Lecture 4 Sep 9, 2015
|
|
- Constance Morton
- 5 years ago
- Views:
Transcription
1 CS 388R: Radomzed Algorthms Fall 205 Prof. Erc Prce Lecture 4 Sep 9, 205 Scrbe: Xagru Huag & Chad Voegele Overvew I prevous lectures, we troduced some basc probablty, the Cheroff boud, the coupo collector problem, ad game tree evaluato. I ths lecture, we wll troduce cocetrato equaltes. 2 Coupo Collector Problem Draw umbers (coupos) depedetly from ] = {, 2,..., }. How log does t take to see all of the umbers? Suppose T s the umber of draws to get the -th ew umber. Let T = T. Fact. The T s are depedet of each other. Fact 2. T follows geometrc dstrbuto wth success probablty, p = +. Fact 3. If X Geometrc(p), EX] = p + ( p) (EX X 2]) EX] = p = p + ( p)( + EX]) EX 2 ] = p 2 + ( p) EX 2 X 2] = p + ( p)e(x + ) 2 ] = p + ( p)(ex 2 ] + 2EX] + ) = p + ( p)ex 2 ] + 2( p)/p + ( p) EX 2 ] = 2 p p 2 V ar(x) = E(X E(X)) 2 ] = EX 2 ] (EX]) 2 V ar(x) = p p 2 p 2
2 Therefore the Coupo Collector Problem, ET ] = ET ] = = = + = H log V art ] ( p 2 = + V art ] = ) 2 = V art ] 2 ( = ) 2 2 π2 6 = O(2 ) 3 Cocetrato Iequaltes, P rt + α] Assume s some falure probablty. Settg α = have Defto 4. Uo Boud ( + ( +, P rt + α ] ) log ) α ad because ( x )x < e, we P rx X 2... X ] P rx ] Usg a uo boud, we have P r T + log log ] = P r T + ] α = P rt + α... T + α ] P rt + α ] Defto 5. Wth Hgh Probablty (w.h.p.) X O(y) w.h.p. c 2, c, s.t. P rx c y] c 2 T = O( log 2 ) wth hgh probablty. 2
3 3. Markov s Iequalty For a o-egatve radom varable T ad ay o-egatve α, I the Coupo Collector Problem, ET ] P rt α] α P rt α] ET ] α α = ET ] P r = H T H ] 3.2 Chebyshev s Iequalty For a radom varable, X, let µ = EX] deote the expectato ad σ 2 = V arx] deote the varace. Startg from Markov s Iequalty, we fd Settg α ασ P r(x µ) 2 α 2 ] E(X µ)]2 α 2 = σ2 α 2 P r(x µ) 2 α 2 σ 2 ] α 2 Takg the square root, we fd P rx µ + ασ] α 2 P rx µ ασ] α 2 Usg ths result the Coupo Collector Problem, gves us P rt H + O()] Settg = log 2 Most of the tme, the typcal devato s O(σ). ( ) P rt H + O( log )] O log 2 P r x µ O(σ)] 3
4 3.3 Momet Method If f s o-egatve, by Markov s equalty, P rf(x µ) f(α)] Ef(X µ)] f(α) For f creasg, Set f = t k, For oe sde, P rx µ α] Ef(X µ)] f(α) P r X µ k α k ] E x µ k ] α k P rx µ + α] E x µ k ] α k Settg = E x µ k ], we have α k P r X µ + E x µ k ] /k ( ) ] /k If we cosder X N(0, σ 2 ), we kow E x k ] (kσ 2 ) k/2 k > 0 whch meas Settg k = log, we get P r X µ + O P r X µ + O ( ( ) )] k /k σ ( log )] 3.4 Momet Geeratg Fucto Defto 6. The momet geeratg fucto, parameterzed by λ, s defed as MGF X (λ) = Ee λ(x µ) ] Assume X s cetered (EX] = 0). e λx = + λx + (λx)2 2 + (λx)3 3! (λx)k k! We ca use parameter λ to adjust the weghts o each term. Whe λ s larger, more weght s o hgher order terms. 4
5 From the dervato of the Momet Method, settg f(x) = e λx, P rx µ + α] MGF X(λ) e λα Fact 7. If X N(0, σ 2 ), MGF X (λ) = Ee λx ] e λ2 σ 2 Usg ths, Set λ = α σ 2, we get If = e α2 2σ 2, we have λ R P rx µ + α] MGF x(λ) e λα = e λ2 σ 2 2 λα = e 2(λσ α σ ) 2 α2 2σ 2 P rx µ + α] e α2 2σ 2 α = σ 2log ( ) Note that ths s the same O log boud as we foud the method of momets, except that ow we kow the costat. 3.5 Subgaussa Varables Clam 8. The followg three statemets are equvalet f we oly care up to a costat for σ (.e., j {, 2, 3}, σ = θ(σ j )) X s subgaussa wth parameter σ,.e. λ R, MGF X (λ) e λ 2 σ 2 2 () P rx µ + t] e t 2 ) E x k ] /k O (σ 3 k Fact 9. The sum of subgaussa varables are subgaussa. 2σ 2 2 (2) X = X X MGF X (λ) = E e λx] ] = E e λ( x ) = E = ] E e λx (by depedece) e λx ] (3) = MGF X (λ) e λ2 σ 2 /2 (by subgaussa) = e λ2 2 ( σ2 ) 5
6 Ths mples X s subgaussa wth parameter σ2. Fact 0. If X 0, ], the X s subgaussa wth σ = /2 by Hoeffdg s Lemma. Let X = X where X 0, ]. X s subgaussa wth σ = /2. Plug ths to (2), we have whch s exactly the Cheroff boud. P rx µ + α] e 2α2 4 Next Class I the Coupo Collector Problem, we had P rt α] ( ) α e α Ths s ot of the form e α2 so we caot use the subgaussa results. We wll relax the subgaussa requremet to subexpoetal ad subgamma. Ths wll lead to Berste s equalty. MGF X (λ) = Ee λx ] e λ2 σ 2 2 λ B Refereces MR] Rajeev Motwa, Prabhakar Raghava Radomzed Algorthms. Cambrdge Uversty Press, , 995. RV] Roma Vershy Itroducto to the o-asymptotc aalyss of radom matrces. CoRR, abs ,
The Occupancy and Coupon Collector problems
Chapter 4 The Occupacy ad Coupo Collector problems By Sarel Har-Peled, Jauary 9, 08 4 Prelmares [ Defto 4 Varace ad Stadard Devato For a radom varable X, let V E [ X [ µ X deote the varace of X, where
More informationLecture 02: Bounding tail distributions of a random variable
CSCI-B609: A Theorst s Toolkt, Fall 206 Aug 25 Lecture 02: Boudg tal dstrbutos of a radom varable Lecturer: Yua Zhou Scrbe: Yua Xe & Yua Zhou Let us cosder the ubased co flps aga. I.e. let the outcome
More informationLaw of Large Numbers
Toss a co tmes. Law of Large Numbers Suppose 0 f f th th toss came up H toss came up T s are Beroull radom varables wth p ½ ad E( ) ½. The proporto of heads s. Itutvely approaches ½ as. week 2 Markov s
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b
CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package
More informationHomework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015
Fall 05 Homework : Solutos Problem : (Practce wth Asymptotc Notato) A essetal requremet for uderstadg scalg behavor s comfort wth asymptotc (or bg-o ) otato. I ths problem, you wll prove some basc facts
More information,m = 1,...,n; 2 ; p m (1 p) n m,m = 0,...,n; E[X] = np; n! e λ,n 0; E[X] = λ.
CS70: Lecture 21. Revew: Dstrbutos Revew: Idepedece Varace; Iequaltes; WLLN 1. Revew: Dstrbutos 2. Revew: Idepedece 3. Varace 4. Iequaltes Markov Chebyshev 5. Weak Law of Large Numbers U[1,...,] : Pr[X
More informationIntroduction to Probability
Itroducto to Probablty Nader H Bshouty Departmet of Computer Scece Techo 32000 Israel e-mal: bshouty@cstechoacl 1 Combatorcs 11 Smple Rules I Combatorcs The rule of sum says that the umber of ways to choose
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More informationComplete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables
Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl
More informationD KL (P Q) := p i ln p i q i
Cheroff-Bouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The Kullback-Lebler dvergece or relatve etroy of P ad Q s defed as m D KL
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationClass 13,14 June 17, 19, 2015
Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral
More information18.657: Mathematics of Machine Learning
8.657: Mathematcs of Mache Learg Lecturer: Phlppe Rgollet Lecture 3 Scrbe: James Hrst Sep. 6, 205.5 Learg wth a fte dctoary Recall from the ed of last lecture our setup: We are workg wth a fte dctoary
More information3. Basic Concepts: Consequences and Properties
: 3. Basc Cocepts: Cosequeces ad Propertes Markku Jutt Overvew More advaced cosequeces ad propertes of the basc cocepts troduced the prevous lecture are derved. Source The materal s maly based o Sectos.6.8
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationLecture 9: Tolerant Testing
Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have
More informationhp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations
HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
More informationChapter 4 Multiple Random Variables
Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More information1 Solution to Problem 6.40
1 Soluto to Problem 6.40 (a We wll wrte T τ (X 1,...,X where the X s are..d. wth PDF f(x µ, σ 1 ( x µ σ g, σ where the locato parameter µ s ay real umber ad the scale parameter σ s > 0. Lettg Z X µ σ we
More informationPTAS for Bin-Packing
CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationDimensionality Reduction and Learning
CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More information9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d
9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,
More informationCIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights
CIS 800/002 The Algorthmc Foudatos of Data Prvacy October 13, 2011 Lecturer: Aaro Roth Lecture 9 Scrbe: Aaro Roth Database Update Algorthms: Multplcatve Weghts We ll recall aga) some deftos from last tme:
More informationCS286.2 Lecture 4: Dinur s Proof of the PCP Theorem
CS86. Lecture 4: Dur s Proof of the PCP Theorem Scrbe: Thom Bohdaowcz Prevously, we have prove a weak verso of the PCP theorem: NP PCP 1,1/ (r = poly, q = O(1)). Wth ths result we have the desred costat
More informationParameter, Statistic and Random Samples
Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose
More informationBayes (Naïve or not) Classifiers: Generative Approach
Logstc regresso Bayes (Naïve or ot) Classfers: Geeratve Approach What do we mea by Geeratve approach: Lear p(y), p(x y) ad the apply bayes rule to compute p(y x) for makg predctos Ths s essetally makg
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationRuntime analysis RLS on OneMax. Heuristic Optimization
Lecture 6 Rutme aalyss RLS o OeMax trals of {,, },, l ( + ɛ) l ( ɛ)( ) l Algorthm Egeerg Group Hasso Platter Isttute, Uversty of Potsdam 9 May T, We wat to rgorously uderstad ths behavor 9 May / Rutme
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationFeature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)
CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More information8.1 Hashing Algorithms
CS787: Advaced Algorthms Scrbe: Mayak Maheshwar, Chrs Hrchs Lecturer: Shuch Chawla Topc: Hashg ad NP-Completeess Date: September 21 2007 Prevously we looked at applcatos of radomzed algorthms, ad bega
More informationExtend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables
ppled Mathematcal Sceces, Vol 4, 00, o 3, 637-64 xted the Borel-Catell Lemma to Sequeces of No-Idepedet Radom Varables olah Der Departmet of Statstc, Scece ad Research Campus zad Uversty of Tehra-Ira der53@gmalcom
More informationChapter 5 Properties of a Random Sample
Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 Revew for the revous lecture Cocets: radom samle, samle mea, samle varace Theorems: roertes of a radom samle, samle mea, samle varace Examles: how
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More informationThe Arithmetic-Geometric mean inequality in an external formula. Yuki Seo. October 23, 2012
Sc. Math. Japocae Vol. 00, No. 0 0000, 000 000 1 The Arthmetc-Geometrc mea equalty a exteral formula Yuk Seo October 23, 2012 Abstract. The classcal Jese equalty ad ts reverse are dscussed by meas of terally
More informationChain Rules for Entropy
Cha Rules for Etroy The etroy of a collecto of radom varables s the sum of codtoal etroes. Theorem: Let be radom varables havg the mass robablty x x.x. The...... The roof s obtaed by reeatg the alcato
More informationENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections
ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty
More informationBounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy
Bouds o the expected etropy ad KL-dvergece of sampled multomal dstrbutos Brado C. Roy bcroy@meda.mt.edu Orgal: May 18, 2011 Revsed: Jue 6, 2011 Abstract Iformato theoretc quattes calculated from a sampled
More informationSpecial Instructions / Useful Data
JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth
More informationTHE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5
THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should
More informationChapter 8: Statistical Analysis of Simulated Data
Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by Marquette Uversty MSCS600 Ageda 8. The Sample
More informationChapter 4 Multiple Random Variables
Revew o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for Chapter 4-5 Notes: Although all deftos ad theorems troduced our lectures ad ths ote are mportat ad you should be famlar wth, but I put those
More informationTHE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA
THE ROYAL STATISTICAL SOCIETY 3 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER I STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad
More informationA tighter lower bound on the circuit size of the hardest Boolean functions
Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the
More informationRademacher Complexity. Examples
Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed
More informationHard Core Predicates: How to encrypt? Recap
Hard Core Predcates: How to ecrypt? Debdeep Mukhopadhyay IIT Kharagpur Recap A ecrypto scheme s secured f for every probablstc adversary A carryg out some specfed kd of attack ad for every polyomal p(.),
More informationLecture 2 - What are component and system reliability and how it can be improved?
Lecture 2 - What are compoet ad system relablty ad how t ca be mproved? Relablty s a measure of the qualty of the product over the log ru. The cocept of relablty s a exteded tme perod over whch the expected
More informationExtreme Value Theory: An Introduction
(correcto d Extreme Value Theory: A Itroducto by Laures de Haa ad Aa Ferrera Wth ths webpage the authors ted to form the readers of errors or mstakes foud the book after publcato. We also gve extesos for
More informationArithmetic Mean Suppose there is only a finite number N of items in the system of interest. Then the population arithmetic mean is
Topc : Probablty Theory Module : Descrptve Statstcs Measures of Locato Descrptve statstcs are measures of locato ad shape that perta to probablty dstrbutos The prmary measures of locato are the arthmetc
More informationMultiple Choice Test. Chapter Adequacy of Models for Regression
Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to
More informationEstimation of Stress- Strength Reliability model using finite mixture of exponential distributions
Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur
More informationComplete Convergence for Weighted Sums of Arrays of Rowwise Asymptotically Almost Negative Associated Random Variables
A^VÇÚO 1 32 ò 1 5 Ï 2016 c 10 Chese Joural of Appled Probablty ad Statstcs Oct., 2016, Vol. 32, No. 5, pp. 489-498 do: 10.3969/j.ss.1001-4268.2016.05.005 Complete Covergece for Weghted Sums of Arrays of
More informationModule 7. Lecture 7: Statistical parameter estimation
Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato
More informationContinuous Distributions
7//3 Cotuous Dstrbutos Radom Varables of the Cotuous Type Desty Curve Percet Desty fucto, f (x) A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Test scores Desty Curve Percet Probablty Desty Fucto, f
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationLecture Note to Rice Chapter 8
ECON 430 HG revsed Nov 06 Lecture Note to Rce Chapter 8 Radom matrces Let Y, =,,, m, =,,, be radom varables (r.v. s). The matrx Y Y Y Y Y Y Y Y Y Y = m m m s called a radom matrx ( wth a ot m-dmesoal dstrbuto,
More informationPr[X (p + t)n] e D KL(p+t p)n.
Cheroff Bouds Wolfgag Mulzer 1 The Geeral Boud Let P 1,..., m ) ad Q q 1,..., q m ) be two dstrbutos o m elemets,.e.,, q 0, for 1,..., m, ad m 1 m 1 q 1. The Kullback-Lebler dvergece or relatve etroy of
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More informationAlgorithms behind the Correlation Setting Window
Algorths behd the Correlato Settg Wdow Itroducto I ths report detaled forato about the correlato settg pop up wdow s gve. See Fgure. Ths wdow s obtaed b clckg o the rado butto labelled Kow dep the a scree
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More informationQ-analogue of a Linear Transformation Preserving Log-concavity
Iteratoal Joural of Algebra, Vol. 1, 2007, o. 2, 87-94 Q-aalogue of a Lear Trasformato Preservg Log-cocavty Daozhog Luo Departmet of Mathematcs, Huaqao Uversty Quazhou, Fua 362021, P. R. Cha ldzblue@163.com
More informationIntroduction to Matrices and Matrix Approach to Simple Linear Regression
Itroducto to Matrces ad Matrx Approach to Smple Lear Regresso Matrces Defto: A matrx s a rectagular array of umbers or symbolc elemets I may applcatos, the rows of a matrx wll represet dvduals cases (people,
More informationExchangeable Sequences, Laws of Large Numbers, and the Mortgage Crisis.
Exchageable Sequeces, Laws of Large Numbers, ad the Mortgage Crss. Myug Joo Sog Advsor: Prof. Ja Madel May 2009 Itroducto The law of large umbers for..d. sequece gves covergece of sample meas to a costat,.e.,
More informationThe expected value of a sum of random variables,, is the sum of the expected values:
Sums of Radom Varables xpected Values ad Varaces of Sums ad Averages of Radom Varables The expected value of a sum of radom varables, say S, s the sum of the expected values: ( ) ( ) S Ths s always true
More information1 Review and Overview
CS9T/STATS3: Statstcal Learg Teory Lecturer: Tegyu Ma Lecture #7 Scrbe: Bra Zag October 5, 08 Revew ad Overvew We wll frst gve a bref revew of wat as bee covered so far I te frst few lectures, we stated
More informationå 1 13 Practice Final Examination Solutions - = CS109 Dec 5, 2018
Chrs Pech Fal Practce CS09 Dec 5, 08 Practce Fal Examato Solutos. Aswer: 4/5 8/7. There are multle ways to obta ths aswer; here are two: The frst commo method s to sum over all ossbltes for the rak of
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationA Study of the Reproducibility of Measurements with HUR Leg Extension/Curl Research Line
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) A Study of the Reproducblty of Measuremets wth HUR Leg Eteso/Curl Research Le A mportat property of measuremets s that the results should
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More informationApplication of Generating Functions to the Theory of Success Runs
Aled Mathematcal Sceces, Vol. 10, 2016, o. 50, 2491-2495 HIKARI Ltd, www.m-hkar.com htt://dx.do.org/10.12988/ams.2016.66197 Alcato of Geeratg Fuctos to the Theory of Success Rus B.M. Bekker, O.A. Ivaov
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationAlgorithms Design & Analysis. Hash Tables
Algorthms Desg & Aalyss Hash Tables Recap Lower boud Order statstcs 2 Today s topcs Drect-accessble table Hash tables Hash fuctos Uversal hashg Perfect Hashg Ope addressg 3 Symbol-table problem Symbol
More informationArora Rao Vazirani Approximation for Expansion
Proof, belefs, ad algorthms through the les of sum-of-squares 1 Arora Rao Vazra Approxmato for Expaso I ths lecture, we cosder the problem of fdg a set wth smallest possble expaso 1 a gve graph. Earler
More informationLikewise, properties of the optimal policy for equipment replacement & maintenance problems can be used to reduce the computation.
Whe solvg a vetory repleshmet problem usg a MDP model, kowg that the optmal polcy s of the form (s,s) ca reduce the computatoal burde. That s, f t s optmal to replesh the vetory whe the vetory level s,
More informationApplying the condition for equilibrium to this equilibrium, we get (1) n i i =, r G and 5 i
CHEMICAL EQUILIBRIA The Thermodyamc Equlbrum Costat Cosder a reversble reacto of the type 1 A 1 + 2 A 2 + W m A m + m+1 A m+1 + Assgg postve values to the stochometrc coeffcets o the rght had sde ad egatve
More informationNP!= P. By Liu Ran. Table of Contents. The P versus NP problem is a major unsolved problem in computer
NP!= P By Lu Ra Table of Cotets. Itroduce 2. Prelmary theorem 3. Proof 4. Expla 5. Cocluso. Itroduce The P versus NP problem s a major usolved problem computer scece. Iformally, t asks whether a computer
More information5 Short Proofs of Simplified Stirling s Approximation
5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More information1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.
Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please
More informationMean is only appropriate for interval or ratio scales, not ordinal or nominal.
Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot
More informationLecture 9. Some Useful Discrete Distributions. Some Useful Discrete Distributions. The observations generated by different experiments have
NM 7 Lecture 9 Some Useful Dscrete Dstrbutos Some Useful Dscrete Dstrbutos The observatos geerated by dfferet eermets have the same geeral tye of behavor. Cosequetly, radom varables assocated wth these
More informationNP!= P. By Liu Ran. Table of Contents. The P vs. NP problem is a major unsolved problem in computer
NP!= P By Lu Ra Table of Cotets. Itroduce 2. Strategy 3. Prelmary theorem 4. Proof 5. Expla 6. Cocluso. Itroduce The P vs. NP problem s a major usolved problem computer scece. Iformally, t asks whether
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationMarcinkiewicz strong laws for linear statistics of ρ -mixing sequences of random variables
Aas da Academa Braslera de Cêcas 2006 784: 65-62 Aals of the Brazla Academy of Sceces ISSN 000-3765 www.scelo.br/aabc Marckewcz strog laws for lear statstcs of ρ -mxg sequeces of radom varables GUANG-HUI
More informationBeam Warming Second-Order Upwind Method
Beam Warmg Secod-Order Upwd Method Petr Valeta Jauary 6, 015 Ths documet s a part of the assessmet work for the subject 1DRP Dfferetal Equatos o Computer lectured o FNSPE CTU Prague. Abstract Ths documet
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationMu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
More informationPART ONE. Solutions to Exercises
PART ONE Soutos to Exercses Chapter Revew of Probabty Soutos to Exercses 1. (a) Probabty dstrbuto fucto for Outcome (umber of heads) 0 1 probabty 0.5 0.50 0.5 Cumuatve probabty dstrbuto fucto for Outcome
More informationInvestigation of Partially Conditional RP Model with Response Error. Ed Stanek
Partally Codtoal Radom Permutato Model 7- vestgato of Partally Codtoal RP Model wth Respose Error TRODUCTO Ed Staek We explore the predctor that wll result a smple radom sample wth respose error whe a
More informationMEASURES OF DISPERSION
MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda
More information