Ex. 2.1 (Davide Basilio Bartolini)
|
|
- Paul Green
- 8 years ago
- Views:
Transcription
1 ECE 54: Elements of Information Theory, Fall 00 Homework Solutions Ex.. (Davide Basilio Bartolini) Text Coin Flips. A fair coin is flipped until the first head occurs. Let X denote the number of flips required. (a) Find the Entropy H(X) in bits (b) A random variable X is drawn according to this distribution. Find an efficient sequence of yes-no questions of the form, Is X contained in the set S?. Compare H(X) to the expected number of questions required to determine X. Solution (a) The random variable X is on the domain X = {,,,...} and it denotes the number of flips needed to get the first head, i.e. + the number of consecutive tails appeared before the first head. Since the coin is said to be fair, we have p( head ) = p( tail ) = and hence (exploiting the independence of the coin flips): p(x = ) = p( head ) = p(x = ) = p( tail ) p( head ) = ( =. ntimes { }} { p(x = n) = p( tail )... p( tail ) p( head ) =... ( = ) ) n from this, it is clear that the probability mass distribution of X is: p X (x) = ( ) x
2 Once the distribution is known, H(X) can be computed from the definition: H(X) = p X (x) log p X (x) x X ( ) x ( ) x = log x= ( ) x ( ) x = log (since the summed expr. equals 0 for x = 0) ( ) x ( ) = x log (property of logarithms) ( ) ( ) x = log x ( ) ( ) x = x = ( ) = [bit] exploiting (k) x k x = ( k) (b) Since the most likely value for X is (p(x = ) = ), the most efficient first question is: Is X =? ; the next question will be Is X =? and so on, until a positive answer is found. If this strategy is used, the random variable Y representing the number of questions will have the same distribution as X and it will be: E [Y ] = y y=0 ( ) y = ( ) = which is exactly equal to the entropy of X. An interpretation of this fact could be that bits (which is the entropy value for X) are the amount of memory required to store the outcomes of the two binary questions which are enough (on average) to get a positive answer on the value of X. Exercise.4 (Matteo Carminati) Entropy of functions of a random variable. Let X be a discrete random variable. Show that the entropy of a function of X is less than or equal to the entropy of X by justifying the following steps: H(X, g(x)) (a) = H(X) + H(g(X) X) (b) = H(X) () H(X, g(x)) (c) = H(g(X)) + H(X g(x)) (d) H(g(X)) ()
3 Thus, H(g(X)) H(X). Solution (a) It comes from entropy s chain rule applied to random variables X and g(x), i.e. H(X, Y ) = H(X) + H(Y X), so H(X, g(x)) = H(X) + H(g(X) X). (b) Intuitively, if g(x) depends only on X and if the value of X is known, g(x) is completely specified and it has a deterministic value. The entropy of a deterministic value is 0, so H(g(X) X) = 0 and H(X) + H(g(X) X) = H(X). (c) Again, this formula comes from the entropy s chain rule, in the form: H(X, Y ) = H(Y ) + H(X Y ). (d) Proving that H(g(X)) + H(X g(x)) H(g(X)) means proving that H(X g(x)) 0: the non-negativity is one of the property of entropy and can be proved from its definition by noting that the logarithm of a probability (a quantity always less than or equal to ) is non-positive. In particular H(X g(x)) = 0 if the knowledge of the value of g(x) allows to totally specify the value of X; otherwise H(X g(x)) > 0 (for example if g(x) is an injective function). Ex..7(a) (Davide Basilio Bartolini) Text Coin weighing. Suppose that one has n coins, among which there may or may not be one counterfeit coin. If there is a counterfeit coin, it may be either heavier or lighter than the other coins. The coins are to be weighed by a balance. Find an upper bound on the number of coins n so that k weighings will find the counterfeit coin (if any) and correctly declare it to be heavier or lighter. Solution Let X be a string of n characters on the alphabet X = {, 0, } n, each of which represents one coin. Each of the characters of X may have three different values (say if the coin is heavier than a normal one, 0 if it is regular, if it is lighter). Since only one of the coins may be counterfeit, X may be a string of all 0 (if all the coins are regular) or may present either a or a at only one position. Thus, the possible configurations for X are n +. Under the hypothesis of a uniform distribution of the probability of which coin is counterfeit, the entropy of X will be: H(X) = log (n + )
4 Now let Z = [Z, Z,..., Z k ] be a random variable representing the weighings; each of the Z i will have three possible values to indicate whether the result of the weighing is balanced, left arm heavier or right arm heavier. The entropy of each Z i will be upper-bounded by the three possible values it can assume: H(Z i ) log, i [, k] and for Z (under the hypothesis of independence of the weighings): H(Z) = H(Z, Z,..., Z k ) (ChainR.) = (Indep.) = k H(Z i ) log i= k H(Z i Z i,..., Z ) i= Since we want to know how many weghings will yield the same amount of information which is given by the configuration of X (i.e. we want to know how many weighings will be needed to find out which coin - if any - is counterfeit), we can write: H(X) = H(Z) log log (n + ) log n + k n k, which is the wanted upper bound. Ex.. (Kenneth Palacio) X Y 0 0 / / 0 / Table : p(x,y) for problem.. Find: (a) H(X), H(Y ). (b) H(X Y ), H(Y X). (c) H(X, Y ). (d) H(Y ) H(Y X). (e) I(X; Y ). (f) Draw a Venn diagram for the quantities in parts (a) through (e). Solution: 4
5 Compute of marginal distributions: p(x) = [, ] p(y) = [, ] (a) H(X), H(Y ). H(X) = log ( ) log ( ) H(X) = 0.98bits (4) () H(Y ) = log ( ) log ( ) H(Y ) = 0.98bits (6) (5) Figure : H(X), H(Y) (b) H(X Y ), H(Y X). H(X Y ) = p(y = i)h(x Y = y) (7) i=0 H(X Y ) = H(X Y = 0) + H(X Y = ) (8) H(X Y ) = H(, 0) + H(/, /) (9) H(X Y ) = / (0) 5
6 H(Y X) = X p(x = i)h(y X = x) () i=0 H(Y X) = H(Y X = 0) + H(Y X = ) H(Y X) = H(/, /) + H(0, ) H(Y X) = / () () (4) Figure : H(X Y ), H(Y X) (c) H(X, Y ). H(X, Y ) =, X p(x, y) log p(x, y) (5),y=0 H(X, Y ) = log H(X, Y ) = bits Figure : H(X,Y) 6 (6) (7)
7 (d) H(Y ) H(Y X). H(Y ) H(Y X) = 0.98 / (8) H(Y ) H(Y X) = 0.54 (9) Figure 4: H(Y ) H(Y X) (e) I(X; Y ). p(x, y) I(X; Y ) = p(x, y) log p(x)p(y) x,y / / / I(X; Y ) = log + log + log (/)(/) (/)(/) (/)(/) I(X; Y ) = X (0) () () Figure 5: I(X;Y) (f) Venn diagram is already shown for each item. Ex..0 (Kenneth Palacio) Run-length coding. Let X,X,..., Xn be (possibly dependent) binary random variables. Suppose that one calculates the run lengths R = (R, R,...) of this sequence (in order as they occur). For example, the sequence X = yields run lengths R = (,,,, ). Compare 7
8 H(X, X,..., X n ), H(R), and H(Xn, R). Show all equalities and inequalities, and bound all the differences. Solution: Lets assume that one random variable Xj (0 < j n) is known, then if R is also known, H(Xj,R) will provide the same information about uncertainty than H(X,X,.. Xj,..,Xn), since the whole sequence of X can be completely recovered from the knowledge of Xj and R. For example, with X5 = and the run lengths R = (,,,, ) it s possible to recover the original sequence as follows: X5 =, R = (,,,, ) leads to recover the sequence: X = It can be concluded that: H(Xj, R) = H(X, X,...Xn) Using the chain rule, H(Xj,R) can be written as: H(Xj, R) = H(R) + H(Xj R) H(Xj R) H(Xj), since conditioning reduces entropy. Then it s possible to write: H(Xj) H(X, X,...Xn) H(R) H(Xj) + H(R) H(X, X,...Xn) Computing H(Xj) = n x p(xj) log Xj, where the distribution of Xj is unknown, it can be assumed to be: a probability of p for Xj=0 and of (-p) for Xj=. It can be observed that the maximum entropy is given when p=/ leading max H(Xj)=. Then: + H(R) H(X, X,...Xn) Considering the results obtained in problem.4, we can write also that: H(R) H(X). Because R is a function of X. 8
An Introduction to Information Theory
An Introduction to Information Theory Carlton Downey November 12, 2013 INTRODUCTION Today s recitation will be an introduction to Information Theory Information theory studies the quantification of Information
More informationLECTURE 4. Last time: Lecture outline
LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding
More informationQuestion: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationRandom variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.
Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()
More informationNational Sun Yat-Sen University CSE Course: Information Theory. Gambling And Entropy
Gambling And Entropy 1 Outline There is a strong relationship between the growth rate of investment in a horse race and the entropy of the horse race. The value of side information is related to the mutual
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,
More informationGambling and Data Compression
Gambling and Data Compression Gambling. Horse Race Definition The wealth relative S(X) = b(x)o(x) is the factor by which the gambler s wealth grows if horse X wins the race, where b(x) is the fraction
More informationThe sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].
Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationBasics of information theory and information complexity
Basics of information theory and information complexity a tutorial Mark Braverman Princeton University June 1, 2013 1 Part I: Information theory Information theory, in its modern format was introduced
More informationLecture 4: AC 0 lower bounds and pseudorandomness
Lecture 4: AC 0 lower bounds and pseudorandomness Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: Jason Perry and Brian Garnett In this lecture,
More information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
More informationCHAPTER 2 Estimating Probabilities
CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a
More informationThe Basics of Graphical Models
The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationSection 6.1 Joint Distribution Functions
Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function
More informationIntroduction to Learning & Decision Trees
Artificial Intelligence: Representation and Problem Solving 5-38 April 0, 2007 Introduction to Learning & Decision Trees Learning and Decision Trees to learning What is learning? - more than just memorizing
More informationLecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationProbability Generating Functions
page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence
More informationDiscrete Math in Computer Science Homework 7 Solutions (Max Points: 80)
Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you
More informationStatistics 100A Homework 3 Solutions
Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win $ for each black ball selected and we
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationMeasuring Intrusion Detection Capability: An Information-Theoretic Approach
Measuring Intrusion Detection Capability: An Information-Theoretic Approach Guofei Gu, Prahlad Fogla, David Dagon, Boris Škorić Wenke Lee Philips Research Laboratories, Netherlands Georgia Institute of
More informationNotes from Week 1: Algorithms for sequential prediction
CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 22-26 Jan 2007 1 Introduction In this course we will be looking
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationInfluences in low-degree polynomials
Influences in low-degree polynomials Artūrs Bačkurs December 12, 2012 1 Introduction In 3] it is conjectured that every bounded real polynomial has a highly influential variable The conjecture is known
More informationInformation Theoretic Analysis of Proactive Routing Overhead in Mobile Ad Hoc Networks
Information Theoretic Analysis of Proactive Routing Overhead in obile Ad Hoc Networks Nianjun Zhou and Alhussein A. Abouzeid 1 Abstract This paper considers basic bounds on the overhead of link-state protocols
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationMAS108 Probability I
1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper
More informationLecture 4: BK inequality 27th August and 6th September, 2007
CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationChapter Objectives. Chapter 9. Sequential Search. Search Algorithms. Search Algorithms. Binary Search
Chapter Objectives Chapter 9 Search Algorithms Data Structures Using C++ 1 Learn the various search algorithms Explore how to implement the sequential and binary search algorithms Discover how the sequential
More informationProbabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I
Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with
More informationST 371 (IV): Discrete Random Variables
ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible
More information2WB05 Simulation Lecture 8: Generating random variables
2WB05 Simulation Lecture 8: Generating random variables Marko Boon http://www.win.tue.nl/courses/2wb05 January 7, 2013 Outline 2/36 1. How do we generate random variables? 2. Fitting distributions Generating
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More information2. Discrete random variables
2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be
More informationWhat is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference
0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures
More informationFor a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )
Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll
More informationCompression techniques
Compression techniques David Bařina February 22, 2013 David Bařina Compression techniques February 22, 2013 1 / 37 Contents 1 Terminology 2 Simple techniques 3 Entropy coding 4 Dictionary methods 5 Conclusion
More informationLikelihood, MLE & EM for Gaussian Mixture Clustering. Nick Duffield Texas A&M University
Likelihood, MLE & EM for Gaussian Mixture Clustering Nick Duffield Texas A&M University Probability vs. Likelihood Probability: predict unknown outcomes based on known parameters: P(x θ) Likelihood: eskmate
More informationMath 431 An Introduction to Probability. Final Exam Solutions
Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationJoint Exam 1/P Sample Exam 1
Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question
More informationSolution for Homework 2
Solution for Homework 2 Problem 1 a. What is the minimum number of bits that are required to uniquely represent the characters of English alphabet? (Consider upper case characters alone) The number of
More informationMath 461 Fall 2006 Test 2 Solutions
Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two
More informationLecture 1: Course overview, circuits, and formulas
Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.
More informationA New Interpretation of Information Rate
A New Interpretation of Information Rate reproduced with permission of AT&T By J. L. Kelly, jr. (Manuscript received March 2, 956) If the input symbols to a communication channel represent the outcomes
More informationMicroeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
More informationPennies and Blood. Mike Bomar
Pennies and Blood Mike Bomar In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics.
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationReading.. IMAGE COMPRESSION- I IMAGE COMPRESSION. Image compression. Data Redundancy. Lossy vs Lossless Compression. Chapter 8.
Reading.. IMAGE COMPRESSION- I Week VIII Feb 25 Chapter 8 Sections 8.1, 8.2 8.3 (selected topics) 8.4 (Huffman, run-length, loss-less predictive) 8.5 (lossy predictive, transform coding basics) 8.6 Image
More informationPrinciple of Data Reduction
Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then
More informationStatistical Testing of Randomness Masaryk University in Brno Faculty of Informatics
Statistical Testing of Randomness Masaryk University in Brno Faculty of Informatics Jan Krhovják Basic Idea Behind the Statistical Tests Generated random sequences properties as sample drawn from uniform/rectangular
More information1 Approximating Set Cover
CS 05: Algorithms (Grad) Feb 2-24, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the set-covering problem consists of a finite set X and a family F of subset of X, such that every elemennt
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
More informationExact Confidence Intervals
Math 541: Statistical Theory II Instructor: Songfeng Zheng Exact Confidence Intervals Confidence intervals provide an alternative to using an estimator ˆθ when we wish to estimate an unknown parameter
More informationIntroduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 4.4 Homework
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 4.4 Homework 4.65 You buy a hot stock for $1000. The stock either gains 30% or loses 25% each day, each with probability.
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Spring 2012 Homework # 9, due Wednesday, April 11 8.1.5 How many ways are there to pay a bill of 17 pesos using a currency with coins of values of 1 peso, 2 pesos,
More informationTHE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok
THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan
More informationMATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...
MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................
More informationOn Directed Information and Gambling
On Directed Information and Gambling Haim H. Permuter Stanford University Stanford, CA, USA haim@stanford.edu Young-Han Kim University of California, San Diego La Jolla, CA, USA yhk@ucsd.edu Tsachy Weissman
More informatione.g. arrival of a customer to a service station or breakdown of a component in some system.
Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be
More informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover
More informationRegular Languages and Finite State Machines
Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries - some review One application formal definition of finite automata Examples 1 Sets A set is an unordered collection
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More informationMULTIVARIATE PROBABILITY DISTRIBUTIONS
MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationMIMO CHANNEL CAPACITY
MIMO CHANNEL CAPACITY Ochi Laboratory Nguyen Dang Khoa (D1) 1 Contents Introduction Review of information theory Fixed MIMO channel Fading MIMO channel Summary and Conclusions 2 1. Introduction The use
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationInverse Functions and Logarithms
Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a one-to-one function if it never takes on the same value twice; that
More informationFinal Mathematics 5010, Section 1, Fall 2004 Instructor: D.A. Levin
Final Mathematics 51, Section 1, Fall 24 Instructor: D.A. Levin Name YOU MUST SHOW YOUR WORK TO RECEIVE CREDIT. A CORRECT ANSWER WITHOUT SHOWING YOUR REASONING WILL NOT RECEIVE CREDIT. Problem Points Possible
More informationDepartment of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
More informationExact Nonparametric Tests for Comparing Means - A Personal Summary
Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationTail inequalities for order statistics of log-concave vectors and applications
Tail inequalities for order statistics of log-concave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.Tomczak-Jaegermann Banff, May 2011 Basic
More information6.4 Logarithmic Equations and Inequalities
6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.
More informationThe Math. P (x) = 5! = 1 2 3 4 5 = 120.
The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct
More informationTowards a Tight Finite Key Analysis for BB84
The Uncertainty Relation for Smooth Entropies joint work with Charles Ci Wen Lim, Nicolas Gisin and Renato Renner Institute for Theoretical Physics, ETH Zurich Group of Applied Physics, University of Geneva
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationSMT 2014 Algebra Test Solutions February 15, 2014
1. Alice and Bob are painting a house. If Alice and Bob do not take any breaks, they will finish painting the house in 20 hours. If, however, Bob stops painting once the house is half-finished, then the
More informationTwo-Stage Stochastic Linear Programs
Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationPropagation of Errors Basic Rules
Propagation of Errors Basic Rules See Chapter 3 in Taylor, An Introduction to Error Analysis. 1. If x and y have independent random errors δx and δy, then the error in z = x + y is δz = δx 2 + δy 2. 2.
More informationNonparametric adaptive age replacement with a one-cycle criterion
Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More informationMathematical Induction. Lecture 10-11
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationCHAPTER 6. Shannon entropy
CHAPTER 6 Shannon entropy This chapter is a digression in information theory. This is a fascinating subject, which arose once the notion of information got precise and quantifyable. From a physical point
More information