Detection and Estimation Theory

Detection and Estimation Theory

ESE 524 Detection and Etimation Theory Joeph A. O Sullivan Samuel C. Sach Profeor Electronic Sytem and Signal Reearch Laboratory Electrical l and Sytem Engineering Wahington Univerity 2 Urbauer Hall 34-935-473

Detection and Estimation Theory

ESE 524 Detection and Estimation Theory Joseph A. O Sullivan Samuel C. Sachs Professor Electronic Systems and Signals Research Laboratory Electrical and Systems Engineering Washington University 2 Urbauer

Detection and Estimation Theory

ESE 524 Detection and Estimation Theory Joseph A. O Sullivan Samuel C. Sachs Professor Electronic Systems and Signals Research Laboratory Electrical and Systems Engineering Washington University 2 Urbauer

Detection and Estimation Theory

ESE 524 Detection and Estimation Theory Joseph A. O Sullivan Samuel C. Sachs Professor Electronic Systems and Signals Research Laboratory Electrical and Systems Engineering Washington University 211 Urbauer

MINITAB Stat Lab 3

MINITAB Stat 20080 Lab 3. Statitical Inference In the previou lab we explained how to make prediction from a imple linear regreion model and alo examined the relationhip between the repone and predictor

Exercises with solutions (Set D)

Exercises with solutions Set D. A fair die is rolled at the same time as a fair coin is tossed. Let A be the number on the upper surface of the die and let B describe the outcome of the coin toss, where

Comparing Means: t-tests for Two Independent Samples

Comparing ean: t-tet for Two Independent Sample Independent-eaure Deign t-tet for Two Independent Sample Allow reearcher to evaluate the mean difference between two population uing data from two eparate

If Y is normally Distributed, then and 2 Y Y 10. σ σ

ull Hypothei Significance Teting V. APS 50 Lecture ote. B. Dudek. ot for General Ditribution. Cla Member Uage Only. Chi-Square and F-Ditribution, and Diperion Tet Recall from Chapter 4 material on: ( )

e st t u(t 2) dt = lim t dt = T 2 2 e st = T e st lim + e st

Math 46, Profeor David Levermore Anwer to Quetion for Dicuion Friday, 7 October 7 Firt Set of Quetion ( Ue the definition of the Laplace tranform to compute Lf]( for the function f(t = u(t t, where u i

Gaussian Random Variables Why we Care

Gaussian Random Variables Why we Care I Gaussian random variables play a critical role in modeling many random phenomena. I By central limit theorem, Gaussian random variables arise from the superposition

Suggested Answers To Exercises. estimates variability in a sampling distribution of random means. About 68% of means fall

Beyond Significance Teting ( nd Edition), Rex B. Kline Suggeted Anwer To Exercie Chapter. The tatitic meaure variability among core at the cae level. In a normal ditribution, about 68% of the core fall

The Moment Method; Convex Duality; and Large/Medium/Small Deviations

Stat 928: Statistical Learning Theory Lecture: 5 The Moment Method; Convex Duality; and Large/Medium/Small Deviations Instructor: Sham Kakade The Exponential Inequality and Convex Duality The exponential

Lecture 7: Testing Distributions

CSE 5: Sublinear (and Streaming) Algorithm Spring 014 Lecture 7: Teting Ditribution April 1, 014 Lecturer: Paul Beame Scribe: Paul Beame 1 Teting Uniformity of Ditribution We return today to property teting

Detection and Estimation Theory

ESE 54 Detection and Estimation heory Joseph A. O Sullivan Samuel C. Sachs Professor Electronic Systems and Signals Research Laboratory Electrical and Systems Engineering Washington University Urbauer

SIMPLE LINEAR REGRESSION

SIMPLE LINEAR REGRESSION In linear regreion, we conider the frequency ditribution of one variable (Y) at each of everal level of a econd variable (). Y i known a the dependent variable. The variable for

ESE 523 Information Theory

ESE 53 Inforation Theory Joseph A. O Sullivan Sauel C. Sachs Professor Electrical and Systes Engineering Washington University 11 Urbauer Hall 10E Green Hall 314-935-4173 (Lynda Marha Answers) jao@wustl.edu

arxiv: v1 [math.ca] 23 Sep 2017

arxiv:709.08048v [math.ca] 3 Sep 07 On the unit ditance problem A. Ioevich Abtract. The Erdő unit ditance conjecture in the plane ay that the number of pair of point from a point et of ize n eparated by

Regression. What is regression? Linear Regression. Cal State Northridge Ψ320 Andrew Ainsworth PhD

Regreion Cal State Northridge Ψ30 Andrew Ainworth PhD What i regreion? How do we predict one variable from another? How doe one variable change a the other change? Caue and effect Linear Regreion A technique

Detection and Estimation Theory

ESE 54 Detection and Estiation Theory Joseph A. O Sullivan Sauel C. Sachs Professor Electronic Systes and Signals Research Laboratory Electrical and Systes Engineering Washington University 11 Urbauer

Uniform Distribution. Uniform Distribution. Uniform Distribution. Graphs of Gamma Distributions. Gamma Distribution. Continuous Distributions CD - 1

Continuou Uniform Special Probability Denitie Definition 6.: The continuou random variable ha a uniform ditribution if it p.d.f. i equal to a contant on it upport. If the upport i the interval [a b] then

ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION. Xiaoqun Wang

Proceeding of the 2008 Winter Simulation Conference S. J. Maon, R. R. Hill, L. Mönch, O. Roe, T. Jefferon, J. W. Fowler ed. ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION Xiaoqun Wang

Digital Communications I: Modulation and Coding Course. Term Catharina Logothetis Lecture 8

Digital Communication I: odulation and Coding Coure Term 3-8 Catharina Logotheti Lecture 8 Lat time we talked about: Some bandpa modulation cheme -A, -SK, -FSK, -QA How to perform coherent and noncoherent

Exponential Family and Maximum Likelihood, Gaussian Mixture Models and the EM Algorithm. by Korbinian Schwinger

Exponential Family and Maximum Likelihood, Gaussian Mixture Models and the EM Algorithm by Korbinian Schwinger Overview Exponential Family Maximum Likelihood The EM Algorithm Gaussian Mixture Models Exponential

EE/Stats 376A: Homework 7 Solutions Due on Friday March 17, 5 pm

EE/Stats 376A: Homework 7 Solutions Due on Friday March 17, 5 pm 1. Feedback does not increase the capacity. Consider a channel with feedback. We assume that all the recieved outputs are sent back immediately

Z a>2 s 1n = X L - m. X L = m + Z a>2 s 1n X L = The decision rule for this one-tail test is

M09_BERE8380_12_OM_C09.QD 2/21/11 3:44 PM Page 1 9.6 The Power of a Tet 9.6 The Power of a Tet 1 Section 9.1 defined Type I and Type II error and their aociated rik. Recall that a repreent the probability

Semiconductor Physics and Devices

EE321 Fall 2015 Semiconductor Phyic and Device November 30, 2015 Weiwen Zou ( 邹卫文 ) Ph.D., Aociate Prof. State Key Lab of advanced optical communication ytem and network, Dept. of Electronic Engineering,

Quantile-quantile plots and the method of peaksover-threshold

Problems in SF2980 2009-11-09 12 6 4 2 0 2 4 6 0.15 0.10 0.05 0.00 0.05 0.10 0.15 Figure 2: qqplot of log-returns (x-axis) against quantiles of a standard t-distribution with 4 degrees of freedom (y-axis).

Listen-and-Talk: Full-duplex Cognitive Radio Networks

Liten-and-Talk: Full-duplex Cognitive Radio Network Yun Liao, Tianyu Wang, Lingyang Song, and Zhu Han State Key Laboratory of Advanced Optical Communication Sytem and Network, School of Electrical Engineering

1 if X v I v (X) = 0 if X < v. e sx p(x)dx

Homework 3 Solution Problem Chernoff Bound: The indicator function I v (X) for a real random variable X i defined a, { if X v I v (X) 0 if X < v For 0 (i) if X < v (ii) if X v E{e X } e v P rx v I v (X)

skipping section 6.6 / 5.6 (generating permutations and combinations) concludes basic counting in Chapter 6 / 5

kiing ection 6.6 / 5.6 generating ermutation and combination conclude baic counting in Chater 6 / 5 on to Chater 7 / 6: Dicrete robability before we go to trickier counting in Chater 8 / 7 age 431-475

Hoeffding, Chernoff, Bennet, and Bernstein Bounds

Stat 928: Statistical Learning Theory Lecture: 6 Hoeffding, Chernoff, Bennet, Bernstein Bounds Instructor: Sham Kakade 1 Hoeffding s Bound We say X is a sub-gaussian rom variable if it has quadratically

P ( N m=na c m) (σ-additivity) exp{ P (A m )} (1 x e x for x 0) m=n P (A m ) 0

MA414 STOCHASTIC ANALYSIS: EXAMINATION SOLUTIONS, 211 Q1.(i) Firt Borel-Cantelli Lemma). A = lim up A n = n m=n A m, o A m=na m for each n. So P (A) P ( m=na m ) m=n P (A m ) (n ) (tail of a convergent

IEOR 3106: Fall 2013, Professor Whitt Topics for Discussion: Tuesday, November 19 Alternating Renewal Processes and The Renewal Equation

IEOR 316: Fall 213, Profeor Whitt Topic for Dicuion: Tueday, November 19 Alternating Renewal Procee and The Renewal Equation 1 Alternating Renewal Procee An alternating renewal proce alternate between

l t t L = t= l t l t t F t t S t,i i t t l t = F t i= S t,i, L = F t S t,i, t= i= F t t S t,i t i E t t t e t,i e, e 3, e 3,2 e 4, e 6, 2 3 4 5 6 x e i =,..., E t t L t,i F t,i l l =,..., L t,i e t,i l

Machine Learning Basics Lecture 7: Multiclass Classification. Princeton University COS 495 Instructor: Yingyu Liang

Machine Learning Basics Lecture 7: Multiclass Classification Princeton University COS 495 Instructor: Yingyu Liang Example: image classification indoor Indoor outdoor Example: image classification (multiclass)

Moments. Raw moment: February 25, 2014 Normalized / Standardized moment:

Moments Lecture 10: Central Limit Theorem and CDFs Sta230 / Mth 230 Colin Rundel Raw moment: Central moment: µ n = EX n ) µ n = E[X µ) 2 ] February 25, 2014 Normalized / Standardized moment: µ n σ n Sta230

Concentration inequalities and tail bounds

Concentration inequalities and tail bounds John Duchi Outline I Basics and motivation 1 Law of large numbers 2 Markov inequality 3 Cherno bounds II Sub-Gaussian random variables 1 Definitions 2 Examples

Source slideplayer.com/fundamentals of Analytical Chemistry, F.J. Holler, S.R.Crouch. Chapter 6: Random Errors in Chemical Analysis

Source lideplayer.com/fundamental of Analytical Chemitry, F.J. Holler, S.R.Crouch Chapter 6: Random Error in Chemical Analyi Random error are preent in every meaurement no matter how careful the experimenter.

TMA4125 Matematikk 4N Spring 2016

Norwegian Univerity of Science and Technology Department of Mathematical Science TMA45 Matematikk 4N Spring 6 Solution to problem et 6 In general, unle ele i noted, if f i a function, then F = L(f denote

Concentration Inequalities

Chapter Concentration Inequalities I. Moment generating functions, the Chernoff method, and sub-gaussian and sub-exponential random variables a. Goal for this section: given a random variable X, how does

Research Article Goodness-of-Fit Based Secure Cooperative Spectrum Sensing for Cognitive Radio Network

e Scientific World Journal, Article ID 75257, 6 page http://dx.doi.org/.55/24/75257 Reearch Article Goodne-of-Fit Baed Secure Cooperative Spectrum Sening for Cognitive Radio Network Hiep Vu-Van and Inoo

Gaussian source Assumptions d = (x-y) 2, given D, find lower bound of I(X;Y)

Gaussian source Assumptions d = (x-y) 2, given D, find lower bound of I(X;Y) E{(X-Y) 2 } D

Maximization of Technical Efficiency of a Normal- Half Normal Stochastic Production Frontier Model

Global Journal of Pure and Applied Mathematic. ISS 973-1768 Volume 13, umber 1 (17), pp. 7353-7364 Reearch India Publication http://www.ripublication.com Maximization of Technical Efficiency of a ormal-

Theory and Applications of Stochastic Systems Lecture Exponential Martingale for Random Walk

Instructor: Victor F. Araman December 4, 2003 Theory and Applications of Stochastic Systems Lecture 0 B60.432.0 Exponential Martingale for Random Walk Let (S n : n 0) be a random walk with i.i.d. increments

Part IA Probability. Definitions. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015

Part IA Probability Definitions Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.

March 1, Florida State University. Concentration Inequalities: Martingale. Approach and Entropy Method. Lizhe Sun and Boning Yang.

Florida State University March 1, 2018 Framework 1. (Lizhe) Basic inequalities Chernoff bounding Review for STA 6448 2. (Lizhe) Discrete-time martingales inequalities via martingale approach 3. (Boning)

Chapter 4: Continuous channel and its capacity

meghdadi@ensil.unilim.fr Reference : Elements of Information Theory by Cover and Thomas Continuous random variable Gaussian multivariate random variable AWGN Band limited channel Parallel channels Flat

μ + = σ = D 4 σ = D 3 σ = σ = All units in parts (a) and (b) are in V. (1) x chart: Center = μ = 0.75 UCL =

Our online Tutor are available 4*7 to provide Help with Proce control ytem Homework/Aignment or a long term Graduate/Undergraduate Proce control ytem Project. Our Tutor being experienced and proficient

Beta Burr XII OR Five Parameter Beta Lomax Distribution: Remarks and Characterizations

Marquette Univerity e-publication@marquette Mathematic, Statitic and Computer Science Faculty Reearch and Publication Mathematic, Statitic and Computer Science, Department of 6-1-2014 Beta Burr XII OR

BMIR Lecture Series on Probability and Statistics Fall 2015 Discrete RVs

Lecture #7 BMIR Lecture Series on Probability and Statistics Fall 2015 Department of Biomedical Engineering and Environmental Sciences National Tsing Hua University 7.1 Function of Single Variable Theorem

1. The F-test for Equality of Two Variances

. The F-tet for Equality of Two Variance Previouly we've learned how to tet whether two population mean are equal, uing data from two independent ample. We can alo tet whether two population variance are

Lecture 4 Topic 3: General linear models (GLMs), the fundamentals of the analysis of variance (ANOVA), and completely randomized designs (CRDs)

Lecture 4 Topic 3: General linear model (GLM), the fundamental of the analyi of variance (ANOVA), and completely randomized deign (CRD) The general linear model One population: An obervation i explained

Minimum Eigenvalue Detection for Spectrum Sensing in Cognitive Radio

International Journal of Electrical and Computer Engineering (IJECE) Vol. 4, No. 4, Augut 014, pp. 63~630 ISSN: 088-8708 63 Minimum Eigenvalue Detection for Spectrum Sening in Cognitive Radio Syed Sajjad

Exponential Distribution and Poisson Process

Exponential Distribution and Poisson Process Stochastic Processes - Lecture Notes Fatih Cavdur to accompany Introduction to Probability Models by Sheldon M. Ross Fall 215 Outline Introduction Exponential

Tests of Statistical Hypotheses with Respect to a Fuzzy Set

Modern pplied cience; Vol 8, No 1; 014 IN 1913-1844 E-IN 1913-185 Publihed by Canadian Center of cience and Education Tet of tatitical Hypothee with Repect to a uzzy et P Pandian 1 & D Kalpanapriya 1 1

Testing the Equality of Two Pareto Distributions

Proceeding of the World Congre on Engineering 07 Vol II WCE 07, July 5-7, 07, London, U.K. Teting the Equality of Two Pareto Ditribution Huam A. Bayoud, Member, IAENG Abtract Thi paper propoe an overlapping-baed

(3) A bilinear map B : S(R n ) S(R m ) B is continuous (for the product topology) if and only if there exist C, N and M such that

The material here can be found in Hörmander Volume 1, Chapter VII but he ha already done almot all of ditribution theory by thi point(!) Johi and Friedlander Chapter 8. Recall that S( ) i a complete metric

Chapter 4. Chapter 4 sections

Chapter 4 sections 4.1 Expectation 4.2 Properties of Expectations 4.3 Variance 4.4 Moments 4.5 The Mean and the Median 4.6 Covariance and Correlation 4.7 Conditional Expectation SKIP: 4.8 Utility Expectation

NEGATIVE z Scores. TABLE A-2 Standard Normal (z) Distribution: Cumulative Area from the LEFT. (continued)

NEGATIVE z Score z 0 TALE A- Standard Normal (z) Ditribution: Cumulative Area from the LEFT z.00.01.0.03.04.05.06.07.08.09-3.50 and lower.0001-3.4.0003.0003.0003.0003.0003.0003.0003.0003.0003.000-3.3.0005.0005.0005.0004.0004.0004.0004.0004.0004.0003-3..0007.0007.0006.0006.0006.0006.0006.0005.0005.0005-3.1.0010.0009.0009.0009.0008.0008.0008.0008.0007.0007-3.0.0013.0013.0013.001.001.0011.0011.0011.0010.0010

Jan Purczyński, Kamila Bednarz-Okrzyńska Estimation of the shape parameter of GED distribution for a small sample size

Jan Purczyńki, Kamila Bednarz-Okrzyńka Etimation of the hape parameter of GED ditribution for a mall ample ize Folia Oeconomica Stetinenia 4()/, 35-46 04 Folia Oeconomica Stetinenia DOI: 0.478/foli-04-003

Math 407: Probability Theory 5/10/ Final exam (11am - 1pm)

Math 407: Probability Theory 5/10/2013 - Final exam (11am - 1pm) Name: USC ID: Signature: 1. Write your name and ID number in the spaces above. 2. Show all your work and circle your final answer. Simplify

Financial Econometrics and Volatility Models Extreme Value Theory

Financial Econometrics and Volatility Models Extreme Value Theory Eric Zivot May 3, 2010 1 Lecture Outline Modeling Maxima and Worst Cases The Generalized Extreme Value Distribution Modeling Extremes Over

10-704: Information Processing and Learning Spring Lecture 8: Feb 5

10-704: Information Processing and Learning Spring 2015 Lecture 8: Feb 5 Lecturer: Aarti Singh Scribe: Siheng Chen Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal

The space complexity of approximating the frequency moments

The space complexity of approximating the frequency moments Felix Biermeier November 24, 2015 1 Overview Introduction Approximations of frequency moments lower bounds 2 Frequency moments Problem Estimate

3. Review of Probability and Statistics

3. Review of Probability and Statistics ECE 830, Spring 2014 Probabilistic models will be used throughout the course to represent noise, errors, and uncertainty in signal processing problems. This lecture

C up (E) C low (E) E 2 E 1 E 0

Spreading in lock-fading hannel. Muriel Médard David N.. Te medardmit.edu Maachuett Intitute of Technoy dteeec.berkeley.edu Univerity of alifornia at erkeley btract We conider wideband fading channel which

U Logo Use Guidelines

Information Theory Lecture 3: Applications to Machine Learning U Logo Use Guidelines Mark Reid logo is a contemporary n of our heritage. presents our name, d and our motto: arn the nature of things. authenticity

Informatics 2B: Learning and Data Lecture 10 Discriminant functions 2. Minimal misclassifications. Decision Boundaries

Overview Gaussians estimated from training data Guido Sanguinetti Informatics B Learning and Data Lecture 1 9 March 1 Today s lecture Posterior probabilities, decision regions and minimising the probability

Math 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential.

Math 8A Lecture 6 Friday May 7 th Epectation Recall the three main probability density functions so far () Uniform () Eponential (3) Power Law e, ( ), Math 8A Lecture 6 Friday May 7 th Epectation Eample

15-388/688 - Practical Data Science: Basic probability. J. Zico Kolter Carnegie Mellon University Spring 2018

15-388/688 - Practical Data Science: Basic probability J. Zico Kolter Carnegie Mellon University Spring 2018 1 Announcements Logistics of next few lectures Final project released, proposals/groups due

Matching Feature Distributions for Robust Speaker Verification

Matching Feature Ditribution for Robut Speaker Verification Marhalleno Skoan, Daniel Mahao Department of Electrical Engineering, Univerity of Cape Town Rondeboch, Cape Town, South Africa mkoan@crg.ee.uct.ac.za

Composite Hypotheses and Generalized Likelihood Ratio Tests

Composite Hypotheses and Generalized Likelihood Ratio Tests Rebecca Willett, 06 In many real world problems, it is difficult to precisely specify probability distributions. Our models for data may involve

EP2200 Queueing theory and teletraffic systems

P00 Queueing theory and teletraffic ytem Lecture 9 M/G/ ytem Vitoria Fodor KTH S/LCN The M/G/ queue Arrival proce memoryle (Poion( Service time general, identical, independent, f(x Single erver M/ r /

The Binomial distribution. Probability theory 2. Example. The Binomial distribution

Probability theory Tron Anders Moger September th 7 The Binomial distribution Bernoulli distribution: One experiment X i with two possible outcomes, probability of success P. If the experiment is repeated

Chapter 7. Basic Probability Theory

Chapter 7. Basic Probability Theory I-Liang Chern October 20, 2016 1 / 49 What s kind of matrices satisfying RIP Random matrices with iid Gaussian entries iid Bernoulli entries (+/ 1) iid subgaussian entries

Bayesian estimation, classification, and denoising based on alpha-stable statistical modeling

Bayeian etimation, claification, and denoiing baed on alpha-table tatitical modeling Panagioti Takalide FORTH-ICS & Computer Science Department Univerity of Crete Heraklion, Greece www.ic.forth.gr/~takalid

Singular Value Analysis of Linear- Quadratic Systems!

Singular Value Analyi of Linear- Quadratic Sytem! Robert Stengel! Optimal Control and Etimation MAE 546! Princeton Univerity, 2017!! Multivariable Nyquit Stability Criterion!! Matrix Norm and Singular

arxiv: v2 [math.nt] 30 Apr 2015

A THEOREM FOR DISTINCT ZEROS OF L-FUNCTIONS École Normale Supérieure arxiv:54.6556v [math.nt] 3 Apr 5 943 Cachan November 9, 7 Abtract In thi paper, we etablih a imple criterion for two L-function L and

1.3 and 3.9: Derivatives of exponential and logarithmic functions

. and.9: Derivative of exponential and logarithmic function Problem Explain what each of the following mean: (a) f (x) Thi denote the invere function of f, f, evauluated at x. (b) f(x ) Thi mean f. x (c)

Performance Analysis of Wald s SPRT with Independent but Non-Stationary Log-Likelihood Ratios

4th International Conference on Information Fuion Chicago, Illinoi, US, July 5-8, 2 Performance nalyi of Wald SPRT with Independent but Non-Stationary Log-Likelihood Ratio Yu Liu and X Rong Li Department

EC381/MN308 Probability and Some Statistics. Lecture 7 - Outline. Chapter Cumulative Distribution Function (CDF) Continuous Random Variables

EC38/MN38 Probability and Some Statitic Yanni Pachalidi yannip@bu.edu, http://ionia.bu.edu/ Lecture 7 - Outline. Continuou Random Variable Dept. of Manufacturing Engineering Dept. of Electrical and Computer

ECE 4400:693 - Information Theory

ECE 4400:693 - Information Theory Dr. Nghi Tran Lecture 8: Differential Entropy Dr. Nghi Tran (ECE-University of Akron) ECE 4400:693 Lecture 1 / 43 Outline 1 Review: Entropy of discrete RVs 2 Differential

Problem Set 8 Solutions

Deign and Analyi of Algorithm April 29, 2015 Maachuett Intitute of Technology 6.046J/18.410J Prof. Erik Demaine, Srini Devada, and Nancy Lynch Problem Set 8 Solution Problem Set 8 Solution Thi problem

Lecture 7 Introduction to Statistical Decision Theory

Lecture 7 Introduction to Statistical Decision Theory I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 20, 2016 1 / 55 I-Hsiang Wang IT Lecture 7

Confusion matrices. True / False positives / negatives. INF 4300 Classification III Anne Solberg The agenda today: E.g., testing for cancer

INF 4300 Claification III Anne Solberg 29.10.14 The agenda today: More on etimating claifier accuracy Cure of dimenionality knn-claification K-mean clutering x i feature vector for pixel i i- The cla label

Chapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued

Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions

Correction of Overlapping Template Matching Test Included in NIST Randomness Test Suite

1788 PAPER Special Section on Information Theory and It Application Correction of Overlapping Template Matching Tet Included in NIST Randomne Tet Suite Kenji HAMANO a), Member and Tohinobu KANEKO b), Fellow

Problem 1. Construct a filtered probability space on which a Brownian motion W and an adapted process X are defined and such that

Stochatic Calculu Example heet 4 - Lent 5 Michael Tehranchi Problem. Contruct a filtered probability pace on which a Brownian motion W and an adapted proce X are defined and uch that dx t = X t t dt +

IEOR 3106: Introduction to Operations Research: Stochastic Models. Professor Whitt. SOLUTIONS to Homework Assignment 2

IEOR 316: Introduction to Operations Research: Stochastic Models Professor Whitt SOLUTIONS to Homework Assignment 2 More Probability Review: In the Ross textbook, Introduction to Probability Models, read

STATISTICS 3A03. Applied Regression Analysis with SAS. Angelo J. Canty

STATISTICS 3A03 Applied Regression Analysis with SAS Angelo J. Canty Office : Hamilton Hall 209 Phone : (905) 525-9140 extn 27079 E-mail : cantya@mcmaster.ca SAS Labs : L1 Friday 11:30 in BSB 249 L2 Tuesday

Estimation Theory. as Θ = (Θ 1,Θ 2,...,Θ m ) T. An estimator

Estimation Theory Estimation theory deals with finding numerical values of interesting parameters from given set of data. We start with formulating a family of models that could describe how the data were

Name: Solutions Exam 2

Name: Solution Exam Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will

Chapter 2 Signal Processing at Receivers: Detection Theory

Chapter Signal Processing at Receivers: Detection Theory As an application of the statistical hypothesis testing, signal detection plays a key role in signal processing at receivers of wireless communication

Lecture 2: Review of Basic Probability Theory

ECE 830 Fall 2010 Statistical Signal Processing instructor: R. Nowak, scribe: R. Nowak Lecture 2: Review of Basic Probability Theory Probabilistic models will be used throughout the course to represent

Alternate Dispersion Measures in Replicated Factorial Experiments

Alternate Diperion Meaure in Replicated Factorial Experiment Neal A. Mackertich The Raytheon Company, Sudbury MA 02421 Jame C. Benneyan Northeatern Univerity, Boton MA 02115 Peter D. Krau The Raytheon

APPM/MATH 4/5520 Solutions to Exam I Review Problems. f X 1,X 2. 2e x 1 x 2. = x 2

APPM/MATH 4/5520 Solutions to Exam I Review Problems. (a) f X (x ) f X,X 2 (x,x 2 )dx 2 x 2e x x 2 dx 2 2e 2x x was below x 2, but when marginalizing out x 2, we ran it over all values from 0 to and so

University of Cambridge Engineering Part IIB Module 3F3: Signal and Pattern Processing Handout 2:. The Multivariate Gaussian & Decision Boundaries

University of Cambridge Engineering Part IIB Module 3F3: Signal and Pattern Processing Handout :. The Multivariate Gaussian & Decision Boundaries..15.1.5 1 8 6 6 8 1 Mark Gales mjfg@eng.cam.ac.uk Lent

Lecture 5: Importance sampling and Hamilton-Jacobi equations

Lecture 5: Importance sampling and Hamilton-Jacobi equations Henrik Hult Department of Mathematics KTH Royal Institute of Technology Sweden Summer School on Monte Carlo Methods and Rare Events Brown University,

EE 508 Lecture 16. Filter Transformations. Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject

EE 508 Lecture 6 Filter Tranformation Lowpa to Bandpa Lowpa to Highpa Lowpa to Band-reject Review from Lat Time Theorem: If the perimeter variation and contact reitance are neglected, the tandard deviation

STAT/MATH 395 A - PROBABILITY II UW Winter Quarter Moment functions. x r p X (x) (1) E[X r ] = x r f X (x) dx (2) (x E[X]) r p X (x) (3)

STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 07 Néhémy Lim Moment functions Moments of a random variable Definition.. Let X be a rrv on probability space (Ω, A, P). For a given r N, E[X r ], if it