Set Theory. Pattern Recognition III. Michal Haindl. Set Operations. Outline
|
|
- Lynn Hicks
- 5 years ago
- Views:
Transcription
1 Set Theory A, B sets e.g. A = {ζ 1,...,ζ n } A = { c x y d} S space (universe) A,B S Outline Pattern Recognition III Michal Haindl Faculty of Information Technology, KTI Czech Technical University in Prague Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Prague, Czech Republic Evropský sociální fond. Praha & EU: Investujeme do vaší budoucnosti MI-ROZ /Z Set Operations c M. Haindl MI-ROZ /17 Outline c M. Haindl MI-ROZ /17 January 16, 2012 Outline A = {1,2,3} B = {3,2,1} sum (union) product (intersection) A 1,...,A n are iff 1 Set Theory 2 Probability c M. Haindl MI-ROZ /17
2 Set Operations Set Operations sum (union) product (intersection) A 1,...,A n are iff sum (union) A = {1,2,3} B = {3,4,5} A B = {1,2,3,4,5} product (intersection) A 1,...,A n are iff Set Operations 2 Set Operations complement Ā = S, S =, AĀ =, A+Ā = S difference A B,A\B A B = A B = A AB De Morgan law A B = Ā B A B = Ā B sum (union) product (intersection) A B = {3} A 1,...,A n are iff c M. Haindl MI-ROZ /17
3 Classical Set Operations 2 P(A) = N A N N A no of favourable outcomes complement Ā = S, S =, AĀ =, A+Ā = S if A i i are disjoint P( A i ) = difference A\B = {1,2} A B,A\B A B = A B = A AB De Morgan law 1654 Blaise Pascal, 1812 Pierre-Simon Laplace Théorie analytique des probabilités A B = Ā B A B = Ā B c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17 Axiomatic Probability 1 P(A) is positive: P(A) 0 2 Probability of certain events equals 1: P(S) = 1 3 If A and B are mutually exclusive, then: P(A+B) = P(A)+P(B) (otherwise P(A+B) = P(A)+P(B) P(AB) n A no of event A appearance Relative Frequency n A P(A) = lim n n definitions: Classical (A priori definition as a ratio of favourable to total number of alternatives.) Axiomatic (measure, A. Kolmogoroff ) Relative frequency (Richard von Mieses ) Probability as a measure of belief (inductive reasoning) A, B events, S space (certain event), impossible event, 0 P(.) 1 A, B mutually exclusive events, A Ā = S c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17
4 Conditional Probability Axiomatic given P(B) > 0 P(A B) = P(AB) P(B) Total Probability A i i mutually exclusive events n A i = S 1 P(A) is positive: P(A) 0 2 Probability of certain events equals 1: P(S) = 1 3 If A and B are mutually exclusive, then: P(A+B) = P(A)+P(B) (otherwise P(A+B) = P(A)+P(B) P(AB) P(B) = P(B A i ) = P(B A i ) Relative Frequency Independent Events P(A, B) = P(A)P(B) def. P(A B) = P(A), P(A 1,...,A n ) = i n A no of event A appearance n A P(A) = lim n n c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17 Conditional Probability Conditional Probability given P(B) > 0 given P(B) > 0 P(A B) = P(AB) P(B) Total Probability A i i mutually exclusive events n A i = S P(A B) = P(AB) P(B) Total Probability A i i mutually exclusive events n A i = S P(B) = P(B A i ) = P(B A i ) P(B) = P(B A i ) = P(B A i ) P(A, B) = P(A)P(B) def. Independent Events P(A, B) = P(A)P(B) def. Independent Events P(A B) = P(A), P(A 1,...,A n ) = i P(A B) = P(A), P(A 1,...,A n ) = i c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17
5 Random Variable Bayes Theorem X : ζ (real /complex)number ζ experiment outcome distribution function of the r.v. X F X (x) = P(X x) F( ) = 0 F(+ ) = 1 nondecreasing function of x F(x 1 ) F(x 2 ) for x 1 < x 2 continuous from the right F(x + ) = F(x) A i i mutually exclusive events P(A i B) = n A i = S P(B A i ) n P(B A i) density function F(x) = x f(t)dt nonnegativity f(x) 0 f(x) = df(x) Thomas Bayes: An Essay Toward Solving a Problem in the Doctrine of Chances, 1764 expected value E{X} = xf(x) = xdf(x) c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17 Random Variable Random Variable X : ζ (real /complex)number ζ experiment outcome distribution function of the r.v. X F X (x) = P(X x) F( ) = 0 F(+ ) = 1 X : ζ (real /complex)number ζ experiment outcome distribution function of the r.v. X F X (x) = P(X x) nondecreasing function of x F(x 1 ) F(x 2 ) for x 1 < x 2 continuous from the right F(x + ) = F(x) density function F(x) = x f(t)dt nonnegativity f(x) 0 expected value f(x) = df(x) E{X} = xf(x) = xdf(x) c M. Haindl MI-ROZ /17 F( ) = 0 F(+ ) = 1 nondecreasing function of x F(x 1 ) F(x 2 ) for x 1 < x 2 continuous from the right F(x + ) = F(x) density function c M. Haindl MI-ROZ - 03 f(x) = df(x) 11/17
6 Dicrete Random Variable Random Variable 2 F X (x) = i P(X = x i) i : x i x (staircase form) expected value E{X} = i x i P(X = x i ) moments µ k = E{X k } central moments µ k = E { (X E{X}) k} F X (x) Mixed Random Variable discontinuous but not of a staircase form µ 2 = σ 2 variance (dispersion) σ = µ 2 standard deviation median x : F( x) 1 2 F( x +0) 1 2 c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17 Conditional Distribution F X (x z) = P(X x z) = f(x z) = df(x z) total probability ( n i P(z i ) = 1) F X (x) = P(X x, Z = z) P(z) F X (x z i )P(z i ) i Dicrete Random Variable F X (x) = i P(X = x i) i : x i x (staircase form) expected value E{X} = i x i P(X = x i ) conditional expected value E{X z} = xf(x z) F X (x) Mixed Random Variable discontinuous but not of a staircase form c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17
7 Normal Distribution Joint Distribution Gaussian N(µ,Σ) f(y) = (2π) n 2 Σ 1 2 exp { 1 } 2 (y µ)t Σ 1 (y µ) Y = {Y 1,...,Y n } E{y} = µ E{(y µ) T (y µ)} = Σ F Y (y) = P(Y 1 y 1,...,Y n y n ) f Y (y) = n F Y (y) y 1,..., y n E{Y} = (E{Y 1 },...,E{Y n }) cov{y i,y j } = E {(Y i E{Y i })(Y j E{Y j })} c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17 Normal Distribution Marginal Distribution Gaussian N(µ,Σ) f(y) = (2π) n 2 Σ 1 2 exp { 1 } 2 (y µ)t Σ 1 (y µ) F Yi (y i ) = F Y (,...,,y i,,...) E{y} = µ E{(y µ) T (y µ)} = Σ Σ regular, positive definite matrix if Σ = diag{σ 1,1,...,Σ n,n } then y 1,...,y n independent ỹ y ỹ N conditional distribution N any lin. combination of y i N F Yi (y i ) = f k (y 1,...,y k ) = yi... f(y 1,...,y i 1,t,y i+1,...,y n ) dy 1,...,dy i 1 dtdy i+1,...,dy n R n k f(y 1,...,y n )dy k+1...dy n c M. Haindl MI-ROZ /17 c M. Haindl MI-ROZ /17
Branch-and-Bound Algorithm. Pattern Recognition XI. Michal Haindl. Outline
Branch-and-Bound Algorithm assumption - can be used if a feature selection criterion satisfies the monotonicity property monotonicity property - for nested feature sets X j related X 1 X 2... X l the criterion
More informationFeature Selection. Pattern Recognition X. Michal Haindl. Feature Selection. Outline
Feature election Outline Pattern Recognition X motivation technical recognition problem dimensionality reduction ց class separability increase ր data compression (e.g. required communication channel capacity)
More informationNotation. Pattern Recognition II. Michal Haindl. Outline - PR Basic Concepts. Pattern Recognition Notions
Notation S pattern space X feature vector X = [x 1,...,x l ] l = dim{x} number of features X feature space K number of classes ω i class indicator Ω = {ω 1,...,ω K } g(x) discriminant function H decision
More informationNeural Nets in PR. Pattern Recognition XII. Michal Haindl. Outline. Neural Nets in PR 2
Neural Nets in PR NM P F Outline Motivation: Pattern Recognition XII human brain study complex cognitive tasks Michal Haindl Faculty of Information Technology, KTI Czech Technical University in Prague
More informationRecitation 2: Probability
Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions
More informationAppendix A : Introduction to Probability and stochastic processes
A-1 Mathematical methods in communication July 5th, 2009 Appendix A : Introduction to Probability and stochastic processes Lecturer: Haim Permuter Scribe: Shai Shapira and Uri Livnat The probability of
More informationAnalysis of Experimental Designs
Analysis of Experimental Designs p. 1/? Analysis of Experimental Designs Gilles Lamothe Mathematics and Statistics University of Ottawa Analysis of Experimental Designs p. 2/? Review of Probability A short
More informationEE4601 Communication Systems
EE4601 Communication Systems Week 2 Review of Probability, Important Distributions 0 c 2011, Georgia Institute of Technology (lect2 1) Conditional Probability Consider a sample space that consists of two
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationPart 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.
More informationData Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2017 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
More informationWhy study probability? Set theory. ECE 6010 Lecture 1 Introduction; Review of Random Variables
ECE 6010 Lecture 1 Introduction; Review of Random Variables Readings from G&S: Chapter 1. Section 2.1, Section 2.3, Section 2.4, Section 3.1, Section 3.2, Section 3.5, Section 4.1, Section 4.2, Section
More informationSingle Maths B: Introduction to Probability
Single Maths B: Introduction to Probability Overview Lecturer Email Office Homework Webpage Dr Jonathan Cumming j.a.cumming@durham.ac.uk CM233 None! http://maths.dur.ac.uk/stats/people/jac/singleb/ 1 Introduction
More informationOrigins of Probability Theory
1 16.584: INTRODUCTION Theory and Tools of Probability required to analyze and design systems subject to uncertain outcomes/unpredictability/randomness. Such systems more generally referred to as Experiments.
More informationProbability Theory. Alea iacta est!
Probability Theory Alea iacta est! "Scientific theories which involve the properties of large aggregates of individuals [...] are liable to misinterpretation as soon as the statistical nature of the argument
More informationStatistika pro informatiku
Statistika pro informatiku prof. RNDr. Roman Kotecký DrSc., Dr. Rudolf Blažek, PhD Katedra teoretické informatiky FIT České vysoké učení technické v Praze MI-SPI, ZS 2011/12, Přednáška 5 Evropský sociální
More informationProbability Review. Yutian Li. January 18, Stanford University. Yutian Li (Stanford University) Probability Review January 18, / 27
Probability Review Yutian Li Stanford University January 18, 2018 Yutian Li (Stanford University) Probability Review January 18, 2018 1 / 27 Outline 1 Elements of probability 2 Random variables 3 Multiple
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Lecture 2 Probability Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 3-1 3.1 Definition Random Experiment a process leading to an uncertain
More informationProbability Theory Review Reading Assignments
Probability Theory Review Reading Assignments R. Duda, P. Hart, and D. Stork, Pattern Classification, John-Wiley, 2nd edition, 2001 (appendix A.4, hard-copy). "Everything I need to know about Probability"
More informationStatistika pro informatiku
Statistika pro informatiku prof. RNDr. Roman Kotecký DrSc., Dr. Rudolf Blažek, PhD Katedra teoretické informatiky FIT České vysoké učení technické v Praze MI-SPI, ZS 2011/12, Přednáška 2 Evropský sociální
More informationIntroduction to Probability and Stocastic Processes - Part I
Introduction to Probability and Stocastic Processes - Part I Lecture 1 Henrik Vie Christensen vie@control.auc.dk Department of Control Engineering Institute of Electronic Systems Aalborg University Denmark
More informationProbability Dr. Manjula Gunarathna 1
Probability Dr. Manjula Gunarathna Probability Dr. Manjula Gunarathna 1 Introduction Probability theory was originated from gambling theory Probability Dr. Manjula Gunarathna 2 History of Probability Galileo
More informationMI-RUB Testing Lecture 10
MI-RUB Testing Lecture 10 Pavel Strnad pavel.strnad@fel.cvut.cz Dept. of Computer Science, FEE CTU Prague, Karlovo nám. 13, 121 35 Praha, Czech Republic MI-RUB, WS 2011/12 Evropský sociální fond Praha
More informationProbability- describes the pattern of chance outcomes
Chapter 6 Probability the study of randomness Probability- describes the pattern of chance outcomes Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long
More informationRandom Signals and Systems. Chapter 1. Jitendra K Tugnait. Department of Electrical & Computer Engineering. James B Davis Professor.
Random Signals and Systems Chapter 1 Jitendra K Tugnait James B Davis Professor Department of Electrical & Computer Engineering Auburn University 2 3 Descriptions of Probability Relative frequency approach»
More informationLecture 11. Probability Theory: an Overveiw
Math 408 - Mathematical Statistics Lecture 11. Probability Theory: an Overveiw February 11, 2013 Konstantin Zuev (USC) Math 408, Lecture 11 February 11, 2013 1 / 24 The starting point in developing the
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationReview of Statistics
Review of Statistics Topics Descriptive Statistics Mean, Variance Probability Union event, joint event Random Variables Discrete and Continuous Distributions, Moments Two Random Variables Covariance and
More informationBayes Theorem. Jan Kracík. Department of Applied Mathematics FEECS, VŠB - TU Ostrava
Jan Kracík Department of Applied Mathematics FEECS, VŠB - TU Ostrava Introduction Bayes theorem fundamental theorem in probability theory named after reverend Thomas Bayes (1701 1761) discovered in Bayes
More informationIntroduction to Information Entropy Adapted from Papoulis (1991)
Introduction to Information Entropy Adapted from Papoulis (1991) Federico Lombardo Papoulis, A., Probability, Random Variables and Stochastic Processes, 3rd edition, McGraw ill, 1991. 1 1. INTRODUCTION
More informationGiven a experiment with outcomes in sample space: Ω Probability measure applied to subsets of Ω: P[A] 0 P[A B] = P[A] + P[B] P[AB] = P(AB)
1 16.584: Lecture 2 : REVIEW Given a experiment with outcomes in sample space: Ω Probability measure applied to subsets of Ω: P[A] 0 P[A B] = P[A] + P[B] if AB = P[A B] = P[A] + P[B] P[AB] P[A] = 1 P[A
More informationLecture 1: Probability Fundamentals
Lecture 1: Probability Fundamentals IB Paper 7: Probability and Statistics Carl Edward Rasmussen Department of Engineering, University of Cambridge January 22nd, 2008 Rasmussen (CUED) Lecture 1: Probability
More informationChapter 2. Probability
2-1 Chapter 2 Probability 2-2 Section 2.1: Basic Ideas Definition: An experiment is a process that results in an outcome that cannot be predicted in advance with certainty. Examples: rolling a die tossing
More informationBrief Review of Probability
Brief Review of Probability Nuno Vasconcelos (Ken Kreutz-Delgado) ECE Department, UCSD Probability Probability theory is a mathematical language to deal with processes or experiments that are non-deterministic
More informationTopic 2: Probability & Distributions. Road Map Probability & Distributions. ECO220Y5Y: Quantitative Methods in Economics. Dr.
Topic 2: Probability & Distributions ECO220Y5Y: Quantitative Methods in Economics Dr. Nick Zammit University of Toronto Department of Economics Room KN3272 n.zammit utoronto.ca November 21, 2017 Dr. Nick
More informationStatistics for Business and Economics
Statistics for Business and Economics Basic Probability Learning Objectives In this lecture(s), you learn: Basic probability concepts Conditional probability To use Bayes Theorem to revise probabilities
More information1 INFO Sep 05
Events A 1,...A n are said to be mutually independent if for all subsets S {1,..., n}, p( i S A i ) = p(a i ). (For example, flip a coin N times, then the events {A i = i th flip is heads} are mutually
More informationDept. of Linguistics, Indiana University Fall 2015
L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 34 To start out the course, we need to know something about statistics and This is only an introduction; for a fuller understanding, you would
More informationBACKGROUND NOTES FYS 4550/FYS EXPERIMENTAL HIGH ENERGY PHYSICS AUTUMN 2016 PROBABILITY A. STRANDLIE NTNU AT GJØVIK AND UNIVERSITY OF OSLO
ACKGROUND NOTES FYS 4550/FYS9550 - EXERIMENTAL HIGH ENERGY HYSICS AUTUMN 2016 ROAILITY A. STRANDLIE NTNU AT GJØVIK AND UNIVERSITY OF OSLO efore embarking on the concept of probability, we will first define
More informationMI-RUB Testing II Lecture 11
MI-RUB Testing II Lecture 11 Pavel Strnad pavel.strnad@fel.cvut.cz Dept. of Computer Science, FEE CTU Prague, Karlovo nám. 13, 121 35 Praha, Czech Republic MI-RUB, WS 2011/12 Evropský sociální fond Praha
More informationProbability and statistics; Rehearsal for pattern recognition
Probability and statistics; Rehearsal for pattern recognition Václav Hlaváč Czech Technical University in Prague Czech Institute of Informatics, Robotics and Cybernetics 166 36 Prague 6, Jugoslávských
More informationStatistical Inference
Statistical Inference Lecture 1: Probability Theory MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 11, 2018 Outline Introduction Set Theory Basics of Probability Theory
More informationUnit 7 Probability M2 13.1,2,4, 5,6
+ Unit 7 Probability M2 13.1,2,4, 5,6 7.1 Probability n Obj.: I will be able to determine the experimental and theoretical probabilities of an event, or its complement, occurring. n Vocabulary o Outcome
More informationM378K In-Class Assignment #1
The following problems are a review of M6K. M7K In-Class Assignment # Problem.. Complete the definition of mutual exclusivity of events below: Events A, B Ω are said to be mutually exclusive if A B =.
More informationProbability Theory. Patrick Lam
Probability Theory Patrick Lam Outline Probability Random Variables Simulation Important Distributions Discrete Distributions Continuous Distributions Most Basic Definition of Probability: number of successes
More informationProbability. Paul Schrimpf. January 23, UBC Economics 326. Probability. Paul Schrimpf. Definitions. Properties. Random variables.
Probability UBC Economics 326 January 23, 2018 1 2 3 Wooldridge (2013) appendix B Stock and Watson (2009) chapter 2 Linton (2017) chapters 1-5 Abbring (2001) sections 2.1-2.3 Diez, Barr, and Cetinkaya-Rundel
More informationCopyright c 2006 Jason Underdown Some rights reserved. choose notation. n distinct items divided into r distinct groups.
Copyright & License Copyright c 2006 Jason Underdown Some rights reserved. choose notation binomial theorem n distinct items divided into r distinct groups Axioms Proposition axioms of probability probability
More informationQuantitative Methods in Economics Conditional Expectations
Quantitative Methods in Economics Conditional Expectations Maximilian Kasy Harvard University, fall 2016 1 / 19 Roadmap, Part I 1. Linear predictors and least squares regression 2. Conditional expectations
More informationi=1 k i=1 g i (Y )] = k
Math 483 EXAM 2 covers 2.4, 2.5, 2.7, 2.8, 3.1, 3.2, 3.3, 3.4, 3.8, 3.9, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.9, 5.1, 5.2, and 5.3. The exam is on Thursday, Oct. 13. You are allowed THREE SHEETS OF NOTES and
More informationPROBABILITY CHAPTER LEARNING OBJECTIVES UNIT OVERVIEW
CHAPTER 16 PROBABILITY LEARNING OBJECTIVES Concept of probability is used in accounting and finance to understand the likelihood of occurrence or non-occurrence of a variable. It helps in developing financial
More informationSTA 291 Lecture 8. Probability. Probability Rules. Joint and Marginal Probability. STA Lecture 8 1
STA 291 Lecture 8 Probability Probability Rules Joint and Marginal Probability STA 291 - Lecture 8 1 Union and Intersection Let A and B denote two events. The union of two events: A B The intersection
More informationPCMI Introduction to Random Matrix Theory Handout # REVIEW OF PROBABILITY THEORY. Chapter 1 - Events and Their Probabilities
PCMI 207 - Introduction to Random Matrix Theory Handout #2 06.27.207 REVIEW OF PROBABILITY THEORY Chapter - Events and Their Probabilities.. Events as Sets Definition (σ-field). A collection F of subsets
More informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationProbability and Applications
Chapter 5 Probability and Applications 5.2 SOME USEFUL DEFINITIONS Random experiment: a process that has an unknown outcome or outcomes that are known only after the process is completed. Event: an outcome
More information1 Presessional Probability
1 Presessional Probability Probability theory is essential for the development of mathematical models in finance, because of the randomness nature of price fluctuations in the markets. This presessional
More informationDeep Learning for Computer Vision
Deep Learning for Computer Vision Lecture 3: Probability, Bayes Theorem, and Bayes Classification Peter Belhumeur Computer Science Columbia University Probability Should you play this game? Game: A fair
More informationProbability Theory. Fourier Series and Fourier Transform are widely used techniques to model deterministic signals.
Probability Theory Introduction Fourier Series Fourier Transform are widely used techniques to model deterministic signals. In a typical communication system, the output of an information source (e.g.
More informationn! (k 1)!(n k)! = F (X) U(0, 1). (x, y) = n(n 1) ( F (y) F (x) ) n 2
Order statistics Ex. 4. (*. Let independent variables X,..., X n have U(0, distribution. Show that for every x (0,, we have P ( X ( < x and P ( X (n > x as n. Ex. 4.2 (**. By using induction or otherwise,
More informationPreliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix
More informationAn event described by a single characteristic e.g., A day in January from all days in 2012
Events Each possible outcome of a variable is an event. Simple event An event described by a single characteristic e.g., A day in January from all days in 2012 Joint event An event described by two or
More informationBinary Decision Diagrams
Binary Decision Diagrams Logic Circuits Design Seminars WS2010/2011, Lecture 2 Ing. Petr Fišer, Ph.D. Department of Digital Design Faculty of Information Technology Czech Technical University in Prague
More informationWeek 2. Review of Probability, Random Variables and Univariate Distributions
Week 2 Review of Probability, Random Variables and Univariate Distributions Probability Probability Probability Motivation What use is Probability Theory? Probability models Basis for statistical inference
More informationPerhaps the simplest way of modeling two (discrete) random variables is by means of a joint PMF, defined as follows.
Chapter 5 Two Random Variables In a practical engineering problem, there is almost always causal relationship between different events. Some relationships are determined by physical laws, e.g., voltage
More informationProbability theory. References:
Reasoning Under Uncertainty References: Probability theory Mathematical methods in artificial intelligence, Bender, Chapter 7. Expert systems: Principles and programming, g, Giarratano and Riley, pag.
More informationProbability Theory for Machine Learning. Chris Cremer September 2015
Probability Theory for Machine Learning Chris Cremer September 2015 Outline Motivation Probability Definitions and Rules Probability Distributions MLE for Gaussian Parameter Estimation MLE and Least Squares
More informationQuick Tour of Basic Probability Theory and Linear Algebra
Quick Tour of and Linear Algebra Quick Tour of and Linear Algebra CS224w: Social and Information Network Analysis Fall 2011 Quick Tour of and Linear Algebra Quick Tour of and Linear Algebra Outline Definitions
More informationWhat is Probability? Probability. Sample Spaces and Events. Simple Event
What is Probability? Probability Peter Lo Probability is the numerical measure of likelihood that the event will occur. Simple Event Joint Event Compound Event Lies between 0 & 1 Sum of events is 1 1.5
More informationPreliminary statistics
1 Preliminary statistics The solution of a geophysical inverse problem can be obtained by a combination of information from observed data, the theoretical relation between data and earth parameters (models),
More informationSummary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016
8. For any two events E and F, P (E) = P (E F ) + P (E F c ). Summary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016 Sample space. A sample space consists of a underlying
More informationReview of Probabilities and Basic Statistics
Alex Smola Barnabas Poczos TA: Ina Fiterau 4 th year PhD student MLD Review of Probabilities and Basic Statistics 10-701 Recitations 1/25/2013 Recitation 1: Statistics Intro 1 Overview Introduction to
More informationFormulas for probability theory and linear models SF2941
Formulas for probability theory and linear models SF2941 These pages + Appendix 2 of Gut) are permitted as assistance at the exam. 11 maj 2008 Selected formulae of probability Bivariate probability Transforms
More informationEE514A Information Theory I Fall 2013
EE514A Information Theory I Fall 2013 K. Mohan, Prof. J. Bilmes University of Washington, Seattle Department of Electrical Engineering Fall Quarter, 2013 http://j.ee.washington.edu/~bilmes/classes/ee514a_fall_2013/
More informationCS 591, Lecture 2 Data Analytics: Theory and Applications Boston University
CS 591, Lecture 2 Data Analytics: Theory and Applications Boston University Charalampos E. Tsourakakis January 25rd, 2017 Probability Theory The theory of probability is a system for making better guesses.
More informationBayesian statistics, simulation and software
Module 1: Course intro and probability brush-up Department of Mathematical Sciences Aalborg University 1/22 Bayesian Statistics, Simulations and Software Course outline Course consists of 12 half-days
More informationV7 Foundations of Probability Theory
V7 Foundations of Probability Theory Probability : degree of confidence that an event of an uncertain nature will occur. Events : we will assume that there is an agreed upon space of possible outcomes
More informationExample: Suppose we toss a quarter and observe whether it falls heads or tails, recording the result as 1 for heads and 0 for tails.
Example: Suppose we toss a quarter and observe whether it falls heads or tails, recording the result as 1 for heads and 0 for tails. (In Mathematical language, the result of our toss is a random variable,
More informationFundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes
Fundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes Klaus Witrisal witrisal@tugraz.at Signal Processing and Speech Communication Laboratory www.spsc.tugraz.at Graz University of
More informationReview: mostly probability and some statistics
Review: mostly probability and some statistics C2 1 Content robability (should know already) Axioms and properties Conditional probability and independence Law of Total probability and Bayes theorem Random
More informationThe enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail}
Random Experiment In random experiments, the result is unpredictable, unknown prior to its conduct, and can be one of several choices. Examples: The Experiment of tossing a coin (head, tail) The Experiment
More informationBMIR 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
More informationPROBABILITY AND RANDOM PROCESSESS
PROBABILITY AND RANDOM PROCESSESS SOLUTIONS TO UNIVERSITY QUESTION PAPER YEAR : JUNE 2014 CODE NO : 6074 /M PREPARED BY: D.B.V.RAVISANKAR ASSOCIATE PROFESSOR IT DEPARTMENT MVSR ENGINEERING COLLEGE, NADERGUL
More informationReview of Statistics I
Review of Statistics I Hüseyin Taştan 1 1 Department of Economics Yildiz Technical University April 17, 2010 1 Review of Distribution Theory Random variables, discrete vs continuous Probability distribution
More information2 (Statistics) Random variables
2 (Statistics) Random variables References: DeGroot and Schervish, chapters 3, 4 and 5; Stirzaker, chapters 4, 5 and 6 We will now study the main tools use for modeling experiments with unknown outcomes
More informationCS 630 Basic Probability and Information Theory. Tim Campbell
CS 630 Basic Probability and Information Theory Tim Campbell 21 January 2003 Probability Theory Probability Theory is the study of how best to predict outcomes of events. An experiment (or trial or event)
More informationIntroduction to Machine Learning
Introduction to Machine Learning Introduction to Probabilistic Methods Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB
More informationLecture 2. Conditional Probability
Math 408 - Mathematical Statistics Lecture 2. Conditional Probability January 18, 2013 Konstantin Zuev (USC) Math 408, Lecture 2 January 18, 2013 1 / 9 Agenda Motivation and Definition Properties of Conditional
More informationProbability. Lecture Notes. Adolfo J. Rumbos
Probability Lecture Notes Adolfo J. Rumbos October 20, 204 2 Contents Introduction 5. An example from statistical inference................ 5 2 Probability Spaces 9 2. Sample Spaces and σ fields.....................
More informationProperties of Summation Operator
Econ 325 Section 003/004 Notes on Variance, Covariance, and Summation Operator By Hiro Kasahara Properties of Summation Operator For a sequence of the values {x 1, x 2,..., x n, we write the sum of x 1,
More informationProbability Theory and Applications
Probability Theory and Applications Videos of the topics covered in this manual are available at the following links: Lesson 4 Probability I http://faculty.citadel.edu/silver/ba205/online course/lesson
More informationMATH 556: PROBABILITY PRIMER
MATH 6: PROBABILITY PRIMER 1 DEFINITIONS, TERMINOLOGY, NOTATION 1.1 EVENTS AND THE SAMPLE SPACE Definition 1.1 An experiment is a one-off or repeatable process or procedure for which (a there is a well-defined
More informationNonlinearOptimization
1/35 NonlinearOptimization Pavel Kordík Department of Computer Systems Faculty of Information Technology Czech Technical University in Prague Jiří Kašpar, Pavel Tvrdík, 2011 Unconstrained nonlinear optimization,
More informationIntroduction to probability theory
Introduction to probability theory Fátima Sánchez Cabo Institute for Genomics and Bioinformatics, TUGraz f.sanchezcabo@tugraz.at 07/03/2007 - p. 1/35 Outline Random and conditional probability (7 March)
More informationChapter 5. Chapter 5 sections
1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationChapter 1: Probability Theory Lecture 1: Measure space and measurable function
Chapter 1: Probability Theory Lecture 1: Measure space and measurable function Random experiment: uncertainty in outcomes Ω: sample space: a set containing all possible outcomes Definition 1.1 A collection
More informationReview Basic Probability Concept
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.9008 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More informationReview (Probability & Linear Algebra)
Review (Probability & Linear Algebra) CE-725 : Statistical Pattern Recognition Sharif University of Technology Spring 2013 M. Soleymani Outline Axioms of probability theory Conditional probability, Joint
More informationExamples of random experiment (a) Random experiment BASIC EXPERIMENT
Random experiment A random experiment is a process leading to an uncertain outcome, before the experiment is run We usually assume that the experiment can be repeated indefinitely under essentially the
More informationLecture 6. Probability events. Definition 1. The sample space, S, of a. probability experiment is the collection of all
Lecture 6 1 Lecture 6 Probability events Definition 1. The sample space, S, of a probability experiment is the collection of all possible outcomes of an experiment. One such outcome is called a simple
More informationIntroduction to Probability Theory
Introduction to Probability Theory Ping Yu Department of Economics University of Hong Kong Ping Yu (HKU) Probability 1 / 39 Foundations 1 Foundations 2 Random Variables 3 Expectation 4 Multivariate Random
More information