Chapter 10 Advanced Topics in Random Processes
|
|
- Britton Newton
- 5 years ago
- Views:
Transcription
1 ery Stark ad Joh W. Woods, Probability, Statistics, ad Radom Variables for Egieers, 4th ed., Pearso Educatio Ic.,. ISBN Chapter Advaced opics i Radom Processes Sectios. Mea-Square (m.s.) Calculus 635 Stochastic Cotiuity ad Derivatives [-] 635 Further Results o m.s. Covergece [-] 645. Mea-Square Stochastic Itegrals 65.3 Mea-Square Stochastic Differetial Equatios Ergodicity [-3] Karhue Lo`eve Expasio [-5] Represetatio of Badlimited ad Periodic Processes 67 Badlimited Processes 67 Badpass Radom Processes 674 WSS Periodic Processes 677 Fourier Series for WSS Processes 68 Summary 68 Appedix Itegral Equatios 68 Existece heorem 683 Problems 686 Refereces 699 B.J. Bazui, Fall 6 of ECE 38
2 .4 Ergodicity From Cooper ad McGillem Ergodicity deals with the problem of determiig the statistics of a esemble based o measuremets from a sample fuctio of the esemble. For ergodic processes, all the statistics ca be determied from a sigle fuctio of the process. his may also be stated based o the time averages. For a ergodic process, the time averages (expected values) equal the esemble averages (expected values). hat is to say, x f x dx lim t dt Note that ergodicity caot exist uless the process is statioary! A Process for Determiig Statioarity ad Ergodicity a) Fid the mea ad the d momet based o the probability b) Fid the time sample mea ad time sample d momet based o time averagig. c) If the meas or d momets are fuctios of time o-statioary d) If the time average mea ad momets are ot equal to the probabilistic mea ad momets or if it is ot statioary, the it is o ergodic. Example Computatios for meas ad d momet x f x dx ad x f x dx x ˆ lim x t dt ad x lim x t dt B.J. Bazui, Fall 6 of ECE 38
3 Back to our textbook Based o the theoretical developmet skipped over i sectio. of the stochastic itegral, we ca defie the time average of a radom process as ˆ lim t dt If this is equivalet to the probabilistic mea, the the R.V. is ergodic i the mea. his provides a way of coectig statistical measuremets with a probabilistic equivalet. I establishig whether or ot this property exists, a mea-square test ca be applied. he mea-square calculus defied i the earlier sectios of the chapter is ecessary to perform this operatio. herefore, to get deeper ito the subject, the material is available! Examples of o ergodic Radom variable processes ad sequeces First t A For A, a radom variable, the time average will be the measured value ad ca ot represet possible values for a esemble based o the pdf. Ay time a amplitude or magitude takes o a radom value for all time the other possible esemble values ca ot be assessed based o a time average. Secod t A cos f t B si f t c c For A ad B radom variables, we have the problem of amplitudes or magitudes existig for all time. Eve if E A EB ad E A B, the process would be ergodic i the mea but ot i power! herefore, defiitios of ergodic i the mea ad ergodic i mea-square ca ad are defied. If we pass these tests, we ca have a WSS R.V. that is also ergodic i mea ad power. B.J. Bazui, Fall 6 3 of ECE 38
4 Defiitio.4- A WSS radom process is ergodic i the mea if the time average coverges to the esemble mea lim t dt as From statistics, we ca defie so we would expect for WSS ad ˆ E lim t dt E M ˆ lim E t E M dt ˆ lim dt lim dt E M E lim Computatio of the variace of the statistical mea has bee previously doe! If we are to determie that this is correct i a mea squared sese, we must determie that lim ˆ lim E M his is where the mea-square calculus derivatio comes i! As may be expected, this is related to the covariace based o K t t dt dt this leads to heorem.4- B.J. Bazui, Fall 6 4 of ECE 38
5 heorem.4- fuctio satisfies A WSS radom process (t) is ergodic i the mea iff the covariace lim K d For ergodicity of higher momets, we have ergodic i mea square ad ergodic i correlatio. Defiitio.4- A WSS radom process is ergodic i the mea square if the time average coverges to the d momet lim t dt R E t as Defiitio.4- A WSS radom process is ergodic i correlatio if the time average coverges to the autocorrelatio as lim * t t dt R as o establish the ecessary ad sufficiet coditios, heorem.4- is provided. he derivatio is based o the covariace of the autocorrealtio, which is a forth power process requirig fourth momet statioarity. B.J. Bazui, Fall 6 5 of ECE 38
6 Example.4 Radom amplitude cosie Where A is N(,) ad theta is U(-pi,pi) From before we ca expect t A cos f t ergodic i the mea ot ergodic i power ot ergodic i correlatio Explicitly t EA cos f t E E t EA Ecos f t ad lim t dt as lim A cos f t A lim cos f t dt dt A lim si f t f A lim si f si f f f f A lim cos si4 f For appropriate, such as 4 ad the summig all periods as ifiity f A f cos lim si4 f B.J. Bazui, Fall 6 6 of ECE 38
7 he text provides a aalysis of the heorem.4- to validate this computatio. o asses ergodic i power lim t dt R E t as lim A cos f t dt A lim cos f t dt From this equatio, the R.V A remais ad we really do ot have to cotiue it caot be ergodic i power. Ad if it is ot ergodic i power, there is o way to be ergodic i correlatio. B.J. Bazui, Fall 6 7 of ECE 38
8 .5 Karhue Lo`eve Expasio his sectio provides a alterate derivatio for the matched filter. It uses a orthoormal expasio whose coefficiets are ucorrelated radom variables. heorem.5 A complete orthoormal basis set is defied for a covariace as K he we ca decompose the radom process usig with t, t t dt t, t t t, for t t t * dt he resultig coefficiets are statistically orthogoal such that With a orthoormal set defied as E * m m m * t t dt m Now based o the theory of itegrals kow as Mercer s theorem, K t t t *, t B.J. Bazui, Fall 6 8 of ECE 38
9 Example.5 4 Matched Filter We defie two hypotheses; there is a sigal preset i oise or just oise. t t m t W W t Usig the theorem preseted, the cotiuous waveforms ca be replaced by simpler scalars as m W W where the m ad W are K-L coefficiets. Now the W represet a idepedet radom sequece. If we take the first basis elemet to be based o m(t), scaled so it is ormalized where c t c m t m t dt he oly will have a relatioship to m(t) such that c W W o compute, usig K-L it is just c t m t dt he zeroth lag of the correlatio of m(t) either with m(t) + oise or oise. he correlatio ca be equivaletly performed usig a filter operatios a matched filter! B.J. Bazui, Fall 6 9 of ECE 38
10 .6 Represetatio of Badlimited ad Periodic Processes I ECE 455 you will be taught about the samplig theorem ad covertig betwee a cotiuous ad discreet time domai. his sectio is providig a similar traslatio betwee the cotiuous autocorrelatio ad power spectral desity to a bad-limited discrete time sample domai. I geeral R R R t t For a badlimited autocorrealtio fuctio with a sample spacig of w f f his is defied as the Nyquist samplig rate for a badlimited sigal. Recostructio of the cotiuous time autocorrealtio is performed as R R t R t R si w w si f f t t he relatioship of the Power Spectral Desity fuctios become t t S w S w, a a w B.J. Bazui, Fall 6 of ECE 38
1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
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 informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationProbability and Statistics
Probability ad Statistics Cotets. Multi-dimesioal Gaussia radom variable. Gaussia radom process 3. Wieer process Why we eed to discuss Gaussia Process The most commo Accordig to the cetral limit theorem,
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 informationTIME-CORRELATION FUNCTIONS
p. 8 TIME-CORRELATION FUNCTIONS Time-correlatio fuctios are a effective way of represetig the dyamics of a system. They provide a statistical descriptio of the time-evolutio of a variable for a esemble
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationAdditional Notes on Power Series
Additioal Notes o Power Series Mauela Girotti MATH 37-0 Advaced Calculus of oe variable Cotets Quick recall 2 Abel s Theorem 2 3 Differetiatio ad Itegratio of Power series 4 Quick recall We recall here
More informationRandom Signals and Noise Winter Semester 2017 Problem Set 12 Wiener Filter Continuation
Radom Sigals ad Noise Witer Semester 7 Problem Set Wieer Filter Cotiuatio Problem (Sprig, Exam A) Give is the sigal W t, which is a Gaussia white oise with expectatio zero ad power spectral desity fuctio
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 informationEntropy Rates and Asymptotic Equipartition
Chapter 29 Etropy Rates ad Asymptotic Equipartitio Sectio 29. itroduces the etropy rate the asymptotic etropy per time-step of a stochastic process ad shows that it is well-defied; ad similarly for iformatio,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 6 9/23/2013. Brownian motion. Introduction
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 6 9/23/203 Browia motio. Itroductio Cotet.. A heuristic costructio of a Browia motio from a radom walk. 2. Defiitio ad basic properties
More informationEE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
More informationIntroduction to Optimization Techniques
Itroductio to Optimizatio Techiques Basic Cocepts of Aalysis - Real Aalysis, Fuctioal Aalysis 1 Basic Cocepts of Aalysis Liear Vector Spaces Defiitio: A vector space X is a set of elemets called vectors
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationOrthogonal Gaussian Filters for Signal Processing
Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios
More informationNotes 27 : Brownian motion: path properties
Notes 27 : Browia motio: path properties Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces:[Dur10, Sectio 8.1], [MP10, Sectio 1.1, 1.2, 1.3]. Recall: DEF 27.1 (Covariace) Let X = (X
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationThe second is the wish that if f is a reasonably nice function in E and φ n
8 Sectio : Approximatios i Reproducig Kerel Hilbert Spaces I this sectio, we address two cocepts. Oe is the wish that if {E, } is a ierproduct space of real valued fuctios o the iterval [,], the there
More informationSession 5. (1) Principal component analysis and Karhunen-Loève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad Karhue-Loève trasformatio Topic 2 of this course explais the image
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationModule 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation
Module 8 Discrete Time Sigals ad Z-Trasforms Objective:To uderstad represetig discrete time sigals, apply z trasform for aalyzigdiscrete time sigals ad to uderstad the relatio to Fourier trasform Itroductio
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationChapter 7: The z-transform. Chih-Wei Liu
Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability
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 informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
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 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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science. BACKGROUND EXAM September 30, 2004.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departmet of Electrical Egieerig ad Computer Sciece 6.34 Discrete Time Sigal Processig Fall 24 BACKGROUND EXAM September 3, 24. Full Name: Note: This exam is closed
More information(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)
Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig
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 informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationChapter 2 Systems and Signals
Chapter 2 Systems ad Sigals 1 Itroductio Discrete-Time Sigals: Sequeces Discrete-Time Systems Properties of Liear Time-Ivariat Systems Liear Costat-Coefficiet Differece Equatios Frequecy-Domai Represetatio
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More 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 informationSection 5.5. Infinite Series: The Ratio Test
Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches
More informationDescribing Function: An Approximate Analysis Method
Describig Fuctio: A Approximate Aalysis Method his chapter presets a method for approximately aalyzig oliear dyamical systems A closed-form aalytical solutio of a oliear dyamical system (eg, a oliear differetial
More informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationLecture 7: Density Estimation: k-nearest Neighbor and Basis Approach
STAT 425: Itroductio to Noparametric Statistics Witer 28 Lecture 7: Desity Estimatio: k-nearest Neighbor ad Basis Approach Istructor: Ye-Chi Che Referece: Sectio 8.4 of All of Noparametric Statistics.
More informationFourier Series and their Applications
Fourier Series ad their Applicatios The fuctios, cos x, si x, cos x, si x, are orthogoal over (, ). m cos mx cos xdx = m = m = = cos mx si xdx = for all m, { m si mx si xdx = m = I fact the fuctios satisfy
More informationMa 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5
Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You
More informationMath 112 Fall 2018 Lab 8
Ma Fall 08 Lab 8 Sums of Coverget Series I Itroductio Ofte e fuctios used i e scieces are defied as ifiite series Determiig e covergece or divergece of a series becomes importat ad it is helpful if e sum
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 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 informationx x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,
Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative
More informationHilbert Space and Least-squares Collocation
Hilbert Space ad Least-squares Collocatio Lecture : Discrete, Mixed Boudary-Value Problem i Physical Geodesy Lecture : Hilbert Spaces, Reproducig Kerels, ad Fuctioals Lecture 3: Miimum Norm Solutio to
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationSlide Set 13 Linear Model with Endogenous Regressors and the GMM estimator
Slide Set 13 Liear Model with Edogeous Regressors ad the GMM estimator Pietro Coretto pcoretto@uisa.it Ecoometrics Master i Ecoomics ad Fiace (MEF) Uiversità degli Studi di Napoli Federico II Versio: Friday
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter
More informationLecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016
Lecture 3 Digital Sigal Processig Chapter 3 z-trasforms Mikael Swartlig Nedelko Grbic Begt Madersso rev. 06 Departmet of Electrical ad Iformatio Techology Lud Uiversity z-trasforms We defie the z-trasform
More informationFrequency Response of FIR Filters
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steady-state
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
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 information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More informationLECTURE 8: ASYMPTOTICS I
LECTURE 8: ASYMPTOTICS I We are iterested i the properties of estimators as. Cosider a sequece of radom variables {, X 1}. N. M. Kiefer, Corell Uiversity, Ecoomics 60 1 Defiitio: (Weak covergece) A sequece
More informationSignal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform
Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the
More information6.003 Homework #12 Solutions
6.003 Homework # Solutios Problems. Which are rue? For each of the D sigals x [] through x 4 [] below), determie whether the coditios listed i the followig table are satisfied, ad aswer for true or F for
More information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationFig. 2. Block Diagram of a DCS
Iformatio source Optioal Essetial From other sources Spread code ge. Format A/D Source ecode Ecrypt Auth. Chael ecode Pulse modu. Multiplex Badpass modu. Spread spectrum modu. X M m i Digital iput Digital
More informationLecture 20: Multivariate convergence and the Central Limit Theorem
Lecture 20: Multivariate covergece ad the Cetral Limit Theorem Covergece i distributio for radom vectors Let Z,Z 1,Z 2,... be radom vectors o R k. If the cdf of Z is cotiuous, the we ca defie covergece
More informationLinear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form
Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity
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 informationMath 210A Homework 1
Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called
More information9.3 Power Series: Taylor & Maclaurin Series
9.3 Power Series: Taylor & Maclauri Series If is a variable, the a ifiite series of the form 0 is called a power series (cetered at 0 ). a a a a a 0 1 0 is a power series cetered at a c a a c a c a c 0
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 informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More information( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!
.8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has
More informationSignal Processing in Mechatronics
Sigal Processig i Mechatroics Zhu K.P. AIS, UM. Lecture, Brief itroductio to Sigals ad Systems, Review of Liear Algebra ad Sigal Processig Related Mathematics . Brief Itroductio to Sigals What is sigal
More information10.2 Infinite Series Contemporary Calculus 1
10. Ifiite Series Cotemporary Calculus 1 10. INFINITE SERIES Our goal i this sectio is to add together the umbers i a sequece. Sice it would take a very log time to add together the ifiite umber of umbers,
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationWeb Appendix O - Derivations of the Properties of the z Transform
M. J. Roberts - 2/18/07 Web Appedix O - Derivatios of the Properties of the z Trasform O.1 Liearity Let z = x + y where ad are costats. The ( z)= ( x + y )z = x z + y z ad the liearity property is O.2
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 informationThe Sample Variance Formula: A Detailed Study of an Old Controversy
The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace
More 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 informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationki, X(n) lj (n) = (ϱ (n) ij ) 1 i,j d.
APPLICATIONES MATHEMATICAE 22,2 (1994), pp. 193 200 M. WIŚNIEWSKI (Kielce) EXTREME ORDER STATISTICS IN AN EQUALLY CORRELATED GAUSSIAN ARRAY Abstract. This paper cotais the results cocerig the wea covergece
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationRegression with an Evaporating Logarithmic Trend
Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,
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 informationFIR Filter Design: Part II
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we cosider how we might go about desigig FIR filters with arbitrary frequecy resposes, through compositio of multiple sigle-peak
More informationIntroduction to Signals and Systems, Part V: Lecture Summary
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
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 informationBrief Review of Functions of Several Variables
Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(
More informationStatistical Fundamentals and Control Charts
Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,
More information