The standard deviation of the mean

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The standard deviation of the mean"

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

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete Mathematics for CS Spring 2008 David Wagner Note 22 CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig

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

Random Variables, Sampling and Estimation

Random Variables, Sampling and Estimation Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig

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

January 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS

January 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we

More information

Distribution of Random Samples & Limit theorems

Distribution of Random Samples & Limit theorems STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample. Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized

More information

Frequentist Inference

Frequentist Inference Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for

More information

4. Partial Sums and the Central Limit Theorem

4. Partial Sums and the Central Limit Theorem 1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems

More information

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n, CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe

More information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.

More information

Lecture 12: September 27

Lecture 12: September 27 36-705: Itermediate Statistics Fall 207 Lecturer: Siva Balakrisha Lecture 2: September 27 Today we will discuss sufficiecy i more detail ad the begi to discuss some geeral strategies for costructig estimators.

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

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite

More information

EE 4TM4: Digital Communications II Probability Theory

EE 4TM4: Digital Communications II Probability Theory 1 EE 4TM4: Digital Commuicatios II Probability Theory I. RANDOM VARIABLES A radom variable is a real-valued fuctio defied o the sample space. Example: Suppose that our experimet cosists of tossig two fair

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

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio

More information

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22 CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first

More information

Basics of Probability Theory (for Theory of Computation courses)

Basics of Probability Theory (for Theory of Computation courses) Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.

More information

STAT Homework 1 - Solutions

STAT Homework 1 - Solutions STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by

More information

MATH/STAT 352: Lecture 15

MATH/STAT 352: Lecture 15 MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet

More information

Understanding Samples

Understanding Samples 1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We

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

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

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

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The

More information

Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 15

Discrete Mathematics and Probability Theory Fall 2013 Vazirani Note 15 CS 70 Discrete Mathematics ad Probability Theory Fall 2013 Vazirai Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model realworld

More information

Discrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15

Discrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15 CS 70 Discrete Mathematics ad Probability Theory Sprig 2012 Alistair Siclair Note 15 Some Importat Distributios The first importat distributio we leared about i the last Lecture Note is the biomial distributio

More information

Stat 319 Theory of Statistics (2) Exercises

Stat 319 Theory of Statistics (2) Exercises Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.

More information

Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions

Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions CS 70 Discrete Mathematics for CS Sprig 2005 Clacy/Wager Notes 21 Some Importat Distributios Questio: A biased coi with Heads probability p is tossed repeatedly util the first Head appears. What is the

More information

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.

This exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam. Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the

More information

Discrete Mathematics and Probability Theory Summer 2015 Chung-Wei Lin Note 16

Discrete Mathematics and Probability Theory Summer 2015 Chung-Wei Lin Note 16 CS 70 Discrete Mathematics ad Probability Theory Summer 2015 Chug-Wei Li Note 16 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple

More information

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio

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

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6) STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated

More information

1.010 Uncertainty in Engineering Fall 2008

1.010 Uncertainty in Engineering Fall 2008 MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval

More information

Math 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency

Math 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency Math 152. Rumbos Fall 2009 1 Solutios to Review Problems for Exam #2 1. I the book Experimetatio ad Measuremet, by W. J. Youde ad published by the by the Natioal Sciece Teachers Associatio i 1962, the

More information

Discrete Mathematics and Probability Theory Fall 2016 Walrand Probability: An Overview

Discrete Mathematics and Probability Theory Fall 2016 Walrand Probability: An Overview CS 70 Discrete Mathematics ad Probability Theory Fall 2016 Walrad Probability: A Overview Probability is a fasciatig theory. It provides a precise, clea, ad useful model of ucertaity. The successes of

More information

April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE

April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece

More information

Statistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons

Statistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons Statistical Aalysis o Ucertaity for Autocorrelated Measuremets ad its Applicatios to Key Comparisos Nie Fa Zhag Natioal Istitute of Stadards ad Techology Gaithersburg, MD 0899, USA Outlies. Itroductio.

More information

Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 15

Discrete Mathematics and Probability Theory Summer 2014 James Cook Note 15 CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model

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

Chapter 5. Inequalities. 5.1 The Markov and Chebyshev inequalities

Chapter 5. Inequalities. 5.1 The Markov and Chebyshev inequalities Chapter 5 Iequalities 5.1 The Markov ad Chebyshev iequalities As you have probably see o today s frot page: every perso i the upper teth percetile ears at least 1 times more tha the average salary. I other

More information

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f. Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,

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

Lecture 2: Monte Carlo Simulation

Lecture 2: Monte Carlo Simulation STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?

More information

A statistical method to determine sample size to estimate characteristic value of soil parameters

A statistical method to determine sample size to estimate characteristic value of soil parameters A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig

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

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples

More information

This section is optional.

This section is optional. 4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore

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

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

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

Stat 421-SP2012 Interval Estimation Section

Stat 421-SP2012 Interval Estimation Section Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible

More information

Asymptotic distribution of products of sums of independent random variables

Asymptotic distribution of products of sums of independent random variables Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege

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

Convergence of random variables. (telegram style notes) P.J.C. Spreij

Convergence of random variables. (telegram style notes) P.J.C. Spreij Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space

More information

Lecture 3. Properties of Summary Statistics: Sampling Distribution

Lecture 3. Properties of Summary Statistics: Sampling Distribution Lecture 3 Properties of Summary Statistics: Samplig Distributio Mai Theme How ca we use math to justify that our umerical summaries from the sample are good summaries of the populatio? Lecture Summary

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

Chapter 11: Asking and Answering Questions About the Difference of Two Proportions

Chapter 11: Asking and Answering Questions About the Difference of Two Proportions Chapter 11: Askig ad Aswerig Questios About the Differece of Two Proportios These otes reflect material from our text, Statistics, Learig from Data, First Editio, by Roxy Peck, published by CENGAGE Learig,

More information

Economics Spring 2015

Economics Spring 2015 1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures

More information

Exponential Families and Bayesian Inference

Exponential Families and Bayesian Inference Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where

More information

6. Sufficient, Complete, and Ancillary Statistics

6. Sufficient, Complete, and Ancillary Statistics Sufficiet, Complete ad Acillary Statistics http://www.math.uah.edu/stat/poit/sufficiet.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 6. Sufficiet, Complete, ad Acillary

More information

Estimation of the Mean and the ACVF

Estimation of the Mean and the ACVF Chapter 5 Estimatio of the Mea ad the ACVF A statioary process {X t } is characterized by its mea ad its autocovariace fuctio γ ), ad so by the autocorrelatio fuctio ρ ) I this chapter we preset the estimators

More information

Efficient GMM LECTURE 12 GMM II

Efficient GMM LECTURE 12 GMM II DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet

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

Module 1 Fundamentals in statistics

Module 1 Fundamentals in statistics Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly

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

STAT Homework 2 - Solutions

STAT Homework 2 - Solutions STAT-36700 Homework - Solutios Fall 08 September 4, 08 This cotais solutios for Homework. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better isight.

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

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

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

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

SDS 321: Introduction to Probability and Statistics

SDS 321: Introduction to Probability and Statistics SDS 321: Itroductio to Probability ad Statistics Lecture 23: Cotiuous radom variables- Iequalities, CLT Puramrita Sarkar Departmet of Statistics ad Data Sciece The Uiversity of Texas at Austi www.cs.cmu.edu/

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

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

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

An Introduction to Randomized Algorithms

An Introduction to Randomized Algorithms A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis

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

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight)

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight) Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........

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

A quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population

A quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate

More information

Advanced Stochastic Processes.

Advanced Stochastic Processes. Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.

More information

Distributions of Functions of. Normal Random Variables Version 27 Jan 2004

Distributions of Functions of. Normal Random Variables Version 27 Jan 2004 Distributios of Fuctios of Normal Radom Variables Versio 27 Ja 2004 The Uit or Stadard) Normal The uit or stadard ormal radom variable U is a ormally distributed variable with mea zero ad variace oe, i.

More information

AMS 216 Stochastic Differential Equations Lecture 02 Copyright by Hongyun Wang, UCSC ( ( )) 2 = E X 2 ( ( )) 2

AMS 216 Stochastic Differential Equations Lecture 02 Copyright by Hongyun Wang, UCSC ( ( )) 2 = E X 2 ( ( )) 2 AMS 216 Stochastic Differetial Equatios Lecture 02 Copyright by Hogyu Wag, UCSC Review of probability theory (Cotiued) Variace: var X We obtai: = E X E( X ) 2 = E( X 2 ) 2E ( X )E X var( X ) = E X 2 Stadard

More information

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio

More information

Sample Size Determination (Two or More Samples)

Sample Size Determination (Two or More Samples) Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie

More information

MASSACHUSETTS 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 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 information

Lecture 19: Convergence

Lecture 19: Convergence Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may

More information

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700 Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen) Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................

More information

Homework 5 Solutions

Homework 5 Solutions Homework 5 Solutios p329 # 12 No. To estimate the chace you eed the expected value ad stadard error. To do get the expected value you eed the average of the box ad to get the stadard error you eed the

More information