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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Abstract The two biased ad ubiased formulae for the sample variace of a radom variable are uder scrutiy. New research result proves that the formula with a smaller divisor is ubiased ad the formula with a larger divisor is biased. This fact agrees with the curret belief i statistics literature. May mathematical proofs for both formulae cotai errors: This is the reaso for the cotroversy ad this ew research. The fial verdict comes from simulatio results. Keywords: Estimator epectatio operator Helmert s trasformatio sample variace. Itroductio Statisticias kow for a log time that there are two formulae for the sample variace of a radom variable differet by a divisor. For decades a umber of statisticias seem to be happy with a formula called the ubiased formula with a smaller divisor. This formula is correct but the proofs for its ubiasedess by a umber of authors are icorrect; for eample see P.G. Hoel 97) ad I. Guttma S.S. Wilks ad J.S. Huter 97). The other formula with a larger divisor called the biased formula is believed wrog but supported by a smaller umber of statisticias; for eample see R.V. Hogg ad A.T. Craig 978). I this paper we will give a detailed study of the formulae with their supported argumets. The paper is orgaized as follows. Sectio oe is the itroductio sectio. I sectio two we preset the formulae for the sample variace. I sectio three we preset three icorrect proofs collected from tetbooks for the ubiased formula. I sectio four we preset some argumets for the biased formula. I sectio five we discuss the Helmert s trasformatio ad its applicatio to the sample variace formula. I sectio si we report the simulatio results which give the verdict as to which formula is the right oe to use. Fially sectio seve cocludes the discussio of the paper. momet kow as the mea is give as t. ) There is o cotroversy about this formula. There is however a cotroversy for the estimator of the secod momet about the mea kow as the variace. There are two estimators kow by two formulae: ad ˆσ ubiased ˆσ biased t ) ) t ) 3) for this statistic called the sample variace. By developig the square i the last equatio we ca write ˆσ biased t ) t t + ) [ t + [ t [ t t ) t t ). From the last equatio statisticias call the variace the secod momet about the mea. The Formulae for the Sample Variace Suppose that we have a radom variable X has a distributio with mea µ ad variace σ which gives values t s t ) i a sample. The estimated value for the first 3 Argumets for the Ubiased Formula I this sectio we will cite three proofs from propoets for Equatio ). I oe old tetbook which the author of this paper ca o loger fid there is a argumet as follows. If we take the epectatio of both sides of the last

2 equatio i the last sectio we have E{ˆσ biased} E{ E{ t ) t } t } E{ ) t } E{ t } ) E{ t }.4) If t s are idepedet observatios the epected values of the cross-products of the secod term o the right had side of the above equatio are zeros ad we obtai E{ˆσ biased} σ σ σ σ σ. The secod proof is take from P.G. Hoel 97) pages 9-9. From the properties of E ad the defiitio of σ it follows that [ E{ˆσ biased} E [ E [ E t ) [ t µ) µ)] t µ) µ) E t µ) E µ) σ σ σ σ σ. The first proof does ot use the mea ad its estimator but the secod proof uses both. The epected value of the estimator or sample variace ˆσ biased is ot the same as the populatio parameter σ so the proofs claim that the estimator ˆσ biased is biased hece the reaso for its ame. The followig argumet i aother tetbook I. Guttma S.S. Wilks ad J.S. Huter 97) pages 8-8 is a more laborious argumet ad stroger proof for the formula give by Equatio ). Suppose that ) is a radom sample of a radom variable X havig mea µ ad variace σ. The the radom variable L c + + c where the c i s are costats has the epectatio µ ad variace σ i c i. I the special case cosider we obtai + + E{ } µ V ar ) σ. The last equatio states that E{ µ) } σ. Now cosider the algebraic idetity i µ) i ) + µ). i i i c i Usig properties of the epectatio operator we have E i µ) E{ i ) } + E µ). 5) i i Ad makig use of a previous result we fid σ E{ i ) } + σ i i ad that meas E{ i ) } )σ. Fially we have i E{ i ) } σ. 6) i The first ad secod proofs try to prove that the estimator ˆσ biased is biased; the third proof the estimator ˆσ ubiased is ubiased. Ufortuately all the proofs are wrog as the discussio i the et sectio will reveal. 4 Argumets for the Biased Formula I this sectio we will give some argumets for the formula give by Equatio 3). These argumets are listed below. 4. Commo Sese Oce i a while a mathematicia gives a proof for his thikig o some problem that is so cotrary to commo sese thikig ad he believes i his proof. But ufortuately his proof fails him ad does ot solve the problem. The story of the Greek mathematicia Zeo of Elea BC) with the solved parado of Achilles ad the Tortoise is oe eample. The story of the sample variace formula might be aother. From commo sese thikig we kow that to obtai the variace we must divide the sum of squares of the values deviated from the mea by the total umber of values.

3 4. The Probability I R.V. Hogg ad A.T. Craig 978) pages 4-5 the authors of the referece advocated for the variace formula give by Equatio 3). These authors eplaied that the ratio / is the probability of each evet for the radom variable X to have the value t. With such a isight it is too clear to see that the correct sample variace formula must have the divisor ot ). This is because the formula is a reflectio of a epectatio operator E which is a probability-weighted average. However these authors could either prove their preferred formula or discredit the other oe. Usig the probability cocept K. Vu 007) claimed that the epectatio operator E ca be represeted as E{) t } lim ) t where ) t is a fuctio of a radom variable with the ide t. Let us apply this techique to fid the ubiased estimator for the mea µ of a distributio. We have lim t lim t. Now assume that i the sample of values t s there is a value ad k is the umber of times t s have this value. The we ca write the last equatio as below lim t lim lim lim E{X} µ. t k P r{x } Whe the value approaches ifiity the rage of epads to cover the rage of the radom variable X ad at the value of ifiity the ratio k / simply becomes the probability desity for the radom variable X to have the value. Ad depedig o the ature of discrete or cotiuous type of the radom variable X the sum o the right had side of the last equatio will remai a sum or chage to a itegral. I either case the right had side simply becomes the epected value of t. For this to be true however the ratio k / must be a probability value. Usig this approach we ca fid the ubiased estimator for the variace easily. We are ow ready to preset ad prove our research result i a theorem. Theorem 4. The ubiased estimator for the variace of some distributio of a radom variable X with probability desity fuctio p) mea µ variace σ ad sample values t s t ) is ˆσ t ) where is the ubiased mea estimator give by Equatio ). Proof. From the equatio of the estimator we write ˆσ t ) t ). 7) Now assume that i the sample of values t s we have some t s with the value ad the umber of these t s is k times. Sice there is oly a fiite umber of values of we ca write ˆσ k ) ) {P r X } ) p. By takig the limit whe approaches ifiity the last equatio becomes σ µ) p) if X is discrete µ) p)d if X is cotiuous E{X EX)) }. At the limit of ifiity the sample becomes the populatio. Sice at the limit the estimator gives the defiitio of the populatio variace the estimator must be a ubiased estimator. This fact proves the theorem. QED Havig prove that the formula with the divisor is ubiased we must ow prove that the formula with the divisor ) is biased or the proofs i the last sectio prove othig. This is what we are goig to do et. 4.3 The Errors of the Icorrect Proofs Now we will ivestigate the proofs i the last sectio. The first proof does ot use the mea µ ad its estimator as the secod ad third proofs so it does ot make the mistake caused by the estimator but it has a flaw that is easy to fid. The flaw is about the defiitio of the variace. The variace is defied as the secod momet about the mea ot about zero. This meas that σ E{ t }. The argumet has E{ t } σ. This ca be oly true if the values t s

4 i the argumet have a zero mea. However if the values t s have a zero mea the the secod term i Equatio 4) will be zero. This will result i the correct epected value σ for the right had side of Equatio 4). The author of the argumet set out to prove somethig that is correct but obtaied a wrog result. While the secod ad third proofs appear to be mathematically flawless they actually prove othig. Their faults are i usig the value. The errors i the secod ad third proofs is i the probability desity distributios of X ad X. They are differet. Both have mea µ; but the variaces are σ ad σ / respectively. Assumig that the probability desity distributio of X is p ); the to obtai the variace about the mea of this variable we must write E{ µ) } µ) p )d σ. We oly ivestigate the case whe X is of cotiuous type; because it is the usual case for to be the average of values t s. But the argumet applies to the case whe X is of discrete type as well. Now if we apply the same procedure takig epectatio to the quatity t µ) we get E{ t µ) } t µ) p )d σ. This is because t has a wrog probability desity value. The result will lead to E{ t ) } σ which is cotrary to what Equatio 6) claims. 4.4 The Number Degrees of Freedom Aother argumet for the ubiased formula comes from the umber of degrees of freedom. Propoets for this formula claim that the sum of squares t ) has oly ) degrees of freedom because oe is lost with the calculatio of the average. I values t ) t there is oe that does ot have a free value; it ca be calculated from the other ) values. This argumet appears to be correct first ad it has may statisticias icludig the author of this paper uder its spell for some time. But it has a error like the formula it supports. The questio here is: How may degrees of freedom are there i the sum of squares t )? To aswer this questio we write the sum of squares as t ) t t ) ad cout the umber of degrees of freedom of the quatity o the right had side of the last equatio. The umber of degrees of freedom must be. Lookig at the sum o the left had side we accept the fact that it has oly ) free quatities t ). However each of these quatities has two degrees of freedom: Oe is give by t ; the other. Therefore the sum should have ) degrees of freedom. However we caot cout oe degree of freedom more tha oce. The result is: We have a total of degrees of freedom. This is why whe we remove the sum shows that it has degrees of freedom as the right had side tells us. 5 The Helmert Trasformatio A brilliat trasformatio that ca be used to solve the variace formula problem is the Helmert s trasformatio. It is a attempt to show the correct umber of degrees of freedom i the sum of squares of a sample variace formula. If we apply the Helmert s trasformatio to the variables ts t ) we obtai a ew set of variables as follows: y y y ) ) y The ew variables y i s i ) have a zero mea ad the same variace σ of the variable i. The Helmert s trasformatio is a orthogoal trasformatio; therefore the Jacobia of the trasformatio is oe ad we have t yt. Therefore we ca write yt yt y t t ) S. Sice the sum S has ) idepedet variables y i s i ) each with variace σ we ca obtai the variace for the radom variable by takig S divided by ). This meas that this is aother argumet for the ubiased formula ).

5 6 Simulatio Results As there are i s i ) i the sum of squares S we ca still argue that the sum has degrees of freedom a earlier argumet). If the variace ca be cosidered as a sum of squares divided by its umber of degrees of freedom the divisor for the sample variace formula should be. This meas that this is aother argumet for the biased formula 3). The cotroversy starts agai with the Helmert s trasformatio. The cotroversy of the variace formula becomes the cotroversy of the umber of degrees of freedom: Is it because of the variable i or is it ) because of the variable y i? As there are o criteria to choose the right umber of degrees of freedom the author of this paper decided to take the proof-of-thepuddig approach to solve this cotroversy with software simulatio. I this simulatio the author of this paper wrote a small software program usig the MATLAB laguage. Te thousad observatios of a stadardized ormally distributed radom variable were created. The observatios were multiplied by a costat to give the radom variable a variace of value 0 the added by aother costat to give it a mea of value 0. Te thousad observatios are a big umber big eough to be cosidered as a populatio. The sample size take from this populatio has various values from to 8. The the sample variaces were calculated from these samples by the two formulae ) ad 3). The variaces were calculated repeatedly te thousad times each time with a ew populatio of a stadardized ormally distributed radom variable to create populatios of the variaces. The the meas of these variace populatios were calculated ad reported. Table is the result of these simulatio rus. While the umber of degrees of freedom i the sum of squares S ca hardly be agreed upo the simulatio results support the ubiased formula ie. Equatio ). This result comes as a surprise to the author of this paper because of his support for the biased formula ad his fidig of the icorrect proofs for the ubiased formula. The result also tells us that the umber of degrees of freedom i the sum of squares S is ) ot. With the Helmert s trasformatio the proof for the ubiasedess of formula ) ca be established with the trasformed variable y i ad the approach used i the theorem i sectio four. Sice the simulatio supports the ubiased formula we must fid the error i the proof for the biased formula. This error ca be foud i Equatio 7). Oe etry i the mea will joi with the variable t to create a ratio value that will ot give a probability value. The proof is perfect for the mea but it fails for the variace with the way it is defied. Table. Sample Variaces Sample Size Ru Variace ˆσ biased ˆσ ubiased Coclusio I this paper the sample variace formulae are studied agai to determie the right oe for computatio. The old belief is the formula with a smaller divisor is the ubiased oe. While the problem appears to be easy the verdict must be decided with simulatio results as there are icorrect proofs for both formulae i cotetio ad o criteria for determiig the umber of degrees of freedom. Simulatio results support the formula with a smaller divisor. Refereces Irwi Guttma Samuel S. Wilks ad J. Stuart Huter 97). Itroductory Egieerig Statistics. Joh Wiley & Sos Ic. New York NY USA ISBN Paul G. Hoel 97). Itroductio to Mathematical Statistics. Joh Wiley & Sos Ic. New York NY USA 4th Editio ISBN Robert V. Hogg ad Alle T. Craig 978). Itroductio to Mathematical Statistics. Macmilla Publishig Co. Ic. New York NY USA Fourth Editio ISBN Ky M. Vu 007). The ARIMA ad VARIMA Time Series: Their Modeligs Aalyses ad Applicatios. AuLac Techologies Ic. Ottawa ON Caada ISBN MATLAB is a trade mark of The MathWorks Ic.

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

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

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

Basis for simulation techniques

Basis for simulation techniques Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios

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

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

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

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio

More information

Sampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals

Sampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals Chapter 6 Studet Lecture Notes 6-1 Busiess Statistics: A Decisio-Makig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 6-1 Chapter Goals After completig this chapter, you should

More information

Subject: Differential Equations & Mathematical Modeling-III

Subject: Differential Equations & Mathematical Modeling-III Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso

More information

Seunghee Ye Ma 8: Week 5 Oct 28

Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

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

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

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

f(x)dx = 1 and f(x) 0 for all x.

f(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 information

JANE PROFESSOR WW Prob Lib1 Summer 2000

JANE PROFESSOR WW Prob Lib1 Summer 2000 JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or

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

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

This is an introductory course in Analysis of Variance and Design of Experiments.

This is an introductory course in Analysis of Variance and Design of Experiments. 1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class

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

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

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

Proof of Fermat s Last Theorem by Algebra Identities and Linear Algebra

Proof of Fermat s Last Theorem by Algebra Identities and Linear Algebra Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,

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

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram. Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios

More information

1 Inferential Methods for Correlation and Regression Analysis

1 Inferential Methods for Correlation and Regression Analysis 1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

More information

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 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 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

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19 CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple

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

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

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

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

Chapter 6 Sampling Distributions

Chapter 6 Sampling Distributions Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to

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

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

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

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y.

Recall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, 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 o-users, x - y. Such studies are sometimes viewed

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

On an Application of Bayesian Estimation

On an Application of Bayesian Estimation O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, Higashi-Osaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma

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

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

Introduction to Probability and Statistics Twelfth Edition

Introduction 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 information

Solutions: Homework 3

Solutions: Homework 3 Solutios: Homework 3 Suppose that the radom variables Y,...,Y satisfy Y i = x i + " i : i =,..., IID where x,...,x R are fixed values ad ",...," Normal(0, )with R + kow. Fid ˆ = MLE( ). IND Solutio: Observe

More information

Variance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Variance 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

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

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

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

Posted-Price, Sealed-Bid Auctions

Posted-Price, Sealed-Bid Auctions Posted-Price, Sealed-Bid Auctios Professors Greewald ad Oyakawa 207-02-08 We itroduce the posted-price, sealed-bid auctio. This auctio format itroduces the idea of approximatios. We describe how well this

More information

A PROBABILITY PRIMER

A PROBABILITY PRIMER CARLETON COLLEGE A ROBABILITY RIMER SCOTT BIERMAN (Do ot quote without permissio) A robability rimer INTRODUCTION The field of probability ad statistics provides a orgaizig framework for systematically

More information

WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT

WHAT 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 information

Series III. Chapter Alternating Series

Series III. Chapter Alternating Series Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with

More information

Lecture 9: September 19

Lecture 9: September 19 36-700: Probability ad Mathematical Statistics I Fall 206 Lecturer: Siva Balakrisha Lecture 9: September 9 9. Review ad Outlie Last class we discussed: Statistical estimatio broadly Pot estimatio Bias-Variace

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

MA131 - Analysis 1. Workbook 2 Sequences I

MA131 - Analysis 1. Workbook 2 Sequences I MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

More information

Parameter, Statistic and Random Samples

Parameter, 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 information

Elementary Statistics

Elementary Statistics Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of

More information

Unbiased Estimation. February 7-12, 2008

Unbiased Estimation. February 7-12, 2008 Ubiased Estimatio February 7-2, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom

More information

Commutativity in Permutation Groups

Commutativity 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 information

Topic 18: Composite Hypotheses

Topic 18: Composite Hypotheses Toc 18: November, 211 Simple hypotheses limit us to a decisio betwee oe of two possible states of ature. This limitatio does ot allow us, uder the procedures of hypothesis testig to address the basic questio:

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

Algebra II Notes Unit Seven: Powers, Roots, and Radicals

Algebra II Notes Unit Seven: Powers, Roots, and Radicals Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

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

Lecture 9: Hierarchy Theorems

Lecture 9: Hierarchy Theorems IAS/PCMI Summer Sessio 2000 Clay Mathematics Udergraduate Program Basic Course o Computatioal Complexity Lecture 9: Hierarchy Theorems David Mix Barrigto ad Alexis Maciel July 27, 2000 Most of this lecture

More information

Monte Carlo Integration

Monte 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 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

Analysis of Algorithms. Introduction. Contents

Analysis of Algorithms. Introduction. Contents Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We

More information

Modeling and Performance Analysis with Discrete-Event Simulation

Modeling and Performance Analysis with Discrete-Event Simulation Simulatio Modelig ad Performace Aalysis with Discrete-Evet Simulatio Chapter 5 Statistical Models i Simulatio Cotets Basic Probability Theory Cocepts Useful Statistical Models Discrete Distributios Cotiuous

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

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

Sequences I. Chapter Introduction

Sequences I. Chapter Introduction Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which

More information

Final Examination Solutions 17/6/2010

Final 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 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:

More information

Teaching Mathematics Concepts via Computer Algebra Systems

Teaching Mathematics Concepts via Computer Algebra Systems Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) E-ISSN: 4767 P-ISSN: - 4759 Volume 4 Issue 7 September. 6 PP-- Teachig Mathematics Cocepts via Computer Algebra Systems Osama Ajami Rashaw,

More information

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

More information

Sequences III. Chapter Roots

Sequences III. Chapter Roots Chapter 4 Sequeces III 4. Roots We ca use the results we ve established i the last workbook to fid some iterestig limits for sequeces ivolvig roots. We will eed more techical expertise ad low cuig tha

More information

NUMERICAL METHODS FOR SOLVING EQUATIONS

NUMERICAL METHODS FOR SOLVING EQUATIONS Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

More information

CHAPTER 5. Theory and Solution Using Matrix Techniques

CHAPTER 5. Theory and Solution Using Matrix Techniques A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

More information

Law of the sum of Bernoulli random variables

Law 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 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

Large holes in quasi-random graphs

Large holes in quasi-random graphs Large holes i quasi-radom graphs Joaa Polcy Departmet of Discrete Mathematics Adam Mickiewicz Uiversity Pozań, Polad joaska@amuedupl Submitted: Nov 23, 2006; Accepted: Apr 10, 2008; Published: Apr 18,

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

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? 1. My Motivation Some Sort of an Introduction

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? 1. My Motivation Some Sort of an Introduction WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I taught Topological Groups at the Göttige Georg August Uiversity. This

More information

Solutions to Odd Numbered End of Chapter Exercises: Chapter 4

Solutions 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 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

Measures of Spread: Variance and Standard Deviation

Measures of Spread: Variance and Standard Deviation Lesso 1-6 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample

More information

K. Grill Institut für Statistik und Wahrscheinlichkeitstheorie, TU Wien, Austria

K. Grill Institut für Statistik und Wahrscheinlichkeitstheorie, TU Wien, Austria MARKOV PROCESSES K. Grill Istitut für Statistik ud Wahrscheilichkeitstheorie, TU Wie, Austria Keywords: Markov process, Markov chai, Markov property, stoppig times, strog Markov property, trasitio matrix,

More information

The coalescent coalescence theory

The coalescent coalescence theory The coalescet coalescece theory Peter Beerli September 1, 009 Historical ote Up to 198 most developmet i populatio geetics was prospective ad developed expectatios based o situatios of today. Most work

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

Ma 530 Infinite Series I

Ma 530 Infinite Series I Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li

More information

Fundamental Concepts: Surfaces and Curves

Fundamental Concepts: Surfaces and Curves UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat

More information

Bootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests

Bootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests Joural of Moder Applied Statistical Methods Volume 5 Issue Article --5 Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed Al-Ha Ebrahem

More information

Introduction to Probability. Ariel Yadin

Introduction to Probability. Ariel Yadin Itroductio to robability Ariel Yadi Lecture 2 *** Ja. 7 ***. Covergece of Radom Variables As i the case of sequeces of umbers, we would like to talk about covergece of radom variables. There are may ways

More information

MATH 10550, EXAM 3 SOLUTIONS

MATH 10550, EXAM 3 SOLUTIONS MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,

More information

5. Likelihood Ratio Tests

5. Likelihood Ratio Tests 1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,

More information

9.3 Power Series: Taylor & Maclaurin Series

9.3 Power Series: Taylor & Maclaurin Series 9.3 Power Series: Taylor & Maclauri Series If is a variable, the a ifiite series of the form 0 is called a power series (cetered at 0 ). a a a a a 0 1 0 is a power series cetered at a c a a c a c a c 0

More information

1 Constructing and Interpreting a Confidence Interval

1 Constructing and Interpreting a Confidence Interval Itroductory Applied Ecoometrics EEP/IAS 118 Sprig 2014 WARM UP: Match the terms i the table with the correct formula: Adrew Crae-Droesch Sectio #6 5 March 2014 ˆ Let X be a radom variable with mea µ ad

More information