DOCSLIB.ORG

Final Exam Review

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Final Exam Review 10-606 Mathematical Foundations for Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University Final Exam Review Matt Gormley Lecture 13 Oct. 15, 2018 1 Reminders • Homework 4: Probability – Out: Thu, Oct. 11 – Due: Mon, Oct. 15 at 11:59pm • Final Exam – Date: Wed, Oct. 17 – Time: 6:30 – 9:30pm – Location: Posner Hall A35 3 EXAM LOGISTICS 4 Final Exam • Time / Location – Date: Wed, Oct 17 – Time: Evening Exam, 6:30pm – 9:30pm – Room: Posner Hall A35 – Seats: There will be assigned seats. Please arrive early. • Logistics – Format of questions: • Multiple choice • True / False (with justification) • Short answers • Interpreting figures • Derivations • Short proofs – No electronic devices – You are allowed to bring one 8½ x 11 sheet of notes (front and back) 5 Final Exam • How to Prepare – Attend this final exam review session – Review prior year’s exams and solutions • We already posted these (see Piazza) • Disclaimer: This year’s 10-606/607 is not the same as prior offerings! – Review this year’s homework problems – Review this year’s quiz problems 6 Final Exam • Advice (for during the exam) – Solve the easy problems first (e.g. multiple choice before derivations) • if a problem seems extremely complicated you’re likely missing something – Don’t leave any answer blank! – If you make an assumption, write it down – If you look at a question and don’t know the answer: • we probably haven’t told you the answer • but we’ve told you enough to work it out • imagine arguing for some answer and see if you like it 7 Topics Covered • Preliminaries • Matrix Calculus – Sets – Scalar derivatives – Types – Partial derivatives – Functions – Vector derivatives • Linear Algebra – Matrix derivatives – Vector spaces – Method of Lagrange multipliers – Matrices and linear operators – Least squares derivation – Linear independence • Probability – Invertability – Events – Eigenvalues and eigenvectors – Disjoint union – Linear equations – Sum rule – Factorizations – Discrete random variables – Matrix Memories – Continuous random variables – Bayes Rule – Conditional, marginal, joint probabilities – Mean and variance 9 Analysis of 10601 Performance No obvious correlations… 10 Analysis of 10601 Performance Correlation between Background Test and Midterm Exam: • Pearson: 0.46 (moderate) • Spearman: 0.43 (moderate) 11 Q&A 12 Agenda 1. Review of probability (didactic) 2. Review of linear algebra / matrix calculus (through application) 20 21 Oh, the Places You’ll Use Probability! Supervised Classification • Naïve Bayes 1 n p(y x ,x ,...,x )= p(y) p(x y) | 1 2 n Z i| i=1 • Logistic regression P (Y = y X = x; θ)=p(y x; θ) | | 2tT(θ 7(x)) = y · 2tT(θy 7(x) y · 22 Oh, the Places You’ll Use Probability! ML TheoryPAC/SLT models for Supervised Learning • (Example:Algo sees Sample training sample Complexity) S: (x1,c*(x1)),…, (xm,c*(xm)), xi i.i.d. from D • Does optimization over S, find hypothesis ℎ ∈ �. • Goal: h has small error over D. ∗ True error: ���� ℎ = Pr (ℎ � ≠ � (�)) �~ � How often ℎ � ≠ �∗(�) over future instances drawn at random from D • But, can only measure: 1 ∗ Training error: ��� ℎ = � ℎ � ≠ � � � � � � � How often ℎ � ≠ �∗(�) over training instances Sample complexity: bound ���� ℎ in terms of ���� ℎ 23 Oh, the Places You’ll Use Probability! Deep Learning (Example: Deep Bi-directional RNN) y1 y2 y3 y4 h1 h2 h3 h4 h1 h2 h3 h4 x1 x2 x3 x4 24 Oh, the Places You’ll Use Probability! Graphical Models • Hidden Markov Model (HMM) <START> n v p d n time flies like an arrow • Conditional Random Field (CRF) <START> ψ0 n ψ2 v ψ4 p ψ6 d ψ8 n ψ1 ψ3 ψ5 ψ7 ψ9 25 Probability Outline • Probability Theory – Sample space, Outcomes, Events – Complement – Disjoint union – Kolmogorov’s Axioms of Probability – Sum rule • Random Variables – Random variables, Probability mass function (pmf), Probability density function (pdf), Cumulative distribution function (cdf) – Examples – Notation – Expectation and Variance – Joint, conditional, marginal probabilities – Independence – Bayes’ Rule • Common Probability Distributions – Beta, Dirichlet, etc. 26 PROBABILITY AND EVENTS 27 Probability of Events Example 1: Flipping a coin Sample Space Ω {Heads, Tails} Outcome ω Ω Example: Heads ∈ Event E Ω Example: {Heads} ⊆ Probability P (E) P({Heads}) = 0.5 P({Tails}) = 0.5 28 Probability Theory: Definitions Probability provides a science for inference about interesting events Sample Space Ω The set of all possible outcomes Outcome ω Ω Possible result of an experiment ∈ Event E Ω Any subset of the sample space Probability ⊆ The non-negative number assigned P (E) to each event in the sample space • Each outcome is unique • Only one outcome can occur per experiment • An outcome can be in multiple events • An elementary event consists of exactly one outcome • A compound event consists of multiple outcomes 29 Probability of Events Example 2: Rolling a 6-sided die Sample Space Ω {1,2,3,4,5,6} Outcome ω Ω Example: 3 ∈ Event E Ω Example: {3} ⊆ (the event “the die came up 3”) Probability P (E) P({3}) = 1/6 P({4}) = 1/6 30 Probability of Events Example 2: Rolling a 6-sided die Sample Space Ω {1,2,3,4,5,6} Outcome ω Ω Example: 3 ∈ Event E Ω Example: {2,4,6} ⊆ (the event “the roll was even”) Probability P (E) P({2,4,6}) = 0.5 P({1,3,5}) = 0.5 31 Probability of Events Example 3: Timing how long it takes a monkey to reproduce Shakespeare Sample Space Ω [0, +∞) Outcome ω Ω Example: 1,433,600 hours ∈ Event E Ω Example: [1, 6] hours Probability ⊆ P([1,6]) = 0.000000000001 P (E) P([1,433,600, +∞)) = 0.99 32 Probability Theory: Definitions • The complement of an event E, denoted ~E, is the event that E does not occur. • P(E) + P(~E) = 1 • All of the following notations equivalently denote the complement of event E Ω E ~E 33 Disjoint Union • Two events A and B are disjoint if Ω A • The disjoint union rule says that if events A B and B are disjoint, then 34 Disjoint Union • The disjoint union rule can be extended to Ω multiple disjoint events • If each pair of events Ai A and Aj are disjoint, B then 35 Non-disjoint Union • Two events A and B are non-disjoint if Ω • We can apply the A disjoint union rule to various disjoint sets: B 36 Kolmogorov’s Axioms 1. P (E) 0, for all events E ≥ 2. P (Ω)=1 3. If E1,E2,...are disjoint, then P (E1 or E2 or ...)=P (E1)+P (E2)+... 37 Kolmogorov’s Axioms All of 1. P (E) 0, for all events E probability can ≥ 2. P (Ω)=1 be derived 3. If E1,E2,...are disjoint, then from just ∞ ∞ these! P Ei = P (Ei) i=1 i=1 In words: 1. Each event has non-negative probability. 2. The probability that some event will occur is one. 3. The probability of the union of many disjoint sets is the sum of their probabilities 38 Sum Rule • For any two events A and B, we have that Ω A B 39 RANDOM VARIABLES 41 Random Variables: Definitions Random Def 1: Variable whose possible values Variable X are the outcomes of a random (capital experiment letters) Value of a x The value taken by a random variable Random (lowercase letters) Variable 42 Random Variables: Definitions Random Def 1: Variable whose possible values Variable X are the outcomes of a random experiment Discrete Random variable whose values come Random X from a countable set (e.g. the natural Variable numbers or {True, False}) Continuous Random variable whose values come Random X from an interval or collection of Variable intervals (e.g. the real numbers or the range (3, 5)) 43 Random Variables: Definitions Random Def 1: Variable whose possible values Variable X are the outcomes of a random experiment Def 2: A measureable function from the sample space to the real numbers: X : Ω E Discrete Random variable whose→ values come Random X from a countable set (e.g. the natural Variable numbers or {True, False}) Continuous Random variable whose values come Random X from an interval or collection of Variable intervals (e.g. the real numbers or the range (3, 5)) 44 Random Variables: Definitions Discrete Random variable whose values come Random X from a countable set (e.g. the natural Variable numbers or {True, False}) Probability p(x) Function giving the probability that mass discrete r.v. X takes value x. function (pmf) p(x):=P (X = x) 45 Random Variables: Definitions Example 2: Rolling a 6-sided die Sample Space Ω {1,2,3,4,5,6} Outcome ω Ω Example: 3 ∈ Event E Ω Example: {3} ⊆ (the event “the die came up 3”) Probability P (E) P({3}) = 1/6 P({4}) = 1/6 46 Random Variables: Definitions Example 2: Rolling a 6-sided die Sample Space Ω {1,2,3,4,5,6} Outcome ω Ω Example: 3 ∈ Event E Ω Example: {3} ⊆ (the event “the die came up 3”) Probability P (E) P({3}) = 1/6 P({4}) = 1/6 Discrete Ran- Example: The value on the top face dom Variable X of the die. Prob. Mass p(3) = 1/6 Function p(x) p(4) = 1/6 (pmf) 47 Random Variables: Definitions Example 2: Rolling a 6-sided die Sample Space Ω {1,2,3,4,5,6} Outcome ω Ω Example: 3 ∈ Event E Ω Example: {2,4,6} ⊆ (the event “the roll was even”) Probability P (E) P({2,4,6}) = 0.5 P({1,3,5}) = 0.5 Discrete Ran- Example: 1 if the die landed on an dom Variable X even number and 0 otherwise Prob.
Load more

Recommended publications
  • Probability and Statistics Lecture Notes
    Probability and Statistics Lecture Notes Antonio Jiménez-Martínez Chapter 1 Probability spaces In this chapter we introduce the theoretical structures that will allow us to assign proba- bilities in a wide range of probability problems. 1.1. Examples of random phenomena Science attempts to formulate general laws on the basis of observation and experiment. The simplest and most used scheme of such laws is: if a set of conditions B is satisfied =) event A occurs. Examples of such laws are the law of gravity, the law of conservation of mass, and many other instances in chemistry, physics, biology... If event A occurs inevitably whenever the set of conditions B is satisfied, we say that A is certain or sure (under the set of conditions B). If A can never occur whenever B is satisfied, we say that A is impossible (under the set of conditions B). If A may or may not occur whenever B is satisfied, then A is said to be a random phenomenon. Random phenomena is our subject matter. Unlike certain and impossible events, the presence of randomness implies that the set of conditions B do not reflect all the necessary and sufficient conditions for the event A to occur. It might seem them impossible to make any worthwhile statements about random phenomena. However, experience has shown that many random phenomena exhibit a statistical regularity that makes them subject to study. For such random phenomena it is possible to estimate the chance of occurrence of the random event. This estimate can be obtained from laws, called probabilistic or stochastic, with the form: if a set of conditions B is satisfied event A occurs m times =) repeatedly n times out of the n repetitions.
    [Show full text]
  • Introduction to Stochastic Processes - Lecture Notes (With 33 Illustrations)
    Introduction to Stochastic Processes - Lecture Notes (with 33 illustrations) Gordan Žitković Department of Mathematics The University of Texas at Austin Contents 1 Probability review 4 1.1 Random variables . 4 1.2 Countable sets . 5 1.3 Discrete random variables . 5 1.4 Expectation . 7 1.5 Events and probability . 8 1.6 Dependence and independence . 9 1.7 Conditional probability . 10 1.8 Examples . 12 2 Mathematica in 15 min 15 2.1 Basic Syntax . 15 2.2 Numerical Approximation . 16 2.3 Expression Manipulation . 16 2.4 Lists and Functions . 17 2.5 Linear Algebra . 19 2.6 Predefined Constants . 20 2.7 Calculus . 20 2.8 Solving Equations . 22 2.9 Graphics . 22 2.10 Probability Distributions and Simulation . 23 2.11 Help Commands . 24 2.12 Common Mistakes . 25 3 Stochastic Processes 26 3.1 The canonical probability space . 27 3.2 Constructing the Random Walk . 28 3.3 Simulation . 29 3.3.1 Random number generation . 29 3.3.2 Simulation of Random Variables . 30 3.4 Monte Carlo Integration . 33 4 The Simple Random Walk 35 4.1 Construction . 35 4.2 The maximum . 36 1 CONTENTS 5 Generating functions 40 5.1 Definition and first properties . 40 5.2 Convolution and moments . 42 5.3 Random sums and Wald’s identity . 44 6 Random walks - advanced methods 48 6.1 Stopping times . 48 6.2 Wald’s identity II . 50 6.3 The distribution of the first hitting time T1 .......................... 52 6.3.1 A recursive formula . 52 6.3.2 Generating-function approach .
    [Show full text]
  • Topic 1: Basic Probability Definition of Sets
    Topic 1: Basic probability ² Review of sets ² Sample space and probability measure ² Probability axioms ² Basic probability laws ² Conditional probability ² Bayes' rules ² Independence ² Counting ES150 { Harvard SEAS 1 De¯nition of Sets ² A set S is a collection of objects, which are the elements of the set. { The number of elements in a set S can be ¯nite S = fx1; x2; : : : ; xng or in¯nite but countable S = fx1; x2; : : :g or uncountably in¯nite. { S can also contain elements with a certain property S = fx j x satis¯es P g ² S is a subset of T if every element of S also belongs to T S ½ T or T S If S ½ T and T ½ S then S = T . ² The universal set ­ is the set of all objects within a context. We then consider all sets S ½ ­. ES150 { Harvard SEAS 2 Set Operations and Properties ² Set operations { Complement Ac: set of all elements not in A { Union A \ B: set of all elements in A or B or both { Intersection A [ B: set of all elements common in both A and B { Di®erence A ¡ B: set containing all elements in A but not in B. ² Properties of set operations { Commutative: A \ B = B \ A and A [ B = B [ A. (But A ¡ B 6= B ¡ A). { Associative: (A \ B) \ C = A \ (B \ C) = A \ B \ C. (also for [) { Distributive: A \ (B [ C) = (A \ B) [ (A \ C) A [ (B \ C) = (A [ B) \ (A [ C) { DeMorgan's laws: (A \ B)c = Ac [ Bc (A [ B)c = Ac \ Bc ES150 { Harvard SEAS 3 Elements of probability theory A probabilistic model includes ² The sample space ­ of an experiment { set of all possible outcomes { ¯nite or in¯nite { discrete or continuous { possibly multi-dimensional ² An event A is a set of outcomes { a subset of the sample space, A ½ ­.
    [Show full text]
  • Negative Probability in the Framework of Combined Probability
    Negative probability in the framework of combined probability Mark Burgin University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA 90095 Abstract Negative probability has found diverse applications in theoretical physics. Thus, construction of sound and rigorous mathematical foundations for negative probability is important for physics. There are different axiomatizations of conventional probability. So, it is natural that negative probability also has different axiomatic frameworks. In the previous publications (Burgin, 2009; 2010), negative probability was mathematically formalized and rigorously interpreted in the context of extended probability. In this work, axiomatic system that synthesizes conventional probability and negative probability is constructed in the form of combined probability. Both theoretical concepts – extended probability and combined probability – stretch conventional probability to negative values in a mathematically rigorous way. Here we obtain various properties of combined probability. In particular, relations between combined probability, extended probability and conventional probability are explicated. It is demonstrated (Theorems 3.1, 3.3 and 3.4) that extended probability and conventional probability are special cases of combined probability. 1 1. Introduction All students are taught that probability takes values only in the interval [0,1]. All conventional interpretations of probability support this assumption, while all popular formal descriptions, e.g., axioms for probability, such as Kolmogorov’s
    [Show full text]
  • Chapter 1 Probability, Random Variables and Expectations
    Chapter 1 Probability, Random Variables and Expectations Note: The primary reference for these notes is Mittelhammer (1999). Other treatments of proba- bility theory include Gallant (1997), Casella and Berger (2001) and Grimmett and Stirzaker (2001). This chapter provides an overview of probability theory as it applied to both discrete and continuous random variables. The material covered in this chap- ter serves as a foundation of the econometric sequence and is useful through- out financial economics. The chapter begins with a discussion of the axiomatic foundations of probability theory and then proceeds to describe properties of univariate random variables. Attention then turns to multivariate random vari- ables and important difference from univariate random variables. Finally, the chapter discusses the expectations operator and moments. 1.1 Axiomatic Probability Probability theory is derived from a small set of axioms – a minimal set of essential assumptions. A deep understanding of axiomatic probability theory is not essential to financial econometrics or to the use of probability and statistics in general, although understanding these core concepts does provide additional insight. The first concept in probability theory is the sample space, which is an abstract concept con- taining primitive probability events. Definition 1.1 (Sample Space). The sample space is a set, Ω, that contains all possible outcomes. Example 1.1. Suppose interest is on a standard 6-sided die. The sample space is 1-dot, 2-dots, . ., 6-dots. Example 1.2. Suppose interest is in a standard 52-card deck. The sample space is then A|, 2|, 3|,..., J |, Q|, K |, A},..., K }, A~,..., K ~, A♠,..., K ♠.
    [Show full text]
  • Introduction to Probability Theory
    Copyright c August 27, 2020 by NEH Introduction to Probability Theory Nathaniel E. Helwig University of Minnesota 1 Experiments and Events The field of \probability theory" is a branch of mathematics that is concerned with describing the likelihood of different outcomes from uncertain processes. Probability theory is the cornerstone of the field of Statistics, which is concerned with assessing the uncertainty of inferences drawn from random samples of data. Thus, we need to understand basics of probability theory to comprehend some of the basic principles used in inferential statistics. Before defining what the word \probability" means, I will introduce some terminology to motivate the need for thinking probabilistically. Definition. A simple experiment is some action that leads to the occurrence of a single outcome s from a set of possible outcomes S. Note that the single outcome s is referred to as a sample point, and the set of possible outcomes S is referred to as the sample space. Example 1. Suppose that you flip a coin n ≥ 2 times and record the number of times you observe a \heads". The sample space is S = f0; 1; : : : ; ng, where s = 0 corresponds to observing no heads and s = n corresponds to observing only heads. Example 2. Suppose that you pick a card at random from a standard deck of 52 playing cards. The sample points are the individual cards in the deck (e.g., the Queen of Spades is one possible sample point), and the sample space is the collection of all 52 cards. Example 3. Suppose that you roll two standard (six-sided) dice and sum the obtained numbers.
    [Show full text]
  • Chapter 2 Conditional Probability and Independence
    CHAPTER 2 CONDITIONAL PROBABILITY AND INDEPENDENCE INTRODUCTION This chapter introduces the important concepts of conditional probability and statistical independence. Conditional probabilities arise when it is known that a certain event has occurred. This knowledge changes the probabilities of events within the sample space of the experiment. Conditioning on an event occurs frequently, and understanding how to work with conditional probabilities and apply them to a particular problem or application is extremely important. In some cases, knowing that a particular event has occurred will not effect the probability of another event, and this leads to the concept of the statistical independence of events that will be developed along with the concept of conditional independence. 2-1 CONDITIONAL PROBABILITY The three probability axioms introduced in the previous chapter provide the foundation upon which to develop a theory of probability. The next step is to understand how the knowledge that a particular event has occurred will change the probabilities that are assigned to the outcomes of an experiment. The concept of a conditional probability is one of the most important concepts in probability and, although a very simple concept, conditional probability is often confusing to students. In situations where a conditional probability is to be used, it is often overlooked or incorrectly applied, thereby leading to an incorrect answer or conclusion. Perhaps the best starting point for the development of conditional probabilities is a simple example that illustrates the context in which they arise. Suppose that we have an electronic device that has a probability pn of still working after n months of continuous operation, and suppose that the probability is equal to 0.5 that the device will still be working after one year (n = 12).
    [Show full text]
  • Basic Probability Theory
    TU Eindhoven Advanced Algorithms (2IL45) — Course Notes Basic probability theory Events. Probability theory is about experiments that can have different outcomes. The possible outcomes are called the elementary events, and the sample space is the set of all elementary events. A subset of the sample space is an event.1 (Note that if the subset is a singleton, then the event is an elementary event.) We call the set of all events defined by a sample space S the event space defined by S, and we denote it by Events(S). As an example, consider the experiment where we flip a coin two times. One possible outcome of the experiment is HT: we first get heads (H) and then tails (T). The sample space is the set {HH,HT,TH,TT }. The subset {HT, T H, T T } is one of the 16 events defined by this sample space, namely “the event that we get at least one T”. Probability distributions. A sample space S comes with a probability distribution, which is a mapping Pr : Events(S) → R such that 1. Pr[A] > 0 for all events A ∈ Events(S). 2. Pr[S] = 1. 3. Pr[A ∪ B] = Pr[A] + Pr[B] for any two events A, B ∈ Events(S) that are mutually exclusive (that is, events A, B such that A ∩ B = ∅). Another way of looking at this is that we assign non-negative probabilities to the elementary events (which sum to 1), and that the probability of an event is the sum of the probabilities of the elementary events it is composed of.
    [Show full text]
  • On the Measurement Problem
    International Journal of Theoretical and Mathematical Physics 2014, 4(5): 202-219 DOI: 10.5923/j.ijtmp.20140405.04 On the Measurement Problem Raed M. Shaiia Department of Physics, Damascus University, Damascus, Syria Abstract In this paper, we will see that we can reformulate the purely classical probability theory, using similar language to the one used in quantum mechanics. This leads us to reformulate quantum mechanics itself using this different way of understanding probability theory, which in turn will yield a new interpretation of quantum mechanics. In this reformulation, we still prove the existence of none classical phenomena of quantum mechanics, such as quantum superposition, quantum entanglement, the uncertainty principle and the collapse of the wave packet. But, here, we provide a different interpretation of these phenomena of how it is used to be understood in quantum physics. The advantages of this formulation and interpretation are that it induces the same experimental results, solves the measurement problem, and reduces the number of axioms in quantum mechanics. Besides, it suggests that we can use new types of Q-bits which are more easily to manipulate. Keywords Measurement problem, Interpretation of quantum mechanics, Quantum computing, Quantum mechanics, Probability theory Measurement reduced the wave packet to a single wave and 1. Introduction a single momentum [3, 4]. The ensemble interpretation: This interpretation states Throughout this paper, Dirac's notation will be used. that the wave function does not apply to an individual In quantum mechanics, for example [1] in the experiment system – or for example, a single particle – but is an of measuring the component of the spin of an electron on abstract mathematical, statistical quantity that only applies the Z-axis, and using as a basis the eigenvectors of σˆ3 , the to an ensemble of similarly prepared systems or particles [5] state vector of the electron before the measurement is: [6].
    [Show full text]
  • KOLMOGOROV's PROBABILITY THEORY in QUANTUM PHYSICS
    KOLMOGOROV’s PROBABILITY THEORY IN QUANTUM PHYSICS D.A. Slavnov Department of Physics, Moscow State University, GSP-2 Moscow 119992, Russia. E-mail: [email protected] A connection between the probability space measurability requirement and the complementarity principle in quantum mechanics is established. It is shown that measurability of the probability space implies that the results of the quantum measurement depend not only on properties of the quantum object under consider- ation but also on classical characteristics of the measuring instruments used. It is also shown that if the measurability requirement is taken into account, then the hypothesis that the objective reality exists does not lead to the Bell inequality. Recently we commemorated the centenary of quantum mechanics. This year we celebrate the Einstein year. Einstein was one of founders of quantum mechanics. Simultaneously he was one of the most known critics of quantum mechanics. First of all he criticized quantum mechanics because in it there is no a mathematical counter-part of a physical reality which becomes apparent in individual experiment. Seventy years ago he wrote [1]: “a) every element of the physical reality must have a counter-part in the complete physical theory; b) if, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there is an element of physical reality corresponding to this physical quantity”. In the report I want to talk about this mathematical counter-part, probability, and quantum measurement. At the same time, when the quoted statement of Einstein was written, Kolmogorov created modern probability theory [2].
    [Show full text]
  • ELEMENTARY PROBABILITY Events and Event Sets. Consider Tossing A
    ELEMENTARY PROBABILITY Events and event sets. Consider tossing a die. There are six possible outcomes, which we shall denote by elements of the set {Ai; i =1, 2,...,6}. A numerical value is assigned to each outcome or event: A1 −→ 1, A2 −→ 2, . A6 −→ 6. The variable x =1, 2,...,6 that is associated with these outcomes is called a random variable. The set of all possible outcomes is S = {A1 ∪A2 ∪···∪A6} is called the sample space. This is distinct from the elementary event set. For example, we can define the event E ⊂ S to be a matter of getting an even number from tossing the die. We may denote this event by E = {A2 ∪ A4 ∪ A6}, which is the event of getting a 2 or a 4 or a 6. These elementary events that constitute E are, of course, mutually exclusive. The collectison of subsets of S that can be formed by taking unions of the events within S, and also by taking intersections of these unions, is described as a sigma field, or a σ-field, and there is also talk of a sigma algebra. The random variable x is a described as a scalar-valued set function defined over the sample space S. There are numerous other random variables that we might care to define in respect of the sample space by identifying other subsets and assigning numerical values to the corresponding events. We shall do so presently. Summary measures of a statistical experiment. Let us toss the die 30 times and let us record the value assumed by the random variable at each toss: 1, 2, 5, 3,...,4, 6, 2, 1.
    [Show full text]
  • Overview of Probability and Statistics
    Overview of Probability and Statistics Michal Reinštein, Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception http://cmp.felk.cvut.cz/˜reinsmic, [email protected] Slides: V. Hlaváč, modified: M. Reinštein, courtesy: T. Brox, V. Franc, M. Navara, M. Urban. Outline of the talk: Probability vs. statistics. Bayes theorem. Random events. Distribution function, density. Probability, joint, conditional. Characteristics of a random variable. Recommended reading 2/39 A. Papoulis: Probability, Random Variables and Stochastic Processes, McGraw Hill, Edition 4, 2002. http://mathworld.wolfram.com/ http://www.statsoft.com/textbook/stathome.html Probability, statistics 3/39 Probability: probabilistic model =⇒ future behavior. • It is a theory (tool) for purposeful decisions when the outcome of future events depends on circumstances we know only partially and the randomness plays a role. • An abstract model of uncertainty description and quantification of the results. Statistics: behavior of the system =⇒ probabilistic representation. • It is a tool for seeking a probabilistic description of real systems based on observing them and testing them. • It provides more: a tool for investigating the world, seeking and testing dependencies which are not apparent. • Two types: descriptive and inference statistics. • Collection, organization and analysis of data. • Generalization from restricted / finite samples. Random events, concepts 4/39 An experiment with random outcome – states of nature, possibilities, experimental results, etc. A sample space is an nonempty set Ω of all possible outcomes of the experiment. An elementary event ω ∈ Ω are elements of the sample space (outcomes of the experiment). A space of events A is composed of the system of all subsets of the sample space Ω.
    [Show full text]