cocludes that the probability p of the coi (i.e, the true mea µ) is x±σ m, where the error bars represet a 68% cofidece iterval, correspodig to a oe stadard deviatio of the mea ucertaity. Clearly, the large is (i.e. more coi flips), the smaller the correspodig stadard error σ m, ad cosequetly the more reliable x is as a estimate of the probability p of the coi. Likewise, the best estimate for is give by Σ 2. By the way, the latter determiatio also has a error associated with it, which I briefly discuss i Sectio 3 of these otes. Refereces ad 2 provide a coget discussio of the differeces betwee stadard deviatio ad the stadard deviatio of the mea. I particular, referece is a superb treatmet of error aalysis writte specifically for physicists at a elemetary level. 3
4 3. The stadard deviatio of the variace Although Σ 2 provides a best estimate of the ukow, this does ot tell us how likely it is that the measured value Σ 2 is close to. After all, if I perform additioal measuremets of x, I would expect the value of the average Σ 2 to chage (although the chage is expected to be small oce is large eough). Thus, what we would really like to kow is the probability distributio of the radom variable Σ 2. Of course, sice we do ot kow i geeral the expectatio value ad variace of x, we also do ot kow i geeral the expectatio value ad variace of Σ 2. Ideed, we have already see that E(Σ 2 ) =, which we do ot kow. Likewise, oe ca compute Var(Σ 2 ). The result depeds o ad o E(x 4 ) which we have ot discussed i this course. However, it may be of some iterest to cosider the case of a ormal distributio, sice the cetral limit theorem ca be applied if is large eough. I this case, it is straightforward to show that E(x 4 ) = 3σ 4, i which case Var(Σ 2 ) depeds oly o σ. The result (obtaied i Appedix E of referece ad Appedix C of referece 3) is: Var(Σ 2 ) = 2(), which agai depeds o the ukow. However, we ca agai employ best estimate for based o our data, amely Σ 2. Thus, we coclude that uder the assumptio that p(x) is the ormal distributio of ukow mea ad variace, the the best estimate of the stadard deviatio of the variace of the radom variable x obtaied from our data is give by σ v, where v = Σ 2 2() = 2() 2 (x i x) 2. As i the case of σ m, we see that σ v also ca be reduced i size by performig more measuremets (i.e. by takig larger). However, i practice σ v (sometimes called the error of the error ) is ot ofte employed i experimetal aalyses. Refereces. Joh R. Taylor, A Itroductio to Error Aalysis: the study of ucertaities i physical measuremets, 2d editio (Uiversity Sciece Books, Sausalito, CA, 997). 2. David L. Streier, Maitaiig Stadards: Differeces betwee the Stadard Deviatio ad the Stadard Error, ad Whe to Use Each, Caadia Joural of Psychiatry, 4 (996) pp Jörg W. Müller, Some Secod Thoughts o Error Statemets, Nuclear Istrumets ad Methods 63 (979)
5 APPENDIX: Proof that E(Σ 2 ) = Startig with eq. (4), we shall compute E(Σ 2 ) = It is coveiet to rewrite the above equatio by otig that after usig E[(x i x) 2 ]. (8) Var(x i x) = E[(x i x) 2 ] [E(x i x)] 2 = E[(x i x) 2 ], Thus, eq. (8) ca be rewritte as E(x i x) = E(x i ) E(x) = µ µ = 0. E(Σ 2 ) = Var(x i x). To evaluate the above expressio, we shall use Var(cx) = c 2 Var(x) ad Var(x + y) = Var(x)+Var(y), where the latter holds uder the assumptio that x ad y are idepedet radom variables. Sice x i ad x are ot idepedet radom variables (sice x cotais x i i its defiitio), we must perform the followig maipulatio, x i x = x i ( ) x i = x i x j. Cosequetly, E(Σ 2 ) = = = = = = [ ( ) ] Var x i x j ( ) 2 Var(x i )+ Var(x 2 j ) ( ) ( ) 2 ( ) [+] = σ2 = σ2, which completes the proof. Note that this computatio justifies the presece of the deomiator factor rather tha i eq. (4). 5
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 informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the zaxis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationStatisticians 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 informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationBinomial 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 informationThe 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 informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationDiscrete probability distributions
Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationDS 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 informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More 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 informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More information71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 6 9/24/2008 DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS Cotets 1. A few useful discrete radom variables 2. Joit, margial, ad
More informationLecture 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 email: za@agh.edu.pl http://home.agh.edu.pl/~za
More informationSampling Distributions, ZTests, Power
Samplig Distributios, ZTests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace
More informationFirst Year Quantitative Comp Exam Spring, Part I  203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I1 Part I  203A A radom variable X is distributed with the margial desity: >
More informationSome 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 informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More information2.2. Central limit theorem.
36.. Cetral limit theorem. The most ideal case of the CLT is that the radom variables are iid with fiite variace. Although it is a special case of the more geeral LidebergFeller CLT, it is most stadard
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI2 (1075) STATISTICAL DECISION MAKING Advaced
More informationLecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables
CSCIB609: A Theorist s Toolkit, Fall 06 Aug 3 Lecture 0: the Cetral Limit Theorem Lecturer: Yua Zhou Scribe: Yua Xie & Yua Zhou Cetral Limit Theorem for iid radom variables Let us say that we wat to aalyze
More informationA 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 informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 00900 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationConfidence 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 informationBHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13
BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the
More informationOutput Analysis and RunLength Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad RuLegth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
More informationKLMED8004 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 informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and nonusers, x  y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad ousers, x  y. Such studies are sometimes viewed
More informationStat 200 Testing Summary Page 1
Stat 00 Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece
More informationStatistics 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 informationVariance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio
More information(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 informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationMASSACHUSETTS 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 informationDISTRIBUTION LAW Okunev I.V.
1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Twoparameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationAsymptotic Results for the Linear Regression Model
Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform largesample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationESTIMATION AND PREDICTION BASED ON KRECORD VALUES FROM NORMAL DISTRIBUTION
STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON KRECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum 695581, Kerala, Idia M. Shy
More informationA LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!
A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3337 HIKARI Ltd, www.mhikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationMath 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 informationIntroduction to Probability and Statistics Twelfth Edition
Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver Itroductio to Probability ad Statistics Twelfth
More informationSolutions to Odd Numbered End of Chapter Exercises: Chapter 4
Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd Numbered Ed of Chapter Exercises: Chapter 4 (This versio July 2, 24) Stock/Watso  Itroductio to Ecoometrics
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More informationSTATISTICAL 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 informationSection 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 informationLinear Regression Models
Liear Regressio Models Dr. Joh MellorCrummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
More informationLecture 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 informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 5766 ON POINTWISE BINOMIAL APPROXIMATION BY wfunctions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationA RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NONPARAMETRIC KSAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider ksample ad chage poit problems for idepedet data i a
More informationMOMENTMETHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMETMETHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationf(x)dx = 1 and f(x) 0 for all x.
OCR Statistics 2 Module Revisio Sheet The S2 exam is 1 hour 30 miutes log. You are allowed a graphics calculator. Before you go ito the exam make sureyou are fully aware of the cotets of theformula booklet
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
More informationLecture 11 October 27
STATS 300A: Theory of Statistics Fall 205 Lecture October 27 Lecturer: Lester Mackey Scribe: Viswajith Veugopal, Vivek Bagaria, Steve Yadlowsky Warig: These otes may cotai factual ad/or typographic errors..
More informationHOMEWORK #10 SOLUTIONS
Math 33  Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous
More informationLecture 4 The Simple Random Walk
Lecture 4: The Simple Radom Walk 1 of 9 Course: M36K Itro to Stochastic Processes Term: Fall 014 Istructor: Gorda Zitkovic Lecture 4 The Simple Radom Walk We have defied ad costructed a radom walk {X }
More informationDimensionfree PACBayesian bounds for the estimation of the mean of a random vector
Dimesiofree PACBayesia bouds for the estimatio of the mea of a radom vector Olivier Catoi CREST CNRS UMR 9194 Uiversité Paris Saclay olivier.catoi@esae.fr Ilaria Giulii Laboratoire de Probabilités et
More informationA Risk Comparison of Ordinary Least Squares vs Ridge Regression
Joural of Machie Learig Research 14 (2013) 15051511 Submitted 5/12; Revised 3/13; Published 6/13 A Risk Compariso of Ordiary Least Squares vs Ridge Regressio Paramveer S. Dhillo Departmet of Computer
More informationElement 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
o
Metrika, Volume 28, 1981, page 257262. 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 informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationA Simplified Binet Formula for kgeneralized Fibonacci Numbers
A Simplified Biet Formula for kgeeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More informationTopic 6 Sampling, hypothesis testing, and the central limit theorem
CSE 103: Probability ad statistics Fall 2010 Topic 6 Samplig, hypothesis testig, ad the cetral limit theorem 61 The biomial distributio Let X be the umberofheadswhe acoiofbiaspistossedtimes The distributio
More informationTRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN
HARDMEKO 004 Hardess Measuremets Theory ad Applicatio i Laboratories ad Idustries  November, 004, Washigto, D.C., USA TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN Koichiro HATTORI, Satoshi
More informationCH19 Confidence Intervals for Proportions. Confidence intervals Construct confidence intervals for population proportions
CH19 Cofidece Itervals for Proportios Cofidece itervals Costruct cofidece itervals for populatio proportios Motivatio Motivatio We are iterested i the populatio proportio who support Mr. Obama. This sample
More informationExpected Norms of ZeroOne Polynomials
DRAFT: Caad. Math. Bull. July 4, 08 :5 File: borwei80 pp. Page Sheet of Caad. Math. Bull. Vol. XX (Y, ZZZZ pp. 0 0 Expected Norms of ZeroOe Polyomials Peter Borwei, KwokKwog Stephe Choi, ad Idris Mercer
More informationTesting Statistical Hypotheses for Compare. Means with Vague Data
Iteratioal Mathematical Forum 5 o. 3 656 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Aswers & Notes TINspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,
More informationRegression with an Evaporating Logarithmic Trend
Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,
More informationClosed book and notes. No calculators. 60 minutes, but essentially unlimited time.
IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 12
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract I this lecture we derive risk bouds for kerel methods. We will start by showig that Soft Margi kerel SVM correspods to miimizig
More informationLaw of the sum of Bernoulli random variables
Law of the sum of Beroulli radom variables Nicolas Chevallier Uiversité de Haute Alsace, 4, rue des frères Lumière 68093 Mulhouse icolas.chevallier@uha.fr December 006 Abstract Let be the set of all possible
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
More information0, otherwise. EX = E(X 1 + X n ) = EX j = np and. Var(X j ) = np(1 p). Var(X) = Var(X X n ) =
PROBABILITY MODELS 35 10. Discrete probability distributios I this sectio, we discuss several wellow discrete probability distributios ad study some of their properties. Some of these distributios, lie
More informationGamma Distribution and Gamma Approximation
Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 45646, USA cheg@cs.uky.edu Abstract
More informationA goodnessoffit test based on the empirical characteristic function and a comparison of tests for normality
A goodessoffit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
More informationRademacher Complexity
EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for
More informationMonte Carlo Integration
Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce
More informationSRC Technical Note June 17, Tight Thresholds for The Pure Literal Rule. Michael Mitzenmacher. d i g i t a l
SRC Techical Note 1997011 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(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
More informationStatistical Theory MT 2009 Problems 1: Solution sketches
Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where
More informationInstructor: 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 informationTesting Statistical Hypotheses with Fuzzy Data
Iteratioal Joural of Statistics ad Systems ISS 973675 Volume 6, umber 4 (), pp. 44449 Research Idia Publicatios http://www.ripublicatio.com/ijss.htm Testig Statistical Hypotheses with Fuzzy Data E. Baloui
More informationPostedPrice, SealedBid Auctions
PostedPrice, SealedBid Auctios Professors Greewald ad Oyakawa 2070208 We itroduce the postedprice, sealedbid auctio. This auctio format itroduces the idea of approximatios. We describe how well this
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More information18.657: Mathematics of Machine Learning
8.657: Mathematics of Machie Learig Lecturer: Philippe Rigollet Lecture 4 Scribe: Cheg Mao Sep., 05 I this lecture, we cotiue to discuss the effect of oise o the rate of the excess risk E(h) = R(h) R(h
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationA PROOF OF THE TWIN PRIME CONJECTURE AND OTHER POSSIBLE APPLICATIONS
A PROOF OF THE TWI PRIME COJECTURE AD OTHER POSSIBLE APPLICATIOS by PAUL S. BRUCKMA 38 Frot Street, #3 aaimo, BC V9R B8 (Caada) email : pbruckma@hotmail.com ABSTRACT : A elemetary proof of the Twi Prime
More informationSimple Linear Regression
Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio
More informationMathematics 170B Selected HW Solutions.
Mathematics 17B Selected HW Solutios. F 4. Suppose X is B(,p). (a)fidthemometgeeratigfuctiom (s)of(x p)/ p(1 p). Write q = 1 p. The MGF of X is (pe s + q), sice X ca be writte as the sum of idepedet Beroulli
More information10 SimulationAssisted Estimation
CB49510DRV CB495/Trai KEY BOARDED August 20, 2002 13:43 Char Cout= 0 10 SimulatioAssisted Estimatio 10.1 Motivatio So far we have examied how to simulate choice probabilities but have ot ivestigated
More information