DOCSLIB.ORG

Probability with Engineering Applications ECE 313 Course Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Probability with Engineering Applications ECE 313 Course Notes Probability with Engineering Applications ECE 313 Course Notes Bruce Hajek Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign January 2017 c 2017 by Bruce Hajek All rights reserved. Permission is hereby given to freely print and circulate copies of these notes so long as the notes are left intact and not reproduced for commercial purposes. Email to [email protected], pointing out errors or hard to understand passages or providing comments, is welcome. Contents 1 Foundations 3 1.1 Embracing uncertainty . .3 1.2 Axioms of probability . .6 1.3 Calculating the size of various sets . 10 1.4 Probability experiments with equally likely outcomes . 13 1.5 Sample spaces with infinite cardinality . 15 1.6 Short Answer Questions . 20 1.7 Problems . 21 2 Discrete-type random variables 25 2.1 Random variables and probability mass functions . 25 2.2 The mean and variance of a random variable . 27 2.3 Conditional probabilities . 32 2.4 Independence and the binomial distribution . 34 2.4.1 Mutually independent events . 34 2.4.2 Independent random variables (of discrete-type) . 36 2.4.3 Bernoulli distribution . 37 2.4.4 Binomial distribution . 38 2.5 Geometric distribution . 41 2.6 Bernoulli process and the negative binomial distribution . 43 2.7 The Poisson distribution{a limit of binomial distributions . 45 2.8 Maximum likelihood parameter estimation . 47 2.9 Markov and Chebychev inequalities and confidence intervals . 50 2.10 The law of total probability, and Bayes formula . 53 2.11 Binary hypothesis testing with discrete-type observations . 60 2.11.1 Maximum likelihood (ML) decision rule . 61 2.11.2 Maximum a posteriori probability (MAP) decision rule . 62 2.12 Reliability . 67 2.12.1 Union bound . 67 2.12.2 Network outage probability . 67 2.12.3 Distribution of the capacity of a flow network . 70 2.12.4 Analysis of an array code . 72 iii iv CONTENTS 2.12.5 Reliability of a single backup . 74 2.13 Short Answer Questions . 74 2.14 Problems . 76 3 Continuous-type random variables 95 3.1 Cumulative distribution functions . 95 3.2 Continuous-type random variables . 100 3.3 Uniform distribution . 103 3.4 Exponential distribution . 104 3.5 Poisson processes . 107 3.5.1 Time-scaled Bernoulli processes . 107 3.5.2 Definition and properties of Poisson processes . 108 3.5.3 The Erlang distribution . 112 3.6 Linear scaling of pdfs and the Gaussian distribution . 113 3.6.1 Scaling rule for pdfs . 113 3.6.2 The Gaussian (normal) distribution . 115 3.6.3 The central limit theorem and the Gaussian approximation . 119 3.7 ML parameter estimation for continuous-type variables . 124 3.8 Functions of a random variable . 125 3.8.1 The distribution of a function of a random variable . 125 3.8.2 Generating a random variable with a specified distribution . 135 3.8.3 The area rule for expectation based on the CDF . 137 3.9 Failure rate functions . 138 3.10 Binary hypothesis testing with continuous-type observations . 140 3.11 Short Answer Questions . 146 3.12 Problems . 148 4 Jointly Distributed Random Variables 161 4.1 Joint cumulative distribution functions . 161 4.2 Joint probability mass functions . 163 4.3 Joint probability density functions . 165 4.4 Independence of random variables . 175 4.4.1 Definition of independence for two random variables . 175 4.4.2 Determining from a pdf whether independence holds . 176 4.5 Distribution of sums of random variables . 178 4.5.1 Sums of integer-valued random variables . 179 4.5.2 Sums of jointly continuous-type random variables . 181 4.6 Additional examples using joint distributions . 184 4.7 Joint pdfs of functions of random variables . 189 4.7.1 Transformation of pdfs under a linear mapping . 189 4.7.2 Transformation of pdfs under a one-to-one mapping . 191 4.7.3 Transformation of pdfs under a many-to-one mapping . 195 4.8 Correlation and covariance . 196 CONTENTS v 4.9 Minimum mean square error estimation . 205 4.9.1 Constant estimators . 205 4.9.2 Unconstrained estimators . 205 4.9.3 Linear estimators . 206 4.10 Law of large numbers and central limit theorem . 211 4.10.1 Law of large numbers . 212 4.10.2 Central limit theorem . 214 4.11 Joint Gaussian distribution . 217 4.11.1 From the standard 2-d normal to the general . 218 4.11.2 Key properties of the bivariate normal distribution . 219 4.11.3 Higher dimensional joint Gaussian distributions . 222 4.12 Short Answer Questions . 223 4.13 Problems . 225 5 Wrap-up 239 6 Appendix 241 6.1 Some notation . 241 6.2 Some sums . 242 6.3 Frequently used distributions . 242 6.3.1 Key discrete-type distributions . 242 6.3.2 Key continuous-type distributions . 243 6.4 Normal tables . 245 6.5 Answers to short answer questions . 247 6.6 Solutions to even numbered problems . 248 vi CONTENTS Preface A key objective of these notes is to convey how to deal with uncertainty in both qualitative and quantitative ways. Uncertainty is typically modeled as randomness. We must make decisions with partial information all the time in our daily lives, for instance when we decide what activities to pursue. Engineers deal with uncertainty in their work as well, often with precision and analysis. A challenge in applying reasoning to real world situations is to capture the main issues in a mathematical model. The notation that we use to frame a problem can be critical to understanding or solving the problem. There are often events, or variables, that need to be given names. Probability theory is widely used to model systems in engineering and scientific applications. These notes adopt the most widely used framework of probability, namely the one based on Kol- mogorov's axioms of probability. The idea is to assume a mathematically solid definition of the model. This structure encourages a modeler to have a consistent, if not completely accurate, model. It also offers a commonly used mathematical language for sharing models and calculations. Part of the process of learning to use the language of probability theory is learning classifications of problems into broad areas. For example, some problems involve finite numbers of possible alternatives, while others concern real-valued measurements. Many problems involve interaction of physically independent processes. Certain laws of nature or mathematics cause some probability distributions, such as the normal bell-shaped distribution often mentioned in popular literature, to frequently appear. Thus, there is an emphasis in these notes on well-known probability distributions and why each of them arises frequently in applications. These notes were written for the undergraduate course, ECE 313: Probability with Engineering Applications, offered by the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign. The official prerequisites of the course ensure that students have had calculus, including Taylor series expansions, integration over regions in the plane, the use of polar coordinates, and some basic linear algebra. The author gratefully acknowledges the students and faculty who have participated in this course through the years. He is particularly grateful to Professor D. V. Sarwate, who first intro- duced the course, and built up much material for it on the website. B. Hajek January 2017 vii CONTENTS 1 Organization Chapter 1 presents an overview of the many applications of probability theory, and then explains the basic concepts of a probability model and the axioms commonly assumed of probability models. Often probabilities are assigned to possible outcomes based on symmetry. For example, when a six sided die is rolled, it is usually assumed that the probability a particular number i shows is 1/6, for 1 ≤ i ≤ 6: For this reason, we also discuss in Chapter 1 how to determine the sizes of various finite sets of possible outcomes. Random variables are introduced in Chapter 2 and examined in the context of a finite, or countably infinite, set of possible outcomes. Notions of expectation (also known as mean), variance, hypothesis testing, parameter estimation, multiple random variables, and well known probability distributions{Poisson, geometric, and binomial, are covered. The Bernoulli process is considered{it provides a simple setting to discuss a long, even infinite, sequence of event times, and provides a tie between the binomial and geometric probability distributions. The focus shifts in Chapter 3 from discrete-type random variables to continuous-type random variables. The chapter takes advantage of many parallels and connections between discrete-type and continuous-type random variables. The most important well known continuous-type distribu- tions.
Load more

Recommended publications
  • Three Problems of the Theory of Choice on Random Sets
    DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 THREE PROBLEMS OF THE THEORY OF CHOICE ON RANDOM SETS B. A. Berezovskiy,* Yu. M. Baryshnikov, A. Gnedin V. Institute of Control Sciences Academy of Sciences of the U.S.S.R, Moscow *Guest of the California Institute of Technology SOCIAL SCIENCE WORKING PAPER 661 December 1987 THREE PROBLEMS OF THE THEORY OF CHOICE ON RANDOM SETS B. A. Berezovskiy, Yu. M. Baryshnikov, A. V. Gnedin Institute of Control Sciences Academy of Sciences of the U.S.S.R., Moscow ABSTRACT This paper discusses three problems which are united not only by the common topic of research stated in thetitle, but also by a somewhat surprising interlacing of the methods and techniques used. In the first problem, an attempt is made to resolve a very unpleasant metaproblem arising in general choice theory: why theconditions of rationality are not really necessary or, in other words, why in every-day life we are quite satisfied with choice methods which are far from being ideal. The answer, substantiated by a number of results, is as follows: situations in which the choice function "misbehaves" are very seldom met in large presentations. the second problem, an overview of our studies is given on the problem of statistical propertiesIn of choice. One of themost astonishing phenomenon found when we deviate from scalar­ extremal choice functions is in stable multiplicity of choice. If our presentation is random, then a random number of alternatives is chosen in it. But how many? The answer isn't trival, and may be sought in many differentdirections.
    [Show full text]
  • The Open Handbook of Formal Epistemology
    THEOPENHANDBOOKOFFORMALEPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. THEOPENHANDBOOKOFFORMAL EPISTEMOLOGY Richard Pettigrew &Jonathan Weisberg,Eds. Published open access by PhilPapers, 2019 All entries copyright © their respective authors and licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. LISTOFCONTRIBUTORS R. A. Briggs Stanford University Michael Caie University of Toronto Kenny Easwaran Texas A&M University Konstantin Genin University of Toronto Franz Huber University of Toronto Jason Konek University of Bristol Hanti Lin University of California, Davis Anna Mahtani London School of Economics Johanna Thoma London School of Economics Michael G. Titelbaum University of Wisconsin, Madison Sylvia Wenmackers Katholieke Universiteit Leuven iii For our teachers Overall, and ultimately, mathematical methods are necessary for philosophical progress. — Hannes Leitgeb There is no mathematical substitute for philosophy. — Saul Kripke PREFACE In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluc- tuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we ex- plore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way.
    [Show full text]
  • The Halász-Székely Barycenter
    THE HALÁSZ–SZÉKELY BARYCENTER JAIRO BOCHI, GODOFREDO IOMMI, AND MARIO PONCE Abstract. We introduce a notion of barycenter of a probability measure re- lated to the symmetric mean of a collection of nonnegative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We study analytic properties of this Halász–Székely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of nonnegative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a se- quence of dynamical observations converges to the Halász–Székely barycenter of the corresponding distribution. 1. Introduction Means have fascinated man for a long time. Ancient Greeks knew the arith- metic, geometric, and harmonic means of two positive numbers (which they may have learned from the Babylonians); they also studied other types of means that can be defined using proportions: see [He, pp. 85–89]. Newton and Maclaurin en- countered the symmetric means (more about them later). Huygens introduced the notion of expected value and Jacob Bernoulli proved the first rigorous version of the law of large numbers: see [Mai, pp. 51, 73]. Gauss and Lagrange exploited the connection between the arithmetico-geometric mean and elliptic functions: see [BB]. Kolmogorov and other authors considered means from an axiomatic point of view and determined when a mean is arithmetic under a change of coordinates (i.e. quasiarithmetic): see [HLP, p.
    [Show full text]
  • TOPIC 0 Introduction
    TOPIC 0 Introduction 1 Review Course: Math 535 (http://www.math.wisc.edu/~roch/mmids/) - Mathematical Methods in Data Science (MMiDS) Author: Sebastien Roch (http://www.math.wisc.edu/~roch/), Department of Mathematics, University of Wisconsin-Madison Updated: Sep 21, 2020 Copyright: © 2020 Sebastien Roch We first review a few basic concepts. 1.1 Vectors and norms For a vector ⎡ 푥 ⎤ ⎢ 1 ⎥ 푥 ⎢ 2 ⎥ 푑 퐱 = ⎢ ⎥ ∈ ℝ ⎢ ⋮ ⎥ ⎣ 푥푑 ⎦ the Euclidean norm of 퐱 is defined as ‾‾푑‾‾‾‾ ‖퐱‖ = 푥2 = √퐱‾‾푇‾퐱‾ = ‾⟨퐱‾‾, ‾퐱‾⟩ 2 ∑ 푖 √ ⎷푖=1 푇 where 퐱 denotes the transpose of 퐱 (seen as a single-column matrix) and 푑 ⟨퐮, 퐯⟩ = 푢 푣 ∑ 푖 푖 푖=1 is the inner product (https://en.wikipedia.org/wiki/Inner_product_space) of 퐮 and 퐯. This is also known as the 2- norm. More generally, for 푝 ≥ 1, the 푝-norm (https://en.wikipedia.org/wiki/Lp_space#The_p- norm_in_countably_infinite_dimensions_and_ℓ_p_spaces) of 퐱 is given by 푑 1/푝 ‖퐱‖ = |푥 |푝 . 푝 (∑ 푖 ) 푖=1 Here (https://commons.wikimedia.org/wiki/File:Lp_space_animation.gif#/media/File:Lp_space_animation.gif) is a nice visualization of the unit ball, that is, the set {퐱 : ‖푥‖푝 ≤ 1}, under varying 푝. There exist many more norms. Formally: 푑 푑 Definition (Norm): A norm is a function ℓ from ℝ to ℝ+ that satisfies for all 푎 ∈ ℝ, 퐮, 퐯 ∈ ℝ (Homogeneity): ℓ(푎퐮) = |푎|ℓ(퐮) (Triangle inequality): ℓ(퐮 + 퐯) ≤ ℓ(퐮) + ℓ(퐯) (Point-separating): ℓ(푢) = 0 implies 퐮 = 0. ⊲ The triangle inequality for the 2-norm follows (https://en.wikipedia.org/wiki/Cauchy– Schwarz_inequality#Analysis) from the Cauchy–Schwarz inequality (https://en.wikipedia.org/wiki/Cauchy– Schwarz_inequality).
    [Show full text]
  • Probabilities, Random Variables and Distributions A
    Probabilities, Random Variables and Distributions A Contents A.1 EventsandProbabilities................................ 318 A.1.1 Conditional Probabilities and Independence . ............. 318 A.1.2 Bayes’Theorem............................... 319 A.2 Random Variables . ................................. 319 A.2.1 Discrete Random Variables ......................... 319 A.2.2 Continuous Random Variables ....................... 320 A.2.3 TheChangeofVariablesFormula...................... 321 A.2.4 MultivariateNormalDistributions..................... 323 A.3 Expectation,VarianceandCovariance........................ 324 A.3.1 Expectation................................. 324 A.3.2 Variance................................... 325 A.3.3 Moments................................... 325 A.3.4 Conditional Expectation and Variance ................... 325 A.3.5 Covariance.................................. 326 A.3.6 Correlation.................................. 327 A.3.7 Jensen’sInequality............................. 328 A.3.8 Kullback–LeiblerDiscrepancyandInformationInequality......... 329 A.4 Convergence of Random Variables . 329 A.4.1 Modes of Convergence . 329 A.4.2 Continuous Mapping and Slutsky’s Theorem . 330 A.4.3 LawofLargeNumbers........................... 330 A.4.4 CentralLimitTheorem........................... 331 A.4.5 DeltaMethod................................ 331 A.5 ProbabilityDistributions............................... 332 A.5.1 UnivariateDiscreteDistributions...................... 333 A.5.2 Univariate Continuous Distributions . 335
    [Show full text]
  • Continuous Probability Distributions Continuous Probability Distributions – a Guide for Teachers (Years 11–12)
    1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Probability and statistics: Module 21 Continuous probability distributions Continuous probability distributions – A guide for teachers (Years 11–12) Professor Ian Gordon, University of Melbourne Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Continuous random variables: basic ideas ....................... 5 Cumulative distribution functions ............................ 6 Probability density functions ..............................
    [Show full text]
  • (Introduction to Probability at an Advanced Level) - All Lecture Notes
    Fall 2018 Statistics 201A (Introduction to Probability at an advanced level) - All Lecture Notes Aditya Guntuboyina August 15, 2020 Contents 0.1 Sample spaces, Events, Probability.................................5 0.2 Conditional Probability and Independence.............................6 0.3 Random Variables..........................................7 1 Random Variables, Expectation and Variance8 1.1 Expectations of Random Variables.................................9 1.2 Variance................................................ 10 2 Independence of Random Variables 11 3 Common Distributions 11 3.1 Ber(p) Distribution......................................... 11 3.2 Bin(n; p) Distribution........................................ 11 3.3 Poisson Distribution......................................... 12 4 Covariance, Correlation and Regression 14 5 Correlation and Regression 16 6 Back to Common Distributions 16 6.1 Geometric Distribution........................................ 16 6.2 Negative Binomial Distribution................................... 17 7 Continuous Distributions 17 7.1 Normal or Gaussian Distribution.................................. 17 1 7.2 Uniform Distribution......................................... 18 7.3 The Exponential Density...................................... 18 7.4 The Gamma Density......................................... 18 8 Variable Transformations 19 9 Distribution Functions and the Quantile Transform 20 10 Joint Densities 22 11 Joint Densities under Transformations 23 11.1 Detour to Convolutions......................................
    [Show full text]
  • Chapter 2: Probability
    16 Chapter 2: Probability The aim of this chapter is to revise the basic rules of probability. By the end of this chapter, you should be comfortable with: • conditional probability, and what you can and can’t do with conditional expressions; • the Partition Theorem and Bayes’ Theorem; • First-Step Analysis for finding the probability that a process reaches some state, by conditioning on the outcome of the first step; • calculating probabilities for continuous and discrete random variables. 2.1 Sample spaces and events Definition: A sample space, Ω, is a set of possible outcomes of a random experiment. Definition: An event, A, is a subset of the sample space. This means that event A is simply a collection of outcomes. Example: Random experiment: Pick a person in this class at random. Sample space: Ω= {all people in class} Event A: A = {all males in class}. Definition: Event A occurs if the outcome of the random experiment is a member of the set A. In the example above, event A occurs if the person we pick is male. 17 2.2 Probability Reference List The following properties hold for all events A, B. • P(∅)=0. • 0 ≤ P(A) ≤ 1. • Complement: P(A)=1 − P(A). • Probability of a union: P(A ∪ B)= P(A)+ P(B) − P(A ∩ B). For three events A, B, C: P(A∪B∪C)= P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C) . If A and B are mutually exclusive, then P(A ∪ B)= P(A)+ P(B).
    [Show full text]
  • HARMONIC ANALYSIS 1. Maximal Function for a Locally Integrable
    HARMONIC ANALYSIS PIOTR HAJLASZ 1. Maximal function 1 n For a locally integrable function f 2 Lloc(R ) the Hardy-Littlewood max- imal function is defined by Z n Mf(x) = sup jf(y)j dy; x 2 R : r>0 B(x;r) The operator M is not linear but it is subadditive. We say that an operator T from a space of measurable functions into a space of measurable functions is subadditive if jT (f1 + f2)(x)j ≤ jT f1(x)j + jT f2(x)j a.e. and jT (kf)(x)j = jkjjT f(x)j for k 2 C. The following integrability result, known also as the maximal theorem, plays a fundamental role in many areas of mathematical analysis. p n Theorem 1.1 (Hardy-Littlewood-Wiener). If f 2 L (R ), 1 ≤ p ≤ 1, then Mf < 1 a.e. Moreover 1 n (a) For f 2 L (R ) 5n Z (1.1) jfx : Mf(x) > tgj ≤ jfj for all t > 0. t Rn p n p n (b) If f 2 L (R ), 1 < p ≤ 1, then Mf 2 L (R ) and p 1=p kMfk ≤ 2 · 5n=p kfk for 1 < p < 1, p p − 1 p kMfk1 ≤ kfk1 : Date: March 28, 2012. 1 2 PIOTR HAJLASZ The estimate (1.1) is called weak type estimate. 1 n 1 n Note that if f 2 L (R ) is a nonzero function, then Mf 62 L (R ). Indeed, R if λ = B(0;R) jfj > 0, then for jxj > R we have Z λ Mf(x) ≥ jfj ≥ n ; B(x;R+jxj) !n(R + jxj) n and the function on the right hand side is not integrable on R .
    [Show full text]
  • Probability (Graduate Class) Lecture Notes
    Probability (graduate class) Lecture Notes Tomasz Tkocz∗ These lecture notes were written for the graduate course 21-721 Probability that I taught at Carnegie Mellon University in Spring 2020. ∗Carnegie Mellon University; [email protected] 1 Contents 1 Probability space 6 1.1 Definitions . .6 1.2 Basic examples . .7 1.3 Conditioning . 11 1.4 Exercises . 13 2 Random variables 14 2.1 Definitions and basic properties . 14 2.2 π λ systems . 16 − 2.3 Properties of distribution functions . 16 2.4 Examples: discrete and continuous random variables . 18 2.5 Exercises . 21 3 Independence 22 3.1 Definitions . 22 3.2 Product measures and independent random variables . 24 3.3 Examples . 25 3.4 Borel-Cantelli lemmas . 28 3.5 Tail events and Kolmogorov's 0 1law .................. 29 − 3.6 Exercises . 31 4 Expectation 33 4.1 Definitions and basic properties . 33 4.2 Variance and covariance . 35 4.3 Independence again, via product measures . 36 4.4 Exercises . 39 5 More on random variables 41 5.1 Important distributions . 41 5.2 Gaussian vectors . 45 5.3 Sums of independent random variables . 46 5.4 Density . 47 5.5 Exercises . 49 6 Important inequalities and notions of convergence 52 6.1 Basic probabilistic inequalities . 52 6.2 Lp-spaces . 54 6.3 Notions of convergence . 59 6.4 Exercises . 63 2 7 Laws of large numbers 67 7.1 Weak law of large numbers . 67 7.2 Strong law of large numbers . 73 7.3 Exercises . 78 8 Weak convergence 81 8.1 Definition and equivalences .
    [Show full text]
  • Probabilistic Models for Shapes As Continuous Curves
    J Math Imaging Vis (2009) 33: 39–65 DOI 10.1007/s10851-008-0104-3 Probabilistic Models for Shapes as Continuous Curves Jeong-Gyoo Kim · J. Alison Noble · J. Michael Brady Published online: 23 July 2008 © Springer Science+Business Media, LLC 2008 Abstract We develop new shape models by defining a stan- 1 Introduction dard shape from which we can explain shape deformation and variability. Currently, planar shapes are modelled using Our work contributes to shape analysis, particularly in med- a function space, which is applied to data extracted from ical image analysis, by defining a standard representation images. We regard a shape as a continuous curve and identi- of shape. Specifically we develop new mathematical mod- fied on the Wiener measure space whereas previous methods els of planar shapes. Our aim is to develop a standard rep- have primarily used sparse sets of landmarks expressed in a resentation of anatomical or biological shapes that can be Euclidean space. The average of a sample set of shapes is used to explain shape variations, whether in normal subjects defined using measurable functions which treat the Wiener or abnormalities due, for example, to disease. In addition, measure as varying Gaussians. Various types of invariance we propose a quasi-score that measures global deformation of our formulation of an average are examined in regard to and provides a generic way to compare statistical methods practical applications of it. The average is examined with of shape analysis. relation to a Fréchet mean in order to establish its valid- As D’Arcy Thompson pointed in [50], there is an impor- ity.
    [Show full text]
  • Lecture 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 Probabilities Law of Total Probability Bayes' Theorem Examples I False Positive Paradox I Monty Hall Problem Summary Konstantin Zuev (USC) Math 408, Lecture 2 January 18, 2013 2 / 9 Motivation and Definition Recall (see Lecture 1) that the sample space is the set of all possible outcomes of an experiment. Suppose we are interested only in part of the sample space, the part where we know some event { call it A { has happened, and we want to know how likely it is that various other events (B; C; D :::) have also happened. What we want is the conditional probability of B given A. Definition If P(A) > 0, then the conditional probability of B given A is P(AB) P(BjA) = P(A) Useful Interpretation: Think of P(BjA) as the fraction of times B occurs among those in which A occurs Konstantin Zuev (USC) Math 408, Lecture 2 January 18, 2013 3 / 9 Properties of Conditional Probabilities Here are some facts about conditional probabilities: 1 For any fixed A such that P(A) > 0, P(·|A) is a probability, i.e. it satisfies the rules of probability: I 0 ≤ P(BjA) ≤ 1 I P(ΩjA) = 1 I P(;jA) = 0 ¯ I P(BjA) + P(BjA) = 1 I P(B + CjA) = P(BjA) + P(CjA) − P(BCjA) 2 Important: The rules of probability apply to events on the left of the bar.
    [Show full text]