# Probability and statistics; Rehearsal for pattern recognition

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Probability and statistics; Rehearsal for pattern recognition Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception hlavac, Courtesy: T. Brox, V. Franc, M. Navara, M. Urban. Probability vs. statistics. Random events. Outline of the talk: Bayes theorem. Distribution function, density. Probability, joint, conditional. Characteristics of a random variable.

2 Recommended reading 2/29 A. Papoulis: Probability, Random Variables and Stochastic Processes, McGraw Hill, Edition 4,

3 Probability, motivating example A lottery ticket is sold for the price EUR 2. 3/29 1 lottery ticket out of 1000 wins EUR Other lottery tickets win nothing. This gives the value of the lottery ticket after the draw. For what price should the lottery ticket be sold before the draw? Only a fool would by the lottery ticket for EUR 2. (Or not?) 1 The value of the lottery ticket before the draw is = EUR 1 = the average value after the draw. The probability theory is used here.. A lottery question: Why are the lottery tickets being bought? Why do lotteries prosper?

4 Statistics, motivating example 4/29 We have assumed so far that the parameters of the probability model are known. However, this is seldom fulfilled. Example Lotto: One typically looses while playing Lotto because the winnings are set according to the number of winners. It is of advantage to bet differently than others. For doing so, it is needed what model do the other use. Example Roulette: Both parties are interested if all the numbers occur with the same probability. More precisely said, what are the differences from the uniform probability distribution. How to learn it? What is the risk of wrong conclusions? Statistics is used here.

5 Probability, statistics 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. 5/29

6 Random events, concepts 6/29 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. A random event A A is and element of the space of events. Note: The concept of a random event was introduced in order to be able to define the probability, probability distribution, etc.

7 Probability, introduction 7/29 Classic. P.S. Laplace, It is not regarded to be the definition of the probability any more. It is merely an estimate of the probability. P (A) N A N Limit (frequency) definition P (A) = lim N N A N z histogram vs. continuous pdf Axiomatic definition (Andrey Kolmogorov 1930)

8 Axiomatic definition of the probability 8/29 Ω - the sample space. A - the space of events. 1. P (A) 0, A A. 2. P (Ω) = If A B = then P (A B) = P (A) + P (B), A A, B B.

9 Probability is a function P, which assigns number from the interval 0, 1 to events and fulfils the following two conditions: 9/29 P (true) = 1, ( ) P A n n N exclusive. = n N P (A n ), if the events A n, n N, are mutually From these conditions, it follows: P (false) = 0, P ( A) = 1 P (A), if A B then P (A) P (B). Note: Strictly speaking, the space of events have to fulfil some additional conditions.

10 Derived relations 10/29 If A B then P (B \ A) = P (B) P (A). The symbol \ denotes the set difference. P (A B) = P (A) + P (B) P (A B).

11 Joint probability, marginalization 11/29 The joint probability P (A, B), also sometimes denoted P (A B), is the probability that events A, B co-occur. The joint probability is symmetric: P (A, B) = P (B, A). Marginalization (the sum rule): P (A) = P (A, B) allows computing the B probability of a single event by summing the joint probabilities over all possible events B. The probability P (A) is called the marginal probability.

12 Contingency table, marginalization Example: orienteering race 12/29 Orienteering competition example, participants Age <= >= 76 Sum Men Women Sum Orienteering competition example, frequency Age <= >= 76 Sum Men 0,061 0,100 0,125 0,092 0,081 0,058 0,033 0,006 0,556 Women 0,053 0,089 0,103 0,083 0,064 0,039 0,014 0,000 0,444 Sum 0,114 0,189 0,228 0,175 0,144 0,097 0,047 0, ,300 Marginal probability P(sex) 0,200 0,100 0, Marginal probability P(Age_group)

13 The conditional probability Let us have the probability representation of a system given by the joint probability P (A, B). 13/29 If an additional information is available that the event B occurred then our knowledge about the probability of the event A changes to P (A B) = P (A, B) P (B), which is the conditional probability of the event A under the condition B. The conditional probability is defined only for P (B) 0. Product rule: P (A, B) = P (A B) P (B) = P (B A) P (A). From the symmetry of the joint probability and the product rule, the Bayes theorem can be derived (to come in a more general formulation for more than two events).

14 Properties of the conditional probability P (true B) = 1, P (false B) = 0. If A = A n and events A 1, A 2,... are mutually exclusive then n N P (A B) = n N P (A n B). 14/29 Events A, B are independent, if and only if P (A B) = P (A). If B A then P (A B) = 1. If B A then P (A B) = 0. Events Bi, i I, constitute a complete system of events if they are mutually exclusive and B i = true. i I A complete system of events has such property that one and only one event of them occurs.

15 Example: conditional probability 15/29 Consider rolling a single dice. What is the probability that the number higher than three comes up (event A) under the conditions that the odd number came up (event B)? Ω = {1, 2, 3, 4, 5, 6}, A = {4, 5, 6}, B = {1, 3, 5} P (A) = P (B) = 1 2 P (A, B) = P ({5}) = 1 6 P (A B) = P (A, B) P (B) = = 1 3

16 The total probability theorem Let B i, i I, be a complete system of events and let it hold i I : P (B i ) 0. 16/29 Then for every event A holds P (A) = i I P (B i ) P (A B i ). Proof: (( ) ( ) P (A) = P B i ) A = P (B i A) i I i I = i I P (B i A) = i I P (B i ) P (A B i ).

17 Bayes theorem Let B i, i I, be a complete system of events and i I : P (B i ) 0. 17/29 For each event A fulfilling the condition P (A) 0 the following holds P (B i A) = P (B i ) P (A B i ) i I P (B i) P (A B i ), where P (B i A) is the posterior probability; P (B i ) is the prior probability; and P (A B i ) are known conditional probabilities of A having observed B i. Proof (exploring the total probability theorem): P (B i A) = P (B i A) P (A) = P (B i ) P (A B i ) i I P (B i) P (A B i ).

18 The importance of the Bayes theorem Bayes theorem is a fundamental rule for machine learning (pattern recognition) as it allows to compute the probability of an output A given observations B i. The conditional probabilities (also likelihoods) P (A Bi) are estimated from experiments or from a statistical model. Having P (A Bi), the aposteriori probabilities P (Bi A) are determined, which serve to estimate optimally which event from B i occurred. It is needed to know the apriori probability P (Bi) to determine aposteriori probability P (B i A). Informally: posterior prior conditional probability of the event having some observations. 18/29 In a similar manner, we define the conditional probability distribution, conditional density of the continuous random variable, etc.

19 Conditional independence 19/29 Random events A, B are conditionally independent under the condition C, if P (A B C) = P (A C) P (B C). Similarly, a conditional independence of more events, random variables, etc. is defined.

20 Independent events Event A, B are independent P (A B) = P (A) P (B). 20/29 Example The dice is rolled once. Events are: A > 3, event B is an odd number. Are these events independent? Ω = {1, 2, 3, 4, 5, 6}, A = {4, 5, 6}, B = {1, 3, 5} P (A) = P (B) = 1 2 P (A B) = P ({5}) = 1 6 P (A) P (B) = = 1 4 P (A B) P (A) P (B) The events are dependent.

21 The random variable The random variable is an arbitrary function X : Ω R, where Ω is a sample space. 21/29 Why is the concept of the random variable introduced? It allows to work with concepts as the distribution function, probability density, expectation (mean value), etc. There are two basic types of random variables: Discrete a countable number of values. Examples: rolling a dice, the count of number of cars passing through a street in a hour. The discrete probability is given as P (X = a i ) = p(a i ), i = 1,..., i p(a i) = 1. Continuous values from some interval, i.e. infinite number of values. Example: the height persons. The continuous probability is given by the distribution function or the probability density function.

22 Distribution function of a random variable Distribution function of the random variable X is a function F : X [0, 1] defined as F (x) = P (X x), where P is a probability. 22/29 Properties: 1. F (x) is a non-decreasing function, i.e. pair x 1 < x 2 it holds F (x 1 ) F (x 2 ). 2. F (X) is continuous from the right, i.e. it holds lim F (x + h) = F (x). h It holds for every distribution function lim F (x) = 0 a x F (x) = 1. Written more concisely: F ( ) = 0, F ( ) = 1. lim x If the possible values of F (x) are from the interval (a, b) then F (a) = 0, F (b) = 1. Any function fulfilling the above three properties can be understood as a distribution function.

23 Continuous distribution function The distribution function F is called (absolutely) continuous if a nonnegative function f (probability density) exists and it holds 23/29 F (x) = x f(u) du for every x X. The probability density fulfills f(x) dx = 1. f(x) Area = 1 x If the derivative of F (x) exists in the point x then F (x) = f(x). For a, b R, a < b, it holds P (a < X < b) = b a f(x) dx = F (b) F (a).

24 Example, normal distribution 24/29 F (x) f(x) = 1 2πσ 2 e x2 2σ 2 Distribution function Probability density

25 The law of large numbers 25/29 The law of large numbers says that if very many independent experiments can be made then it is almost certain that the relative frequency will converge to the theoretical value of the probability density. Jakob Bernoulli, Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis, 1713, Chapter 4.

26 Expectation 26/29 (Mathematical) expectation = the average of a variable under the probability distribution. Continuous definition: E(x) = x f(x) dx. Discrete definition: E(x) = x P (x). x The expectation can be estimated from a number of samples by E(x) 1 N x i. The approximation becomes exact for N. i Expectation over multiple variables: Ex(x, y) = Conditional expectation: E(x y) = x f(x y) dx. (x, y) f(x) dx

27 Basic characteristics of a random variable 27/29 Continuous distribution Discrete distribution Expectation E(x) = x f(x) dx E(x) = x k-th (general) moment E(x k ) = x k f(x) dx E(x) = x k-th central moment E(x k ) = E(x)) (x k f(x) dx E(x) = x Dispersion D(x) = E(x)) (x 2 f(x) dx E(x) = x x P (x) x k P (x) (x E(x)) k P (x) (x E(x)) 2 P (x)

28 Covariance 28/29 The mutual covariance σ xy of two random variables X, Y is σ xy = E ((X µ x )(Y µ y )). The covariance matrix Σ of n variables X 1,..., X n is Σ = σ σ 1n.... σ n1... σ 2 n The covariance matrix is symmetric (i.e. Σ = Σ ) and positive-semidefinite (as the covariance matrix is real valued, the positive-semidefinite means that xmx 0 for all x R).

29 Quantiles, median 29/29 The p-quantile Qp: P (X < Qp) = p. The median is the p-quantile for p = 12, i.e. P (X < Qp) = 12. Note: Median is often used as a replacement for the mean value in robust statistics.

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### Introduction 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

### Chapter ML:IV. IV. Statistical Learning. Probability Basics Bayes Classification Maximum a-posteriori Hypotheses

Chapter ML:IV IV. Statistical Learning Probability Basics Bayes Classification Maximum a-posteriori Hypotheses ML:IV-1 Statistical Learning STEIN 2005-2015 Area Overview Mathematics Statistics...... Stochastics

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### The basics of probability theory. Distribution of variables, some important distributions

The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a

### Outline. Random Variables. Examples. Random Variable

Outline Random Variables M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno Random variables. CDF and pdf. Joint random variables. Correlated, independent, orthogonal. Correlation,

### Probability for Estimation (review)

Probability for Estimation (review) In general, we want to develop an estimator for systems of the form: x = f x, u + η(t); y = h x + ω(t); ggggg y, ffff x We will primarily focus on discrete time linear

### Lecture 8: Continuous random variables, expectation and variance

Lecture 8: Continuous random variables, expectation and variance Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde

### Chapter 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,

### Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

### Summary of Probability

Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

### Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1

Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

### A Tutorial on Probability Theory

Paola Sebastiani Department of Mathematics and Statistics University of Massachusetts at Amherst Corresponding Author: Paola Sebastiani. Department of Mathematics and Statistics, University of Massachusetts,

### Probability distributions

Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.14-2.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,

### Welcome to Stochastic Processes 1. Welcome to Aalborg University No. 1 of 31

Welcome to Stochastic Processes 1 Welcome to Aalborg University No. 1 of 31 Welcome to Aalborg University No. 2 of 31 Course Plan Part 1: Probability concepts, random variables and random processes Lecturer:

### Probabilities and Random Variables

Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments so

### Exploratory Data Analysis

Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

### THE CENTRAL LIMIT THEOREM TORONTO

THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................

### Probability Theory. Florian Herzog. A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T..

Probability Theory A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T.. Florian Herzog 2013 Probability space Probability space A probability space W is a unique triple W = {Ω, F,

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Continuous Distributions

MAT 2379 3X (Summer 2012) Continuous Distributions Up to now we have been working with discrete random variables whose R X is finite or countable. However we will have to allow for variables that can take

### An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

### Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

### MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

### BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential

### Experimental Uncertainty and Probability

02/04/07 PHY310: Statistical Data Analysis 1 PHY310: Lecture 03 Experimental Uncertainty and Probability Road Map The meaning of experimental uncertainty The fundamental concepts of probability 02/04/07

### Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

### Probability and Expected Value

Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

### Question: 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

### Chapter 4 Expected Values

Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### L10: Probability, statistics, and estimation theory

L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian

### Lecture 2: Probability

Lecture 2: Probability Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 Sample Spaces 3 Event 4 The Probability

### Chapter 1. Probability, Random Variables and Expectations. 1.1 Axiomatic Probability

Chapter 1 Probability, Random Variables and Expectations Note: The primary reference for these notes is Mittelhammer (1999. Other treatments of probability theory include Gallant (1997, Casella & Berger

### Pattern Recognition. Probability Theory

Pattern Recognition Probability Theory Probability Space 2 is a three-tuple with: the set of elementary events algebra probability measure -algebra over is a system of subsets, i.e. ( is the power set)

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### Joint 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

### Fuzzy Probability Distributions in Bayesian Analysis

Fuzzy Probability Distributions in Bayesian Analysis Reinhard Viertl and Owat Sunanta Department of Statistics and Probability Theory Vienna University of Technology, Vienna, Austria Corresponding author:

### Chapter 5. Discrete Probability Distributions

Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable

10-601 Machine Learning http://www.cs.cmu.edu/afs/cs/academic/class/10601-f10/index.html Course data All up-to-date info is on the course web page: http://www.cs.cmu.edu/afs/cs/academic/class/10601-f10/index.html

### IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics

### Toss a coin twice. Let Y denote the number of heads.

! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.

### The 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

### 1 Prior Probability and Posterior Probability

Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which

### Set theory, and set operations

1 Set theory, and set operations Sayan Mukherjee Motivation It goes without saying that a Bayesian statistician should know probability theory in depth to model. Measure theory is not needed unless we

### Lecture 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.

### Probabilities. 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

### M2L1. Random Events and Probability Concept

M2L1 Random Events and Probability Concept 1. Introduction In this lecture, discussion on various basic properties of random variables and definitions of different terms used in probability theory and

### Nonparametric 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

### MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

### MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

### Probability - Part I. Definition : A random experiment is an experiment or a process for which the outcome cannot be predicted with certainty.

Probability - Part I Definition : A random experiment is an experiment or a process for which the outcome cannot be predicted with certainty. Definition : The sample space (denoted S) of a random experiment

### Slides for Risk Management

Slides for Risk Management Introduction to the modeling of assets Groll Seminar für Finanzökonometrie Prof. Mittnik, PhD Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik,

### Elementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.

Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers

### For 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

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli

### P (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )

Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =

### 2.6. Probability. In general the probability density of a random variable satisfies two conditions:

2.6. PROBABILITY 66 2.6. Probability 2.6.. Continuous Random Variables. A random variable a real-valued function defined on some set of possible outcomes of a random experiment; e.g. the number of points

### Lecture 2 : Basics of Probability Theory

Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different

### Summary 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

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### What Is Probability?

1 What Is Probability? The idea: Uncertainty can often be "quantified" i.e., we can talk about degrees of certainty or uncertainty. This is the idea of probability: a higher probability expresses a higher

### Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

### Introduction to probability theory in the Discrete Mathematics course

Introduction to probability theory in the Discrete Mathematics course Jiří Matoušek (KAM MFF UK) Version: Oct/18/2013 Introduction This detailed syllabus contains definitions, statements of the main results

### Random Variables, Expectation, Distributions

Random Variables, Expectation, Distributions CS 5960/6960: Nonparametric Methods Tom Fletcher January 21, 2009 Review Random Variables Definition A random variable is a function defined on a probability

### Contents. TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics. Yuming Jiang. Basic Concepts Random Variables

TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics Yuming Jiang 1 Some figures taken from the web. Contents Basic Concepts Random Variables Discrete Random Variables Continuous Random

### Basics of Probability

Basics of Probability 1 Sample spaces, events and probabilities Begin with a set Ω the sample space e.g., 6 possible rolls of a die. ω Ω is a sample point/possible world/atomic event A probability space

### Example. Two fair dice are tossed and the two outcomes recorded. As is familiar, we have

Lectures 9-10 jacques@ucsd.edu 5.1 Random Variables Let (Ω, F, P ) be a probability space. The Borel sets in R are the sets in the smallest σ- field on R that contains all countable unions and complements

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### M2S1 Lecture Notes. G. A. Young http://www2.imperial.ac.uk/ ayoung

M2S1 Lecture Notes G. A. Young http://www2.imperial.ac.uk/ ayoung September 2011 ii Contents 1 DEFINITIONS, TERMINOLOGY, NOTATION 1 1.1 EVENTS AND THE SAMPLE SPACE......................... 1 1.1.1 OPERATIONS

### Exercises with solutions (1)

Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### 6. Distribution and Quantile Functions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 6. Distribution and Quantile Functions As usual, our starting point is a random experiment with probability measure P on an underlying sample spac

### Random Variate Generation (Part 3)

Random Variate Generation (Part 3) Dr.Çağatay ÜNDEĞER Öğretim Görevlisi Bilkent Üniversitesi Bilgisayar Mühendisliği Bölümü &... e-mail : cagatay@undeger.com cagatay@cs.bilkent.edu.tr Bilgisayar Mühendisliği

### Lecture 4: Random Variables

Lecture 4: Random Variables 1. Definition of Random variables 1.1 Measurable functions and random variables 1.2 Reduction of the measurability condition 1.3 Transformation of random variables 1.4 σ-algebra

### Basic Probability Concepts

page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes

### Statistics 100A Homework 4 Solutions

Chapter 4 Statistics 00A Homework 4 Solutions Ryan Rosario 39. A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is then replaced and another ball is drawn.

### INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

### Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies

### International College of Economics and Finance Syllabus Probability Theory and Introductory Statistics

International College of Economics and Finance Syllabus Probability Theory and Introductory Statistics Lecturer: Mikhail Zhitlukhin. 1. Course description Probability Theory and Introductory Statistics

### Chapter 3: The basic concepts of probability

Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording

### In the situations that we will encounter, we may generally calculate the probability of an event

What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

### Probability. 4.1 Sample Spaces

Probability 4.1 Sample Spaces For a random experiment E, the set of all possible outcomes of E is called the sample space and is denoted by the letter S. For the coin-toss experiment, S would be the results

### Probability OPRE 6301

Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### 1. (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

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### The Alternating Series Test

The Alternating Series Test So far we have considered mostly series all of whose terms are positive. If the signs of the terms alternate, then testing convergence is a much simpler matter. On the other

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.

### Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation

Statistical Foundations: and Central Tendency and and Lecture 4 September 5, 2006 Psychology 790 Lecture #4-9/05/2006 Slide 1 of 26 Today s Lecture Today s Lecture Where this Fits central tendency/location

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### What 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

### 2DI36 Statistics. 2DI36 Part II (Chapter 7 of MR)

2DI36 Statistics 2DI36 Part II (Chapter 7 of MR) What Have we Done so Far? Last time we introduced the concept of a dataset and seen how we can represent it in various ways But, how did this dataset came

### Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

### Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall