Lecture 2: Concentration Bounds
|
|
- Rodney Whitehead
- 5 years ago
- Views:
Transcription
1 CSE 52: Desig ad Aalysis of Algorithms I Sprig 206 Lecture 2: Cocetratio Bouds Lecturer: Shaya Oveis Ghara March 30th Scribe: Syuzaa Sargsya Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for formal publicatios. Laws of large umbers imply for a sequece of i.i.d. radom variables X, X 2,... with mea µ, the sample average, (X + X X ), coverges to µ as goes to ifiity. Cocetratio bouds provide a quatitative distace betwee the sample average ad the expectatio. I this lecture we review several of these fudametal iequalities. I the ext few lectures we will see applicatios of these iequalities i desigig radomized algorithms. Let D be a distributio. Suppose we wat to estimate the mea E[X] of D ad we oly have access to idepedet samples of D, X D. Oe way to estimate the mea is to idepedetly draw samples X, X 2,..., X from the distributio ad retur the empirical mea: X i. By law of large umbers the empriical mea coverges to E [ [X]] as. I this lecture we will prove bouds o the umber of samples oe eeds to obtai a estimate of the mea withi ɛ-additive error. 2. Markov s Iequality Markov s Iequality: For ay oegative radom variable (R.V.) X ad ay umber k, P[X k] E[X] k. Proof. E[X] = i i P [X = i] i k i P [X = i] k P[X = i] k P [X k]. i k For example, for k = 3 2E[X], we ca write P [X 32 ] E[X] E[X] 3 2 E[X] = 2 3. (2.) Example: Suppose the average grade of CSE 52 is 2.0 (out of 4.0). Give a lower boud o the fractio of studets who received a grede at most 3.0. We assume that a grade ca be ay real umber betwee 0.0 ad 4.0. I this example E [X] = 2.0. Takig k = 3.0 = 3 2E[X] we get that at least /3 of the studets received grade at most 3.0. It turs out that if the oly thig that we kow about X is its expectatio the Markov s iequality will be the best boud we ca hope for. For a tight example cosider the followig sceario; assume k E[X] ad let 2-
2 2-2 Lecture 2: Cocetratio Bouds where ɛ is very close to 0. X = { k + ɛ E[X] k w.p. 0 w.p. E[X] k Applicatio. We use Markov s iequality to prove a upper boud o the umber of fixed poits of a radom permutatio. Recall that a permutatio is a oe to oe ad oto map σ : {, 2,..., } {, 2,..., }. We say i is a fixed poit of σ iff σ(i) = i. Claim 2.. With probability at least /k a uiformly radom permutatio σ has at most k fixed poits. Proof. The trick is to defie the right radom variable ad the use the Markov s iequality. Defie X i = I{σ(i) = i} ad X = X i. Observe that X is the umber of fixed poits of σ. We ca write dow the expectatio of X usig the liearity of expectatio. E [X] = E [X i ] = P [X i ] =. The secod equality uses that fact that the expectatio of a idicator radom variable is equal to its probability. The last equality holds sice σ is a uiform permutatio, i.e. P [X i ] =. Thus, by Markov s iequality P [X k] /k. 2.2 Chebyshev s Iequality Recall the defiitio of the variace: Var(X) := E [X E [X]] 2 = E [ X 2 + (E [X]) 2 2XE [X] ] = E[X 2 ] + (E[X]) 2 2E[XE[X]] = E[X 2 ] (E[X]) 2. (2.2) The secod ad the third equalities follow from the liearity of expectaio. Note that sice (X E[X]) 2 is a oegative radom variable, E[X 2 ] (E[X]) 2. The stadard deviatio of radom variable X is defied as σ(x) := Var(X). Chabishev s iequality: For ay radom variable X ad ay ɛ > 0, or equivaletly for ay umber k > 0, P [ X E[X] ɛ] Var(X) ɛ 2 P [ X E[X] kσ] k 2 We ca read the above iequality as follows: For ay radom variable X with probability at least 90%, X is withi three stadard deviatio of its expectatio.
3 Lecture 2: Cocetratio Bouds 2-3 Proof. Let Y := (X E[X]) 2 0. By Markov s iequality P[Y ɛ 2 ] E[Y ] ɛ 2 By the defiitio of Y, Or, equivaletly, P [ (X E[X]) 2 ɛ 2] Var(X) ɛ 2. P [ X E[X] ɛ] Var(X) ɛ 2. Next, we describe two applicatios of Chebyshev s iequality. Applicatio 2. Pollig. Cosider a large set of idividuals each votig 0 or o a presidecy cadidate, ad let p be the expectatio. We see that usig oly O(/ɛ 2 ) idepedet samples from the set we ca estimate p withi ad eps-additive error. Let X, X 2,..., X be the votes of idepedetly chose idividuals i this society. Observe that, for each i, { with probabilityp X i = 0 with probability p Defie a R.V. X = Xi. Obviously, E [X] = E [X i ] = p. We use the Chebyshev s iequality to show that for = O(/ɛ 2 ) w.h.p. X is withi a additive distace ɛ of p. To use Chebyshev s iequality, we first eed to upper boud the variace of X. We use the followig lemma to calculate the variace of sum of idepedet radom variables. Lemma 2.2. Let X, X 2,..., X be pairwise idepedet radom variables. This meas that for ay i j, E [X i X j ] = E [X i ] E [X j ]. For X = X +... X, we have, Var(X) = i = Var(X i ). Proof. By (2.2), Var(X) = E[X 2 ] (E[X]) 2 = i,j E[X i X j ] i,j E[X i ]E[X j ], where the secod equality follows by liearity of expectatio. By pairwise idepedece property, for ay i j, E[X i X j ] = E[X i ]E[X j ]. Therefore, the above expressio simplifies to, Var(X) = E[X 2 i ] i (E[X i ]) 2 = Var(X i ).
4 2-4 Lecture 2: Cocetratio Bouds I the pollig example, we ca write, Var(X) = Var(X i /) = 2 Var(X i ). Recall that X i is a Beroulli radom variable with prior p. We have, Var(X i ) = E[X 2 i ] E[X i] 2. Obviously, E[X i ] = p. I additio, E[X 2 i ] = 2 p ( p) = p. So, Var(X i ) = p p 2 /4 ad Now, by Chebyshev s iequality Var(X) 2 4 = 4. P[ X p ɛ] 4 ɛ 2 = 4ɛ 2 This meas that for = 3/ɛ 2, X approximates p withi a additive error of ɛ with 90% probability. Applicatio 3. Birthday Paradox. Let X,..., X {, 2,..., N} chose idepedetly ad uiformly at radom. How large should be to get a collisio, i.e., to get X i = X j for some i j? We show that if < N the w.h.p. there is o collisio. Ad, if > C. the with probability at least /C 2 there is a collisio. Defie a R.V. Y ij = I(X i = X j ) ad let Y = i,j Y ij. Note that Y ij s are depedet radom variables but they are pairwise idepedet. This crucial fact allows us to use Lemma 2.2 to calculate the variace of Y. Observe that Y is a itegral radom variables which couts the umber of collisios. So, we are iterested i P[Y ]. We start by calculatig the first momet of Y. ( 2) By Markov s iequality E[Y ] = i<j P[Y ] E[Y ] E[Y ij ] = i<j P[Y ij ] = = ( 2) N 2 2N, Therefore, if N with probability at least /2 there is o collisios. Now, let us study the case where N. Here, we use the Chebyshev s iequality. First, observe that sice Y is a itegral radom variable, By Chebyshev s iequality, Therefore Usig pairwise idepedece of Y ij s, we get Var(Y ) = i<j P[Y = 0] P[ Y E[Y ] E[Y ]]. N. P[ Y E[Y ] E[Y ]] Var(Y ) (E[Y ]) 2. P[Y ] = PY = 0 Var(Y ) (E[Y ]) 2 Var(Y ij ) = i<j ( N ) N 2 i,j N ( 2) N.
5 Lecture 2: Cocetratio Bouds 2-5 Therefore, P[Y ] Var(Y ) E [Y ] 2 ( ) 2 /N ( ( 2 ) = N ) 2N /N) 2 2. So, for C N, there is a collisio with probability at least 2/C 2. ( Cheroff Bouds Cetral Limit Theorems i their geeral form state for a sequece i.i.d. radom variables X, X 2,... with bouded mea µ ad variace σ 2, ( ) X i µ N(0, σ 2 ) Cheroff types boud provide a quatitative boud o this covergece. Recall that Chebyshev s boud imply that the probability that a R.V. X is at distace kσ from the mea is /k 2. Roughly speakig, Cheroff types of bouds imply that for a suitable R.V. X this probability is exp(ω(k)). We start by describig the Hoeffdig s boud. Hoeffdig s Iequality: Let X,..., X be a sequece of idepedet variables where for each i, a i X i b i. The, [ [ ] ] P X i E X i ɛ 2 exp ( 22 ɛ 2 ) (ai b i ) 2. I the pollig example we had X i {0, } for each i, ad X,..., X are idepedet radom variables with E[X i ] = p. Therefore, by the Hoeffdig s iequality we get [ ] P X i p ɛ 2 exp ( 22 ɛ 2 ) = 2 exp( 2ɛ 2 ) So, for ay δ > 0, X i is withi additive error ɛ of p with probability at least δ if log δ ɛ 2. Applicatio 4. Ubiased radom walk o a lie. Cosider a particle which does a ubiased radom walk o the real lie. It starts at zero ad i each time step it moves oe step ahead or oe step back, i.e., from positio i with probability /2 it goes to i + ad with the remaiig probability it goes to i. We wat to see how far from the origi the particle will be at time. We ca simulate this variable b a sequece X,..., X of idepedet radom variables where for each i, { with probability/2 X i = with probability/2 Let X = X + X X. We wat to prove a upper boud o X. Sice E[X] = 0, by Hoeffdig iequality, [ ] P X 0 ɛ 2 exp ( 22 ɛ 2 ) 4
6 2-6 Lecture 2: Cocetratio Bouds so if ɛ = log()/, with probability at least /, we have X log()/, or i other words, with probability at least /, X log(). That is, with high probability the particle is at distace log() from the origi. I the ext lecture, we show that the particle has distace at least from the origi w.h.p..
Learning Theory: Lecture Notes
Learig Theory: Lecture Notes Kamalika Chaudhuri October 4, 0 Cocetratio of Averages Cocetratio of measure is very useful i showig bouds o the errors of machie-learig algorithms. We will begi with a basic
More informationn 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 information7.1 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 information7.1 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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationGlivenko-Cantelli Classes
CS28B/Stat24B (Sprig 2008 Statistical Learig Theory Lecture: 4 Gliveko-Catelli Classes Lecturer: Peter Bartlett Scribe: Michelle Besi Itroductio This lecture will cover Gliveko-Catelli (GC classes ad itroduce
More informationLecture 3: August 31
36-705: Itermediate Statistics Fall 018 Lecturer: Siva Balakrisha Lecture 3: August 31 This lecture will be mostly a summary of other useful expoetial tail bouds We will ot prove ay of these i lecture,
More informationAn 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 informationLecture Chapter 6: Convergence of Random Sequences
ECE5: Aalysis of Radom Sigals Fall 6 Lecture Chapter 6: Covergece of Radom Sequeces Dr Salim El Rouayheb Scribe: Abhay Ashutosh Doel, Qibo Zhag, Peiwe Tia, Pegzhe Wag, Lu Liu Radom sequece Defiitio A ifiite
More informationEcon 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 informationRademacher Complexity
EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for
More informationDistribution 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 informationLecture 01: the Central Limit Theorem. 1 Central Limit Theorem for i.i.d. random variables
CSCI-B609: A Theorist s Toolkit, Fall 06 Aug 3 Lecture 0: the Cetral Limit Theorem Lecturer: Yua Zhou Scribe: Yua Xie & Yua Zhou Cetral Limit Theorem for iid radom variables Let us say that we wat to aalyze
More informationLecture 4: April 10, 2013
TTIC/CMSC 1150 Mathematical Toolkit Sprig 01 Madhur Tulsiai Lecture 4: April 10, 01 Scribe: Haris Agelidakis 1 Chebyshev s Iequality recap I the previous lecture, we used Chebyshev s iequality to get a
More informationLecture 2 February 8, 2016
MIT 6.854/8.45: Advaced Algorithms Sprig 206 Prof. Akur Moitra Lecture 2 February 8, 206 Scribe: Calvi Huag, Lih V. Nguye I this lecture, we aalyze the problem of schedulig equal size tasks arrivig olie
More informationIntroduction 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 informationProblem Set 2 Solutions
CS271 Radomess & Computatio, Sprig 2018 Problem Set 2 Solutios Poit totals are i the margi; the maximum total umber of poits was 52. 1. Probabilistic method for domiatig sets 6pts Pick a radom subset S
More informationLecture 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 informationChapter 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 informationThis 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 informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Lecture 16
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Lecture 16 Variace Questio: Let us retur oce agai to the questio of how may heads i a typical sequece of coi flips. Recall that we
More informationAgnostic Learning and Concentration Inequalities
ECE901 Sprig 2004 Statistical Regularizatio ad Learig Theory Lecture: 7 Agostic Learig ad Cocetratio Iequalities Lecturer: Rob Nowak Scribe: Aravid Kailas 1 Itroductio 1.1 Motivatio I the last lecture
More informationLecture 6: Coupon Collector s problem
Radomized Algorithms Lecture 6: Coupo Collector s problem Sotiris Nikoletseas Professor CEID - ETY Course 2017-2018 Sotiris Nikoletseas, Professor Radomized Algorithms - Lecture 6 1 / 16 Variace: key features
More informationLecture 10: Universal coding and prediction
0-704: Iformatio Processig ad Learig Sprig 0 Lecture 0: Uiversal codig ad predictio Lecturer: Aarti Sigh Scribes: Georg M. Goerg Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved
More informationDiscrete 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 informationConvergence 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 informationLecture 12: November 13, 2018
Mathematical Toolkit Autum 2018 Lecturer: Madhur Tulsiai Lecture 12: November 13, 2018 1 Radomized polyomial idetity testig We will use our kowledge of coditioal probability to prove the followig lemma,
More informationBasics 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 informationMath 216A Notes, Week 5
Math 6A Notes, Week 5 Scribe: Ayastassia Sebolt Disclaimer: These otes are ot early as polished (ad quite possibly ot early as correct) as a published paper. Please use them at your ow risk.. Thresholds
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationNotes 19 : Martingale CLT
Notes 9 : Martigale CLT Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: [Bil95, Chapter 35], [Roc, Chapter 3]. Sice we have ot ecoutered weak covergece i some time, we first recall
More informationSDS 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 informationRandomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)
Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black
More informationProbability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
More informationThis 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 informationJanuary 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 information2.2. Central limit theorem.
36.. Cetral limit theorem. The most ideal case of the CLT is that the radom variables are iid with fiite variace. Although it is a special case of the more geeral Lideberg-Feller CLT, it is most stadard
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationCS 330 Discussion - Probability
CS 330 Discussio - Probability March 24 2017 1 Fudametals of Probability 11 Radom Variables ad Evets A radom variable X is oe whose value is o-determiistic For example, suppose we flip a coi ad set X =
More informationMATH/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 informationECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002
ECE 330:541, Stochastic Sigals ad Systems Lecture Notes o Limit Theorems from robability Fall 00 I practice, there are two ways we ca costruct a ew sequece of radom variables from a old sequece of radom
More informationAdvanced 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 informationTopic 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 informationTopic 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 informationProbability and Random Processes
Probability ad Radom Processes Lecture 5 Probability ad radom variables The law of large umbers Mikael Skoglud, Probability ad radom processes 1/21 Why Measure Theoretic Probability? Stroger limit theorems
More information4. 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 informationST5215: Advanced Statistical Theory
ST525: Advaced Statistical Theory Departmet of Statistics & Applied Probability Tuesday, September 7, 2 ST525: Advaced Statistical Theory Lecture : The law of large umbers The Law of Large Numbers The
More informationLecture 2: April 3, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 2: April 3, 203 Scribe: Shubhedu Trivedi Coi tosses cotiued We retur to the coi tossig example from the last lecture agai: Example. Give,
More informationLecture 4: Unique-SAT, Parity-SAT, and Approximate Counting
Advaced Complexity Theory Sprig 206 Lecture 4: Uique-SAT, Parity-SAT, ad Approximate Coutig Prof. Daa Moshkovitz Scribe: Aoymous Studet Scribe Date: Fall 202 Overview I this lecture we begi talkig about
More informationSolutions to HW Assignment 1
Solutios to HW: 1 Course: Theory of Probability II Page: 1 of 6 Uiversity of Texas at Austi Solutios to HW Assigmet 1 Problem 1.1. Let Ω, F, {F } 0, P) be a filtered probability space ad T a stoppig time.
More informationLecture 3 : Random variables and their distributions
Lecture 3 : Radom variables ad their distributios 3.1 Radom variables Let (Ω, F) ad (S, S) be two measurable spaces. A map X : Ω S is measurable or a radom variable (deoted r.v.) if X 1 (A) {ω : X(ω) A}
More informationEcon 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 informationMassachusetts Institute of Technology
Solutios to Quiz : Sprig 006 Problem : Each of the followig statemets is either True or False. There will be o partial credit give for the True False questios, thus ay explaatios will ot be graded. Please
More informationLet 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 informationNotes 5 : More on the a.s. convergence of sums
Notes 5 : More o the a.s. covergece of sums Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: Dur0, Sectios.5; Wil9, Sectio 4.7, Shi96, Sectio IV.4, Dur0, Sectio.. Radom series. Three-series
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013. Large Deviations for i.i.d. Random Variables
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013 Large Deviatios for i.i.d. Radom Variables Cotet. Cheroff boud usig expoetial momet geeratig fuctios. Properties of a momet
More information1 = δ2 (0, ), Y Y n nδ. , T n = Y Y n n. ( U n,k + X ) ( f U n,k + Y ) n 2n f U n,k + θ Y ) 2 E X1 2 X1
8. The cetral limit theorems 8.1. The cetral limit theorem for i.i.d. sequeces. ecall that C ( is N -separatig. Theorem 8.1. Let X 1, X,... be i.i.d. radom variables with EX 1 = ad EX 1 = σ (,. Suppose
More information2 Banach spaces and Hilbert spaces
2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud
More informationMathematics 170B Selected HW Solutions.
Mathematics 17B Selected HW Solutios. F 4. Suppose X is B(,p). (a)fidthemometgeeratigfuctiom (s)of(x p)/ p(1 p). Write q = 1 p. The MGF of X is (pe s + q), sice X ca be writte as the sum of idepedet Beroulli
More informationf X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probablity that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c} Pr(X c) = Pr({s S X(s)
More informationEE 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 informationMa 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 informationEmpirical Process Theory and Oracle Inequalities
Stat 928: Statistical Learig Theory Lecture: 10 Empirical Process Theory ad Oracle Iequalities Istructor: Sham Kakade 1 Risk vs Risk See Lecture 0 for a discussio o termiology. 2 The Uio Boud / Boferoi
More informationStat 400: Georgios Fellouris Homework 5 Due: Friday 24 th, 2017
Stat 400: Georgios Fellouris Homework 5 Due: Friday 4 th, 017 1. A exam has multiple choice questios ad each of them has 4 possible aswers, oly oe of which is correct. A studet will aswer all questios
More informationOn Random Line Segments in the Unit Square
O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationVariance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio
More informationStatistical Properties of OLS estimators
1 Statistical Properties of OLS estimators Liear Model: Y i = β 0 + β 1 X i + u i OLS estimators: β 0 = Y β 1X β 1 = Best Liear Ubiased Estimator (BLUE) Liear Estimator: β 0 ad β 1 are liear fuctio of
More informationSTAT 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 informationSTAT 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 informationLecture 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 informationMA131 - 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 informationUnderstanding 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 informationAda Boost, Risk Bounds, Concentration Inequalities. 1 AdaBoost and Estimates of Conditional Probabilities
CS8B/Stat4B Sprig 008) Statistical Learig Theory Lecture: Ada Boost, Risk Bouds, Cocetratio Iequalities Lecturer: Peter Bartlett Scribe: Subhrasu Maji AdaBoost ad Estimates of Coditioal Probabilities We
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationSequences 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 informationFrequentist 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 informationSTAT 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 information1 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 informationDiscrete 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 informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationEECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1
EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum
More informationECE 6980 An Algorithmic and Information-Theoretic Toolbox for Massive Data
ECE 6980 A Algorithmic ad Iformatio-Theoretic Toolbo for Massive Data Istructor: Jayadev Acharya Lecture # Scribe: Huayu Zhag 8th August, 017 1 Recap X =, ε is a accuracy parameter, ad δ is a error parameter.
More informationLecture 9: Expanders Part 2, Extractors
Lecture 9: Expaders Part, Extractors Topics i Complexity Theory ad Pseudoradomess Sprig 013 Rutgers Uiversity Swastik Kopparty Scribes: Jaso Perry, Joh Kim I this lecture, we will discuss further the pseudoradomess
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationLaw of the sum of Bernoulli random variables
Law of the sum of Beroulli radom variables Nicolas Chevallier Uiversité de Haute Alsace, 4, rue des frères Lumière 68093 Mulhouse icolas.chevallier@uha.fr December 006 Abstract Let be the set of all possible
More information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece
More informationLECTURE 8: ASYMPTOTICS I
LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece
More informationLecture 7: October 18, 2017
Iformatio ad Codig Theory Autum 207 Lecturer: Madhur Tulsiai Lecture 7: October 8, 207 Biary hypothesis testig I this lecture, we apply the tools developed i the past few lectures to uderstad the problem
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 3 9//203 Large deviatios Theory. Cramér s Theorem Cotet.. Cramér s Theorem. 2. Rate fuctio ad properties. 3. Chage of measure techique.
More informationTopic 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 informationElements of Statistical Methods Lots of Data or Large Samples (Ch 8)
Elemets of Statistical Methods Lots of Data or Large Samples (Ch 8) Fritz Scholz Sprig Quarter 2010 February 26, 2010 x ad X We itroduced the sample mea x as the average of the observed sample values x
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationLecture 1. Statistics: A science of information. Population: The population is the collection of all subjects we re interested in studying.
Lecture Mai Topics: Defiitios: Statistics, Populatio, Sample, Radom Sample, Statistical Iferece Type of Data Scales of Measuremet Describig Data with Numbers Describig Data Graphically. Defiitios. Example
More informationMATH 112: HOMEWORK 6 SOLUTIONS. Problem 1: Rudin, Chapter 3, Problem s k < s k < 2 + s k+1
MATH 2: HOMEWORK 6 SOLUTIONS CA PRO JIRADILOK Problem. If s = 2, ad Problem : Rudi, Chapter 3, Problem 3. s + = 2 + s ( =, 2, 3,... ), prove that {s } coverges, ad that s < 2 for =, 2, 3,.... Proof. The
More informationFinal Review for MATH 3510
Fial Review for MATH 50 Calculatio 5 Give a fairly simple probability mass fuctio or probability desity fuctio of a radom variable, you should be able to compute the expected value ad variace of the variable
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationAsymptotic 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