The method of types. PhD short course Information Theory and Statistics Siena, September, Mauro Barni University of Siena
|
|
- Meredith Pearson
- 5 years ago
- Views:
Transcription
1 PhD short course Iformatio Theory ad Statistics Siea, September, 2014 The method of types Mauro Bari Uiversity of Siea
2 Outlie of the course Part 1: Iformatio theory i a utshell Part 2: The method of types ad its relatioship with statistics Part 3: Iformatio theory ad large deviatio theory Part 4: Iformatio theory ad hypothesis testig Part 5: Applicatio to adversarial sigal processig
3 Outlie of Part 2 The method of types Defiitios Basic properties with proof of theorems Law of large umbers Source codig, Uiversal source codig
4 Type or empirical probability Type, or empirical probability, of a sequece P x (a) = N(a x ) a X Set with all the types with deomiator P = all types with deomiator '! 1 if X = {0,1} P 5 = ( 0,1), 5, 4 $! 2 # &, " 5% 5, 3 $! 3 # &, " 5% 5, 2 $! 4 # &, " 5% 5, 1 $ ( # &, 1, 0 ) " 5% ( ) * +,
5 Type class Type class: all the sequeces havig the same type T(P) = { x X : P x = P} Example: x 5 = 01100! P x 5 = 3 5, 2 $ # & " 5% T ( P ) x 5 = ') ( *) 11000,10100,10010,10001, , 01001, 00110, 00101, ), -)
6 Number of types The umber of types grows polyomially with Theorem The umber of types with deomiator is upper bouded by: P ( +1) X Proof. Obvious.
7 Probability of a sequece Theorem The probability that a sequece x = x is emitted by a DMS source with pmf Q is Q(x) = 2 ( H (P x ) +D(P x Q) ) if P x = Q Q(x) = 2 H (P x ) H (Q) = 2 Remember The larger the KL distace from the type of x ad Q the lower the probability.
8 Probability of a sequece Proof. i Q(x) = Q(x i ) = a X Q(a) N (a x) = Q(a) P x (a) = 2 P x (a)logq(a) a X a X a X = 2 [P x (a)logq(a) P x (a)log P x (a)+p x (a)log P x (a)] = 2 a " P x (a)log P x (a) Q(a) +P % $ x (a)log P x (a)' # & = 2 [ H (P x )+D(P x Q) ]
9 Examples Probability of a specific sequece with /2 tails ad heads Fair coi Biased coi with P(H) = 1/3, P(T) = 2/3 Same as above with /3 heads Fair coi Biased coi with P(H) = 1/3, P(T) = 2/3
10 Size of a type class Theorem The size of a type class T(P) ca be bouded as follows: 1 ( +1) X 2 H (P) T(P) 2 H (P) Remember The size of a type class grows expoetially with growig rate equal to the etropy of the type.
11 Size of a type class Proof. (upper boud) Give P P cosider the probability that a source with pmf P emits a sequece i T(P). We have 1 P(x) = 2 x T (P) x T (P) H (P) H (P) = T(P) 2 H (P) T(P) 2
12 Size of a type class Proof. (lower boud)! T(P) = # " P(a 1 )... P(a X ) $ & =! % 1! 2! X!! # " e $ & % T(P)!! $ # & " e % " 1 1 $ # e " $ # e % ' & % 1 ' & X " $ # Stirlig approximatio after some algebra X e % X ' & T(P) 1 ( +1) X 2 H (P)
13 Probability of a type class Theorem The probability that a DMS with pmf Q emits a sequece belogig to T(P) ca be bouded as follows: 1 ( +1) X 2 D(P Q) Q(T(P)) 2 D(P Q) Remember The larger the KL distace betwee P ad Q the smaller the probability. If P=Q, the probability teds to 1 expoetially fast
14 Probability of a type class Proof. Q(T(P)) = Q(x) = 2 x T (P) x T (P) (H (P)+D(P Q)) (H (P)+D(P Q)) = T(P) 2 By rememberig the bouds o the size of T(P): 1 ( +1) X 2 D(P Q) Q(T(P)) 2 D(P Q)
15 I summary P ( +1) X Q(x) = 2 [D(P x Q)+H (P x )] H (P) T(P) 2 Q(T(P)) 2 D(P Q)
16 Iformatio Theory ad Statistics
17 Law of large umbers The law of large umbers provides the lik betwee Iformatio Theory ad Statistics. The weak form of the LLN states that Give a sequece of iid radom variables X i X = 1 ε > 0 i=1 X i lim Pr{ X µ X > ε} = 0 Stadard proof is based o Chebyshev iequality. LLN ca be easily exteded to relative frequecies ad probabilities (for discrete radom variables).
18 Law of large umbers (IT perspective) Q(T(P)) 2 D(P Q) Whe grows the oly type class with a o-egligible probability is Q Theorem (law of large umbers) T ε Q = { x : D(P x Q) ε} P(x T Q ε ) = Q(T(P)) 2 D(P Q) 2 ε P:D(P Q)>ε P:D(P Q)>ε P:D(P Q)>ε ( +1) X 2 ε = 2 # ε X $ % log(+1) & ' ( That teds to 0 whe teds to ifiity
19 Source codig (achievability) Source codig theorem (Shao 48) Give a DMS source Q, ay rate R such that R = H(Q)+ε is achievable (for ay ε > 0) Code sequeces of icreasig leght. Code efficietly oly the sequeces i T(Q), sice the others will (almost) ever occur. To do that we eed oly H(Q) bits.
20 Source codig: rigorous proof Choose a small ε ad defie T ε Q = {x : D(P x Q) ε} By the cotiuity of D d(p x,q) ε ' which 0 if ε 0 By the cotiuity of H H(P x ) H(Q)+ε '' which 0 if ε ' 0 1. Code sequeces i T Q ε by coutig them i T Q ε 2. Code sequeces ot i T Q ε by coutig them i X
21 Source codig: rigorous proof The average umber of bits is L Pr{T Q ε }[H(Q)+ ε ''+ X log( +1)]+ (1 Pr{T Q ε }) log( X ) L log( +1) H(Q)ε ''+ X +δ log( X ) That ca be made arbitrarily small by icreasig ad by properly choosig ε ad δ
22 Uiversal source codig What if Q is ot kow? The suprisig result is that we ca still code at ayrate larger tha the Etropy. Observe a sequece of emitted symbols ad estimate Q, the trasmit iformatio about the type ad the idex of the sequece withi the type
23 Uiversal source codig (rigorous proof) Choose a arbitrarily small ε ad let T ε Q = { x : D(P x Q) ε}. Give a sequece x use X log( +1) bits to idicate its type ad H(P x ) to idex x withi the type. The average umber of bits used by the code is: X log( +1) X log( +1) + Q(x )H(P x ) + Q(x )H(P x ) x T Q ε x T Q ε +Q(x T Q ε )log X +Q(x T Q ε )[H(Q)+δ] H(Q)+δ ' Beig ε ad δ (ad hece δ ) arbitrarily small, ay rate larger tha H(Q) ca be obtaied.
24 Chael codig The method of types ca be used to prove may other results i IT icludig the chael codig theorem Outside the scope of this course
25 Refereces 1. T. M. Cover ad J. A. Thomas, Elemets of Iformatio Theory, Wiley 2. I. Csiszar, The method of types, IEEE Tras. If. Theory, vol.44, o.6, pp , Oct I. Csiszar ad P. C Shields, Iformatio Theory ad Statistics; a Tutorial, Foudatios ad Treds i Commu. ad If. Theory, 2004, NOW Pubisher Ic.
Information Theory and Hypothesis Testing
Summer School on Game Theory and Telecommunications Campione, 7-12 September, 2014 Information Theory and Hypothesis Testing Mauro Barni University of Siena September 8 Review of some basic results linking
More informationEE 4TM4: Digital Communications II Information Measures
EE 4TM4: Digital Commuicatios II Iformatio Measures Defiitio : The etropy H(X) of a discrete radom variable X is defied by We also write H(p) for the above quatity. Lemma : H(X) 0. H(X) = x X Proof: 0
More informationIT and large deviation theory
PhD short course Information Theory and Statistics Siena, 15-19 September, 2014 IT and large deviation theory Mauro Barni University of Siena Outline of the short course Part 1: Information theory in a
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationInformation Theory and Statistics Lecture 4: Lempel-Ziv code
Iformatio Theory ad Statistics Lecture 4: Lempel-Ziv code Łukasz Dębowski ldebowsk@ipipa.waw.pl Ph. D. Programme 203/204 Etropy rate is the limitig compressio rate Theorem For a statioary process (X i)
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 informationEntropy and Ergodic Theory Lecture 5: Joint typicality and conditional AEP
Etropy ad Ergodic Theory Lecture 5: Joit typicality ad coditioal AEP 1 Notatio: from RVs back to distributios Let (Ω, F, P) be a probability space, ad let X ad Y be A- ad B-valued discrete RVs, respectively.
More informationLecture 6: Source coding, Typicality, and Noisy channels and capacity
15-859: Iformatio Theory ad Applicatios i TCS CMU: Sprig 2013 Lecture 6: Source codig, Typicality, ad Noisy chaels ad capacity Jauary 31, 2013 Lecturer: Mahdi Cheraghchi Scribe: Togbo Huag 1 Recap Uiversal
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
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 informationExpectation 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 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 informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More 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 informationShannon s noiseless coding theorem
18.310 lecture otes May 4, 2015 Shao s oiseless codig theorem Lecturer: Michel Goemas I these otes we discuss Shao s oiseless codig theorem, which is oe of the foudig results of the field of iformatio
More informationLecture 15: Strong, Conditional, & Joint Typicality
EE376A/STATS376A Iformatio Theory Lecture 15-02/27/2018 Lecture 15: Strog, Coditioal, & Joit Typicality Lecturer: Tsachy Weissma Scribe: Nimit Sohoi, William McCloskey, Halwest Mohammad I this lecture,
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 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 informationPlease do NOT write in this box. Multiple Choice. Total
Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should
More informationAsymptotic Coupling and Its Applications in Information Theory
Asymptotic Couplig ad Its Applicatios i Iformatio Theory Vicet Y. F. Ta Joit Work with Lei Yu Departmet of Electrical ad Computer Egieerig, Departmet of Mathematics, Natioal Uiversity of Sigapore IMS-APRM
More informationLecture 11: Channel Coding Theorem: Converse Part
EE376A/STATS376A Iformatio Theory Lecture - 02/3/208 Lecture : Chael Codig Theorem: Coverse Part Lecturer: Tsachy Weissma Scribe: Erdem Bıyık I this lecture, we will cotiue our discussio o chael codig
More informationInfinite 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 informationRefinement of Two Fundamental Tools in Information Theory
Refiemet of Two Fudametal Tools i Iformatio Theory Raymod W. Yeug Istitute of Network Codig The Chiese Uiversity of Hog Kog Joit work with Siu Wai Ho ad Sergio Verdu Discotiuity of Shao s Iformatio Measures
More informationCHAPTER 1 SEQUENCES AND INFINITE SERIES
CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More 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 informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
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 informationPower series are analytic
Power series are aalytic Horia Corea 1 1 The expoetial ad the logarithm For every x R we defie the fuctio give by exp(x) := 1 + x + x + + x + = x. If x = 0 we have exp(0) = 1. If x 0, cosider the series
More informationSeries 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 informationLecture 27. Capacity of additive Gaussian noise channel and the sphere packing bound
Lecture 7 Ageda for the lecture Gaussia chael with average power costraits Capacity of additive Gaussia oise chael ad the sphere packig boud 7. Additive Gaussia oise chael Up to this poit, we have bee
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More information6.3 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 informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationLecture 7: Channel coding theorem for discrete-time continuous memoryless channel
Lecture 7: Chael codig theorem for discrete-time cotiuous memoryless chael Lectured by Dr. Saif K. Mohammed Scribed by Mirsad Čirkić Iformatio Theory for Wireless Commuicatio ITWC Sprig 202 Let us first
More informationLecture 5: April 17, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 5: April 7, 203 Scribe: Somaye Hashemifar Cheroff bouds recap We recall the Cheroff/Hoeffdig bouds we derived i the last lecture idepedet
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationSelf-normalized deviation inequalities with application to t-statistic
Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric
More informationInformation Theory Tutorial Communication over Channels with memory. Chi Zhang Department of Electrical Engineering University of Notre Dame
Iformatio Theory Tutorial Commuicatio over Chaels with memory Chi Zhag Departmet of Electrical Egieerig Uiversity of Notre Dame Abstract A geeral capacity formula C = sup I(; Y ), which is correct for
More informationLecture 2: Concentration Bounds
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
More information5.1 A mutual information bound based on metric entropy
Chapter 5 Global Fao Method I this chapter, we exted the techiques of Chapter 2.4 o Fao s method the local Fao method) to a more global costructio. I particular, we show that, rather tha costructig a local
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 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 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 informationEntropies & Information Theory
Etropies & Iformatio Theory LECTURE I Nilajaa Datta Uiversity of Cambridge,U.K. For more details: see lecture otes (Lecture 1- Lecture 5) o http://www.qi.damtp.cam.ac.uk/ode/223 Quatum Iformatio Theory
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
More informationPower series are analytic
Power series are aalytic Horia Corea 1 1 Fubii s theorem for double series Theorem 1.1. Let {α m }, be a real sequece idexed by two idices. Assume that the series α m is coverget for all ad C := ( α m
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
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 informationSieve Estimators: Consistency and Rates of Convergence
EECS 598: Statistical Learig Theory, Witer 2014 Topic 6 Sieve Estimators: Cosistecy ad Rates of Covergece Lecturer: Clayto Scott Scribe: Julia Katz-Samuels, Brado Oselio, Pi-Yu Che Disclaimer: These otes
More informationLecture 8: Convergence of transformations and law of large numbers
Lecture 8: Covergece of trasformatios ad law of large umbers Trasformatio ad covergece Trasformatio is a importat tool i statistics. If X coverges to X i some sese, we ofte eed to check whether g(x ) coverges
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 informationOn a Smarandache problem concerning the prime gaps
O a Smaradache problem cocerig the prime gaps Felice Russo Via A. Ifate 7 6705 Avezzao (Aq) Italy felice.russo@katamail.com Abstract I this paper, a problem posed i [] by Smaradache cocerig the prime gaps
More informationExponential Functions and Taylor Series
Expoetial Fuctios ad Taylor Series James K. Peterso Departmet of Biological Scieces ad Departmet of Mathematical Scieces Clemso Uiversity March 29, 207 Outlie Revistig the Expoetial Fuctio Taylor Series
More information4.1 Data processing inequality
ECE598: Iformatio-theoretic methods i high-dimesioal statistics Sprig 206 Lecture 4: Total variatio/iequalities betwee f-divergeces Lecturer: Yihog Wu Scribe: Matthew Tsao, Feb 8, 206 [Ed. Mar 22] Recall
More informationChapter 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 informationInformation Theory and Coding
Sol. Iformatio Theory ad Codig. The capacity of a bad-limited additive white Gaussia (AWGN) chael is give by C = Wlog 2 ( + σ 2 W ) bits per secod(bps), where W is the chael badwidth, is the average power
More informationSeunghee 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 informationExponential Functions and Taylor Series
MATH 4530: Aalysis Oe Expoetial Fuctios ad Taylor Series James K. Peterso Departmet of Biological Scieces ad Departmet of Mathematical Scieces Clemso Uiversity March 29, 2017 MATH 4530: Aalysis Oe Outlie
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 informationSignal Processing. Lecture 02: Discrete Time Signals and Systems. Ahmet Taha Koru, Ph. D. Yildiz Technical University.
Sigal Processig Lecture 02: Discrete Time Sigals ad Systems Ahmet Taha Koru, Ph. D. Yildiz Techical Uiversity 2017-2018 Fall ATK (YTU) Sigal Processig 2017-2018 Fall 1 / 51 Discrete Time Sigals Discrete
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 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 informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
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 informationThe Boolean Ring of Intervals
MATH 532 Lebesgue Measure Dr. Neal, WKU We ow shall apply the results obtaied about outer measure to the legth measure o the real lie. Throughout, our space X will be the set of real umbers R. Whe ecessary,
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 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 informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
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 informationMA131 - Analysis 1. Workbook 9 Series III
MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationUNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY
UNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY Structure 2.1 Itroductio Objectives 2.2 Relative Frequecy Approach ad Statistical Probability 2. Problems Based o Relative Frequecy 2.4 Subjective Approach
More informationLast Lecture. Wald Test
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 22 Hyu Mi Kag April 9th, 2013 Is the exact distributio of LRT statistic typically easy to obtai? How about its asymptotic distributio? For testig
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More 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 informationSequences 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 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 informationECE 564/645 - Digital Communication Systems (Spring 2014) Final Exam Friday, May 2nd, 8:00-10:00am, Marston 220
ECE 564/645 - Digital Commuicatio Systems (Sprig 014) Fial Exam Friday, May d, 8:00-10:00am, Marsto 0 Overview The exam cosists of four (or five) problems for 100 (or 10) poits. The poits for each part
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 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 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 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 informationLecture 4 February 16, 2016
MIT 6.854/18.415: Advaced Algorithms Sprig 16 Prof. Akur Moitra Lecture 4 February 16, 16 Scribe: Be Eysebach, Devi Neal 1 Last Time Cosistet Hashig - hash fuctios that evolve well Radom Trees - routig
More informationLecture 11: Pseudorandom functions
COM S 6830 Cryptography Oct 1, 2009 Istructor: Rafael Pass 1 Recap Lecture 11: Pseudoradom fuctios Scribe: Stefao Ermo Defiitio 1 (Ge, Ec, Dec) is a sigle message secure ecryptio scheme if for all uppt
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 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 informationMATH 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 informationA 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 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 informationMath 299 Supplement: Real Analysis Nov 2013
Math 299 Supplemet: Real Aalysis Nov 203 Algebra Axioms. I Real Aalysis, we work withi the axiomatic system of real umbers: the set R alog with the additio ad multiplicatio operatios +,, ad the iequality
More informationThe Maximum-Likelihood Decoding Performance of Error-Correcting Codes
The Maximum-Lielihood Decodig Performace of Error-Correctig Codes Hery D. Pfister ECE Departmet Texas A&M Uiversity August 27th, 2007 (rev. 0) November 2st, 203 (rev. ) Performace of Codes. Notatio X,
More informationContinuous Functions
Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio
More informationMath 104: Homework 2 solutions
Math 04: Homework solutios. A (0, ): Sice this is a ope iterval, the miimum is udefied, ad sice the set is ot bouded above, the maximum is also udefied. if A 0 ad sup A. B { m + : m, N}: This set does
More informationRates of Convergence by Moduli of Continuity
Rates of Covergece by Moduli of Cotiuity Joh Duchi: Notes for Statistics 300b March, 017 1 Itroductio I this ote, we give a presetatio showig the importace, ad relatioship betwee, the modulis of cotiuity
More informationA Note on the Kolmogorov-Feller Weak Law of Large Numbers
Joural of Mathematical Research with Applicatios Mar., 015, Vol. 35, No., pp. 3 8 DOI:10.3770/j.iss:095-651.015.0.013 Http://jmre.dlut.edu.c A Note o the Kolmogorov-Feller Weak Law of Large Numbers Yachu
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 information