Société de Calcul Mathématique, S. A. Algorithmes et Optimisation
|
|
- Hannah Francis
- 5 years ago
- Views:
Transcription
1 Société de Calcul Mathématique S A Algorithme et Optimiatio Radom amplig of proportio Berard Beauzamy Jue 2008 From time to time we fid a problem i which we do ot deal with value but with proportio For itace : A total budget ha to be pet ad we wat to tudy the repartitio betwee everal good ; A certai amout of pollutio i detected ad we wat to tudy the repartitio betwee everal factor I order to fid iteretig or dagerou ituatio oe uually trie to imulate variou poibilitie Sice o iformatio i ow a priori about the factor the law ued for uch a imulatio hould be a uiform law Decriptio of the problem Mathematically peaig thi mea : Simulate radom variable uch that : each 0 ad the law of the vector i the uiform law o the et : C x x ; x 0 x Uiform law mea that for a ample the umber of time we fall ito a ubet A C deped oly o the meaure of A It mea alo that the oit deity of the vector i cotat i the et C that i doe ot deped o the particular value of x x i C Siège ocial et bureaux : Faubourg Sait Hooré Pari Tel : Fax : Société Aoyme au capital de Euro RCS : Pari B SIRET : APE : 729Z
2 The dicrete verio of the uiform law i much eaier to udertad Fix a deomiator ay M We wat poitive iteger uch that M o our proportio are ad we wat that all repartitio have the ame probability M M idepedetly of the value of For itace tae 3 M 4 We have the poibilitie : ad each of thee 5 poibilitie hould have probability /5 2 Warig We have the ame warig a previouly : to geerate uiform variable ad the replace them by o 0 i wrog becaue we do ot obtai thi way a uiform law o C Thi i clear o the followig example (dimeio : A P P2 C Q Q2 O B 2
3 Aume we tae radom variable ( with uiform law o the egmet 0 The the umber of time the poit with coordiate ( fall ito the triagle OPP 2 i proportioal to the area of that triagle But the ormalized poit 2 i i the egmet Q Q2 if ad oly if the poit ( i i the 2 2 triagle OPP 2 But the area of the triagle deped upo it orietatio : for give legth of Q Q2 we have a larger triagle if it cotai the poit C ad maller at the extremitie whe the triagle cotai A or B So thi cotructio doe ot lead to a uiform law o the egmet AB 3 Geeratio of uiform variable o C The correct cotructio i a follow We follow the boo "No Uiform Radom Variate Geeratio" by Luc Devroye chapter 5 theorem 22 Theorem - Let be idepedet radom variable with expoetial law The the variable Y Y defied by : Y Y are poitive have um ad follow a uiform law o C Note : i order to geerate radom variable with expoetial law o hould geerate radom variable with uiform law Y ad the tae Log Y So aume you wat to geerate a ample of N 000 proportio for 40 good do the followig : for to N for to geerate tae Log for Y compute Y Y idepedet variable with uiform law o 0 replace each ext ext by 3
4 Proof of the Theorem Let S to implify the otatio Let ad Z Let ZS ( ) computed a a coditioal probability owig Z that i : f z be the deity of the oit law of the uple Z S S Z Z z x x ZS Thi deity ca be f ( z ) f ( ) f ( z) () where fs Z() deote the coditioal probability deity of S owig Z ad fz () z i the deity of Z Sice the variable are idepedet ad follow a expoetial law we have : f () z e Z x x ( The law of S owig x x i eay to fid Ideed P S a x x P a x x x x ad therefore : fs Z() e x x (3) for x x 0 otherwie We deduce from () ( (3) that : fzs ( z ) e (4) for x x 0 otherwie Now we compute the oit deity of variable i the deity (4) We et : S Thi i obtaied by a chage of S S Let U S S x x y y ; we get : fus ( u ) e (5) The deity of U ca be obtaied from the above formula itegratig with repect to We get : 4
5 f ( u) f ( u ) d e d! U (6) U S 0 0 ad we ee that thi value i cotat o the whole et x 0 x 0 x x But ow ice the deity of S S i cotat o i the deity of S S S S S S S (7) Thi coclude the proof of the Theorem Remar A pacage i Matlab due to Roger Stafford geerate radom umber with uiform law ad fixed um : it ca be foud o the Itered uder the ame of radfixedumm Thi pacage ue aother approach more complicated tha the approach decribed here (uig expoetial variable) but it ca olve more geeral problem : fid x i with a xi b ad xi S i Acowledgemet We tha Paul Deheuvel for idicatig the boo by Luc Devroye Luc Devroye for hi commet about the proof ad Roger Stafford for hi commet about the Matlab pacage 5
Random Walks in the Plane: An Energy-Based Approach. Three prizes offered by SCM
Société de Calcul Mathématique S Mathematical Modellig Compay, Corp Tools for decisio help sice 995 Radom Wals i the Plae: Eergy-Based pproach Three prizes offered by SCM Berard Beauzamy ugust 6 I. Itroductio
More informationSTRONG DEVIATION THEOREMS FOR THE SEQUENCE OF CONTINUOUS RANDOM VARIABLES AND THE APPROACH OF LAPLACE TRANSFORM
Joural of Statitic: Advace i Theory ad Applicatio Volume, Number, 9, Page 35-47 STRONG DEVIATION THEORES FOR THE SEQUENCE OF CONTINUOUS RANDO VARIABLES AND THE APPROACH OF LAPLACE TRANSFOR School of athematic
More informationComments on Discussion Sheet 18 and Worksheet 18 ( ) An Introduction to Hypothesis Testing
Commet o Dicuio Sheet 18 ad Workheet 18 ( 9.5-9.7) A Itroductio to Hypothei Tetig Dicuio Sheet 18 A Itroductio to Hypothei Tetig We have tudied cofidece iterval for a while ow. Thee are method that allow
More informationFig. 1: Streamline coordinates
1 Equatio of Motio i Streamlie Coordiate Ai A. Soi, MIT 2.25 Advaced Fluid Mechaic Euler equatio expree the relatiohip betwee the velocity ad the preure field i ivicid flow. Writte i term of treamlie coordiate,
More informationM227 Chapter 9 Section 1 Testing Two Parameters: Means, Variances, Proportions
M7 Chapter 9 Sectio 1 OBJECTIVES Tet two mea with idepedet ample whe populatio variace are kow. Tet two variace with idepedet ample. Tet two mea with idepedet ample whe populatio variace are equal Tet
More informationu t u 0 ( 7) Intuitively, the maximum principles can be explained by the following observation. Recall
Oct. Heat Equatio M aximum priciple I thi lecture we will dicu the maximum priciple ad uiquee of olutio for the heat equatio.. Maximum priciple. The heat equatio alo ejoy maximum priciple a the Laplace
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 informationx z Increasing the size of the sample increases the power (reduces the probability of a Type II error) when the significance level remains fixed.
] z-tet for the mea, μ If the P-value i a mall or maller tha a pecified value, the data are tatitically igificat at igificace level. Sigificace tet for the hypothei H 0: = 0 cocerig the ukow mea of a populatio
More information20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE
20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE If the populatio tadard deviatio σ i ukow, a it uually will be i practice, we will have to etimate it by the ample tadard deviatio. Sice σ i ukow,
More informationStatistical Inference Procedures
Statitical Iferece Procedure Cofidece Iterval Hypothei Tet Statitical iferece produce awer to pecific quetio about the populatio of iteret baed o the iformatio i a ample. Iferece procedure mut iclude a
More informationSTA 4032 Final Exam Formula Sheet
Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace
More informationHeat Equation: Maximum Principles
Heat Equatio: Maximum Priciple Nov. 9, 0 I thi lecture we will dicu the maximum priciple ad uiquee of olutio for the heat equatio.. Maximum priciple. The heat equatio alo ejoy maximum priciple a the Laplace
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationLECTURE 13 SIMULTANEOUS EQUATIONS
NOVEMBER 5, 26 Demad-upply ytem LETURE 3 SIMULTNEOUS EQUTIONS I thi lecture, we dicu edogeeity problem that arie due to imultaeity, i.e. the left-had ide variable ad ome of the right-had ide variable are
More informationState space systems analysis
State pace ytem aalyi Repreetatio of a ytem i tate-pace (tate-pace model of a ytem To itroduce the tate pace formalim let u tart with a eample i which the ytem i dicuio i a imple electrical circuit with
More informationBrief Review of Linear System Theory
Brief Review of Liear Sytem heory he followig iformatio i typically covered i a coure o liear ytem theory. At ISU, EE 577 i oe uch coure ad i highly recommeded for power ytem egieerig tudet. We have developed
More informationChapter 9. Key Ideas Hypothesis Test (Two Populations)
Chapter 9 Key Idea Hypothei Tet (Two Populatio) Sectio 9-: Overview I Chapter 8, dicuio cetered aroud hypothei tet for the proportio, mea, ad tadard deviatio/variace of a igle populatio. However, ofte
More informationCHAPTER 6. Confidence Intervals. 6.1 (a) y = 1269; s = 145; n = 8. The standard error of the mean is = s n = = 51.3 ng/gm.
} CHAPTER 6 Cofidece Iterval 6.1 (a) y = 1269; = 145; = 8. The tadard error of the mea i SE ȳ = = 145 8 = 51.3 g/gm. (b) y = 1269; = 145; = 30. The tadard error of the mea i ȳ = 145 = 26.5 g/gm. 30 6.2
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 informationIsolated Word Recogniser
Lecture 5 Iolated Word Recogitio Hidde Markov Model of peech State traitio ad aligmet probabilitie Searchig all poible aligmet Dyamic Programmig Viterbi Aligmet Iolated Word Recogitio 8. Iolated Word Recogier
More informationSOLUTION: The 95% confidence interval for the population mean µ is x ± t 0.025; 49
C22.0103 Sprig 2011 Homework 7 olutio 1. Baed o a ample of 50 x-value havig mea 35.36 ad tadard deviatio 4.26, fid a 95% cofidece iterval for the populatio mea. SOLUTION: The 95% cofidece iterval for the
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
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 informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
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 informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More 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 informationChapter 1 ASPECTS OF MUTIVARIATE ANALYSIS
Chapter ASPECTS OF MUTIVARIATE ANALYSIS. Itroductio Defiitio Wiipedia: Multivariate aalyi MVA i baed o the tatitical priciple of multivariate tatitic which ivolve obervatio ad aalyi of more tha oe tatitical
More information8.6 Order-Recursive LS s[n]
8.6 Order-Recurive LS [] Motivate ti idea wit Curve Fittig Give data: 0,,,..., - [0], [],..., [-] Wat to fit a polyomial to data.., but wic oe i te rigt model?! Cotat! Quadratic! Liear! Cubic, Etc. ry
More informationWe will look for series solutions to (1) around (at most) regular singular points, which without
ENM 511 J. L. Baai April, 1 Frobeiu Solutio to a d order ODE ear a regular igular poit Coider the ODE y 16 + P16 y 16 + Q1616 y (1) We will look for erie olutio to (1) aroud (at mot) regular igular poit,
More informationTESTS OF SIGNIFICANCE
TESTS OF SIGNIFICANCE Seema Jaggi I.A.S.R.I., Library Aveue, New Delhi eema@iari.re.i I applied ivetigatio, oe i ofte itereted i comparig ome characteritic (uch a the mea, the variace or a meaure of aociatio
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationChapter Vectors
Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it
More informationIntroEcono. Discrete RV. Continuous RV s
ItroEcoo Aoc. Prof. Poga Porchaiwiekul, Ph.D... ก ก e-mail: Poga.P@chula.ac.th Homepage: http://pioeer.chula.ac.th/~ppoga (c) Poga Porchaiwiekul, Chulalogkor Uiverity Quatitative, e.g., icome, raifall
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 informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationA Tail Bound For Sums Of Independent Random Variables And Application To The Pareto Distribution
Applied Mathematic E-Note, 9009, 300-306 c ISSN 1607-510 Available free at mirror ite of http://wwwmaththuedutw/ ame/ A Tail Boud For Sum Of Idepedet Radom Variable Ad Applicatio To The Pareto Ditributio
More informationSTUDENT S t-distribution AND CONFIDENCE INTERVALS OF THE MEAN ( )
STUDENT S t-distribution AND CONFIDENCE INTERVALS OF THE MEAN Suppoe that we have a ample of meaured value x1, x, x3,, x of a igle uow quatity. Aumig that the meauremet are draw from a ormal ditributio
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationTable 12.1: Contingency table. Feature b. 1 N 11 N 12 N 1b 2 N 21 N 22 N 2b. ... a N a1 N a2 N ab
Sectio 12 Tests of idepedece ad homogeeity I this lecture we will cosider a situatio whe our observatios are classified by two differet features ad we would like to test if these features are idepedet
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More 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 informationCryptographic Hash Functions and Message Authentication Codes. Reading: Chapter 4 of Katz & Lindell
Cryptographic Hah Fuctio ad Meage Autheticatio Code Readig: Chapter 4 of Katz & Lidell 1 Hah fuctio A fuctio mappig from a domai to a maller rage (thu ot ijective). Applicatio: Fat looup (hah table) Error
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
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 informationCONFIDENCE INTERVALS STUDY GUIDE
CONFIDENCE INTERVALS STUDY UIDE Last uit, we discussed how sample statistics vary. Uder the right coditios, sample statistics like meas ad proportios follow a Normal distributio, which allows us to calculate
More information6. Sufficient, Complete, and Ancillary Statistics
Sufficiet, Complete ad Acillary Statistics http://www.math.uah.edu/stat/poit/sufficiet.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 6. Sufficiet, Complete, ad Acillary
More 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 informationUse of the. Experimental Probabilistic Hypersurface. in Epidemiology
Société de Calcul Mathématique, S. A. Algorithmes et Optimisatio Use of the Experimetal Probabilistic Hypersurface i Epidemiology Rapport rédigé par Olga Zeydia, Igéieur de Recherche, Société de Calcul
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 informationCS284A: Representations and Algorithms in Molecular Biology
CS284A: Represetatios ad Algorithms i Molecular Biology Scribe Notes o Lectures 3 & 4: Motif Discovery via Eumeratio & Motif Represetatio Usig Positio Weight Matrix Joshua Gervi Based o presetatios by
More information18.05 Problem Set 9, Spring 2014 Solutions
18.05 Problem Set 9, Sprig 2014 Solutio Problem 1. (10 pt.) (a) We have x biomial(, θ), o E(X) =θ ad Var(X) = θ(1 θ). The rule-of-thumb variace i jut 4. So the ditributio beig plotted are biomial(250,
More informationSection 9.2. Tests About a Population Proportion 12/17/2014. Carrying Out a Significance Test H A N T. Parameters & Hypothesis
Sectio 9.2 Tests About a Populatio Proportio P H A N T O M S Parameters Hypothesis Assess Coditios Name the Test Test Statistic (Calculate) Obtai P value Make a decisio State coclusio Sectio 9.2 Tests
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 information1. (a) If u (I : R J), there exists c 0 in R such that for all q 0, cu q (I : R J) [q] I [q] : R J [q]. Hence, if j J, for all q 0, j q (cu q ) =
Math 615, Witer 2016 Problem Set #5 Solutio 1. (a) If u (I : R J), there exit c 0 i R uch that for all q 0, cu q (I : R J) [q] I [q] : R J [q]. Hece, if j J, for all q 0, j q (cu q ) = c(ju) q I [q], o
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 informationKinetics of Complex Reactions
Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet
More informationRandom Graphs and Erdős Magic Joel Spencer Asperen, October 30- November 1, 2018
Radom Graphs ad Erdős Magic Joel Specer Aspere, October 30- November 1, 2018 A Special Note: These otes are oly a approximatio to the actual lectures. There is some extra material here, ad there will almost
More informationIf, for instance, we were required to test whether the population mean μ could be equal to a certain value μ
STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More information10-716: Advanced Machine Learning Spring Lecture 13: March 5
10-716: Advaced Machie Learig Sprig 019 Lecture 13: March 5 Lecturer: Pradeep Ravikumar Scribe: Charvi Ratogi, Hele Zhou, Nicholay opi Note: Lae template courtey of UC Berkeley EECS dept. Diclaimer: hee
More informationELEC 372 LECTURE NOTES, WEEK 4 Dr. Amir G. Aghdam Concordia University
ELEC 37 LECTURE NOTES, WEE 4 Dr Amir G Aghdam Cocordia Uiverity Part of thee ote are adapted from the material i the followig referece: Moder Cotrol Sytem by Richard C Dorf ad Robert H Bihop, Pretice Hall
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 informationMISCELLANEOUS SEQUENCES & SERIES QUESTIONS
MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Questio (***+) Evaluate the followig sum 30 r ( 2) 4r 78. 3 MP2-V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio
More informationConfidence Interval for one population mean or one population proportion, continued. 1. Sample size estimation based on the large sample C.I.
Cofidece Iterval for oe populatio mea or oe populatio proportio, cotiued 1. ample size estimatio based o the large sample C.I. for p ˆ(1 ˆ) ˆ(1 ˆ) From the iterval ˆ p p Z p ˆ, p Z p p L legh of your 100(1
More informationExplicit scheme. Fully implicit scheme Notes. Fully implicit scheme Notes. Fully implicit scheme Notes. Notes
Explicit cheme So far coidered a fully explicit cheme to umerically olve the diffuio equatio: T + = ( )T + (T+ + T ) () with = κ ( x) Oly table for < / Thi cheme i ometime referred to a FTCS (forward time
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
More informationApplied Mathematical Sciences, Vol. 9, 2015, no. 3, HIKARI Ltd,
Applied Mathematical Sciece Vol 9 5 o 3 7 - HIKARI Ltd wwwm-hiaricom http://dxdoiorg/988/am54884 O Poitive Defiite Solutio of the Noliear Matrix Equatio * A A I Saa'a A Zarea* Mathematical Sciece Departmet
More informationLecture 23: Minimal sufficiency
Lecture 23: Miimal sufficiecy Maximal reductio without loss of iformatio There are may sufficiet statistics for a give problem. I fact, X (the whole data set) is sufficiet. If T is a sufficiet statistic
More informationRandom Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.
Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)
More informationLecture 14: Graph Entropy
15-859: Iformatio Theory ad Applicatios i TCS Sprig 2013 Lecture 14: Graph Etropy March 19, 2013 Lecturer: Mahdi Cheraghchi Scribe: Euiwoog Lee 1 Recap Bergma s boud o the permaet Shearer s Lemma Number
More informationChapter 7, Solution 1C.
hapter 7, Solutio 1. he velocity of the fluid relative to the immered olid body ufficietly far away from a body i called the free-tream velocity,. he uptream or approach velocity i the velocity of the
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
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 informationREVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION
REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I liear regreio, we coider the frequecy ditributio of oe variable (Y) at each of everal level of a ecod variable (X). Y i kow a the depedet variable.
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More information(7 One- and Two-Sample Estimation Problem )
34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:
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 informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationM A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
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 informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
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 informationVIII. Interval Estimation A. A Few Important Definitions (Including Some Reminders)
VIII. Iterval Etimatio A. A Few Importat Defiitio (Icludig Some Remider) 1. Poit Etimate - a igle umerical value ued a a etimate of a parameter.. Poit Etimator - the ample tatitic that provide the poit
More informationA quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population
A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate
More informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationChapter 8 Part 2. Unpaired t-test With Equal Variances With Unequal Variances
Chapter 8 Part Upaired t-tet With Equal Variace With Uequal Variace December, 008 Goal: To eplai that the choice of the two ample t-tet deped o whether the ample are depedet or idepedet ad for the idepedet
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationAMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
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 informationThe multiplicative structure of finite field and a construction of LRC
IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio
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 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 informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationBINOMIAL PREDICTORS. + 2 j 1. Then n + 1 = The row of the binomial coefficients { ( n
BINOMIAL PREDICTORS VLADIMIR SHEVELEV arxiv:0907.3302v2 [math.nt] 22 Jul 2009 Abstract. For oegative itegers, k, cosider the set A,k = { [0, 1,..., ] : 2 k ( ). Let the biary epasio of + 1 be: + 1 = 2
More information