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

Save this PDF as:

Size: px
Start display at page:

Download "Welcome to Stochastic Processes 1. Welcome to Aalborg University No. 1 of 31"

## Transcription

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

2 Welcome to Aalborg University No. 2 of 31 Course Plan Part 1: Probability concepts, random variables and random processes Lecturer: Ernest Nlandu Kamavuako Lecture 1: Introduction to probability and random variables Lecture 2: Moments and multi dimensional random variables Lecture 3: Introduction to random processes Lecture 4: Stationarity We expect the student to have a deep learning

3 Welcome to Aalborg University No. 3 of 31 Course Plan Part 2: Random processes Lecturer: Samuel Schmidt Lecture 5: Some random processes (basic knowledge) Lecture 6: Power spectral density Lecture 7: Linear time-invariant systems (Filtering) Lecture 8: Ergodicity Lecture 9: Covariance matrix (if time allows) Lecture 6-8: Deep learning

4 Welcome to Aalborg University No. 4 of 31 Learning outcomes Outcomes: Have knowledge on stochastic processes in general Have knowledge about cross- and auto-correlation of stochastic processes. Have deep knowledge about power spectral analysis of stationary stochastic processes and ergodicity. Explain the defining properties of various stochastic processes Exam: Written

5 Welcome to Aalborg University No. 5 of 31 Lecture 1: Introduction to probability and random variables A deterministic signal can be derived by mathematical expressions. A deterministic model (or system) will always produce the same output from a given starting condition or initial state. In this course: stochastic (random) signals or processes Counterpart to a deterministic process Described in a probabilistic way Given initial condition, many realizations of the process Teaching style: lectures (2 h) and exercises (2 h)

6 Welcome to Aalborg University No. 6 of 31 Concepts of probability A prequisite for understanding the main content of the course. Some definitions: A set is a collecton of objects. a A Given 2 sets A and B, Intersection: A B Union: A B A a b c Elements or members

7 Welcome to Aalborg University No. 7 of 31 Probability theory Probability provides mathematical models for random phenomena and experiments. Some random phenomena Its origin lies in observations associated with games of chance

8 Welcome to Aalborg University No. 8 of 31 Some definitions Sample space S is a set of all possible distinct outcomes of interest in a particular experiment Tossing a coin. S = {head, tail} Toss a fair coin n times and heads come up n H times. The relative frequency of heads is n H /n and will be very closed to 1/2 An EVENT is a particular outcome or a combination of outcomes

9 Welcome to Aalborg University No. 9 of 31 Classical definition of probability Define the probability of an event A as: P A = N A N where N is the number of possible outcomes of the random experiment and N A is the number of outcomes favorable to the event A. For example: A 6-sided die has 6 outcomes. 3 of them are even, Thus P(even) = 3/6

10 Welcome to Aalborg University No. 10 of 31 Problems with this classical definition Here the assumption is that all outcomes are equally likely (probable). Thus, the concept of probability is used to define probability itself! Cannot be used as basis for a mathematical theory. In many random experiments, the outcomes are not equally likely. The definition doesn t work when the number of possible outcomes is infinite.

11 Welcome to Aalborg University No. 11 of 31 Relative frequency definition of probability The probability of en event A is defined as: P A n = lim A n n where n A is the number of times A occurs in n performances of the experiment An experiment has been conducted and the probability is defined based on the observations.

12 Welcome to Aalborg University No. 12 of 31 Problems with Relative frequency definition of probability P A n = lim A n n How do we assert that such limit exists? We often cannot perform the experiment multiple times (e.g. Limited time in the lab)

13 Welcome to Aalborg University No. 13 of 31 Axiomatic Definition of Probability It provides rules for assigning probabilities to events in a mathematically consistent way Elements of axiomatic definition: Set of all possible outcomes of the random experiment S (Sample space) Set of events, which are subsets of S

14 Welcome to Aalborg University No. 14 of 31 Axiomatic Definition of Probability A probability law (measure or function) that assigns probabilities to events such that: o P(A) 0 o P(S) =1 o If A and B are disjoint events (mutually exclusive), i.e. A B =, then P(A B) = P(A) + P(B) That is if A happens, B cannot occur.

15 Welcome to Aalborg University No. 15 of 31 Advantages The theory provides mathematically precise ways for dealing with experiments with infinite number of possible outcomes, defining random variables, etc. The theory does not deal with what the values of the probabilities are or how they are obtained. Any assignment of probabilities that satisfies the axioms is legitimate.

16 Welcome to Aalborg University No. 16 of 31 Some Useful Properties 0 P(A) 1 P P A = 0 : probability of impossible event = 1 P A, A the complement of A If A and B are two events, then P A B = P A + P B P A B If the sample space consits of n mutually exclusive events such that S = A 1 A 2 A n, then P S = P A 1 + P A P A n = 1

17 Welcome to Aalborg University No. 17 of 31 Joint and marginal probability Joint probability: is the likelihood of two events occurring together. Joint probability is the probability of event A occurring at the same time event B occurs. It is P(A B) or P(AB). Marginal probability: is the probability of one event, ignoring any information about the other event. Thus P(A) and P(B) are marginal probabilities of events A and B

18 Welcome to Aalborg University No. 18 of 31 Conditional probability Let A and B be two events. The probability of event B given that event A has occured is called the conditional probability P A B = P(A B) P(B) If the occurance of B has no effect on A, we say A and B are indenpendent events. In this case P A B = P(A) Combining both, we get P A B are indenpendent = P A P B, when A and B

19 Welcome to Aalborg University No. 19 of 31 Relationship involving Joint, marginal and conditional Probabilities P AB = P A B P B = P B A P(A) If AB =, then P AUB C = P A C + P B C If B 1, B 2,, B n are mutually exclusive and exhaustive events, then P A = P A B j P(B j ) n j=1

20 Welcome to Aalborg University No. 20 of 31 Example An automatic diagnostic system classifies patients into: Event A: the patient has the disease Event B: the patient is healthy. The probability that a patient belongs to group A is P(A) = 0.05 and that of group B is P(B) = The probability of the system to make an error is P(E) = Question 1: Find the probability of A detected with error: P(A,E) Question 2: Let P(E A) = 0.02 and P(E B) = 0.1, Find P(A,E) and P(B,E)

21 Welcome to Aalborg University No. 21 of 31 Random variables A random variable is a function, which maps events or outcomes (e.g., the possible results of rolling two dice: {1, 1}, {1, 2}, etc.) to real numbers (e.g., their sum). A random variable can be thought of as a quantity whose value is not fixed, but which can take on different values. A probability distribution is used to describe the probabilities of different values occurring.

22 Welcome to Aalborg University No. 22 of 31 Random variables Notations: Random variables with capital letters: X, Y,..., Z Real value of the random variable by lowercase letters (x, y,, z) Types: Continuous random variables: maps outcomes to values of an uncountable set. the probability of any specific value is zero Discrete random variable: maps outcomes to values of a countable set. Each value has probability 0. P(x i ) = P(X = x i ) Mixed random variables

23 Welcome to Aalborg University No. 23 of 31 Continuous random variables Distribution function: By definition F X (x) P(X x) Properties: 1. F X (x) is either increasing or remains constant. 2. lim x F X x = 0 3. lim x + F X x = 1 4. F X x 1 F X x 2 if x 1 x 2 5. P a X b = F X b F X (a)

24 Welcome to Aalborg University No. 24 of 31 Distribution functions 1 F X (x) Cumulative distribution function (CDF) for a normal distribution 0 x 0

25 Welcome to Aalborg University No. 25 of 31 Probability density function (pdf) Definition: Properties: 1. f X x 0 f X x = df X(x) dx b 2. f X x a dx = F X x b a = F X b F X a = P(a X b) + 3. f X x dx = F X x + = 1 Thus integration of f X (x) gives probability

26 Welcome to Aalborg University No. 26 of 31 Example We have observed the firing of action potentials (AP) in different experiments and noticed that: 1. a motor neuron discharges its first AP at the time instant t = The second AP happens between 100ms and 200ms with T describing its position. 3. All have the same probability: uniform distribution Question: Find the cdf and the pdf

27 Welcome to Aalborg University No. 27 of 31 Take home message 1. Some definitions in probability: sample space, event, set, etc definitions of probability: Classical, relative frequency and axiomial 3. Joint, marginal and conditional probability 4. Random variables: a. Definition b. Cumulative distribution function (CDF) c. Probability dnesity function (PDF) d. Properties of CDF and PDF

28 Welcome to Aalborg University No. 28 of 31 Exercises Course website: If requested: Username: sp1 Password: st6 Culture & Media Dansk Engelsk Engelsk Sprog og Internationale Studier Erhvervssprog og International Erhvervskommunikation Filosofi, Videnskabsteori, Komparativ filosofi Humanistisk Informatik Informationsteknologi, Multimedier (cand.it.) International and Intercultural Communication It og læring, organisatorisk omstilling (cand.it.) Kultur, Kommunikation og Globalisering Læring og Pædagogik Musik Musikterapi Psykologi Spansk Spansk Sprog og Internationale Studier Sprog og International Virksomhedskommunikation Turisme Tysk

### 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

### 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

### 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

### Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance

### 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

### 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

### 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

### Definition of Random Variable A random variable is a function from a sample space S into the real numbers.

.4 Random Variable Motivation example In an opinion poll, we might decide to ask 50 people whether they agree or disagree with a certain issue. If we record a for agree and 0 for disagree, the sample space

### Basics of Probability

Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.

### Probability Review. ICPSR Applied Bayesian Modeling

Probability Review ICPSR Applied Bayesian Modeling Random Variables Flip a coin. Will it be heads or tails? The outcome of a single event is random, or unpredictable What if we flip a coin 10 times? How

### 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.

### 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

### 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.

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

### MASSACHUSETTS 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

### 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

### Math/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

### Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab

Lecture 2 Probability BMIR Lecture Series in Probability and Statistics Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab 2.1 1 Sample Spaces and Events Random

### Events. Independence. Coin Tossing. Random Phenomena

Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,

### 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

### 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.

### Probability and statistics; Rehearsal for pattern recognition

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

### 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

### 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

### 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

### Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments

### 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,

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### A Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability - 1. Probability and Counting Rules

Probability and Counting Rules researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people in this random sample

### 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,

### STAT 270 Probability Basics

STAT 270 Probability Basics Richard Lockhart Simon Fraser University Spring 2015 Surrey 1/28 Purposes of These Notes Jargon: experiment, sample space, outcome, event. Set theory ideas and notation: intersection,

### Lecture 4: Probability Spaces

EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 4: Probability Spaces Lecturer: Dr. Krishna Jagannathan Scribe: Jainam Doshi, Arjun Nadh and Ajay M 4.1 Introduction

### MAS108 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

### CONTINGENCY (CROSS- TABULATION) TABLES

CONTINGENCY (CROSS- TABULATION) TABLES Presents counts of two or more variables A 1 A 2 Total B 1 a b a+b B 2 c d c+d Total a+c b+d n = a+b+c+d 1 Joint, Marginal, and Conditional Probability We study methods

### 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

### 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,

### Statistics in Geophysics: Introduction and Probability Theory

Statistics in Geophysics: Introduction and Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/32 What is Statistics? Introduction Statistics is the

### Chapter 10: Introducing Probability

Chapter 10: Introducing Probability Randomness and Probability So far, in the first half of the course, we have learned how to examine and obtain data. Now we turn to another very important aspect of Statistics

### Joint distributions Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014

Joint distributions Math 17 Probability and Statistics Prof. D. Joyce, Fall 14 Today we ll look at joint random variables and joint distributions in detail. Product distributions. If Ω 1 and Ω are sample

### Chapter 4: Probabilities and Proportions

Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 4: Probabilities and Proportions Section 4.1 Introduction In the real world,

### 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

### Assigning Probabilities

What is a Probability? Probabilities are numbers between 0 and 1 that indicate the likelihood of an event. Generally, the statement that the probability of hitting a target- that is being fired at- is

### People have thought about, and defined, probability in different ways. important to note the consequences of the definition:

PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models

### The study of probability has increased in popularity over the years because of its wide range of practical applications.

6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,

### 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

### 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

### 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

### AP Stats - Probability Review

AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

### Lecture 7: Continuous Random Variables

Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider

### Probability and Random Variables. Generation of random variables (r.v.)

Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly

### Part III. Lecture 3: Probability and Stochastic Processes. Stephen Kinsella (UL) EC4024 February 8, 2011 54 / 149

Part III Lecture 3: Probability and Stochastic Processes Stephen Kinsella (UL) EC4024 February 8, 2011 54 / 149 Today Basics of probability Empirical distributions Properties of probability distributions

### 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

### 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

### PROBABILITY NOTIONS. Summary. 1. Random experiment

PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non

### Probability: 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

### 33 Probability: Some Basic Terms

33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been

### 7.1 Sample space, events, probability

7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.

### Chapter 5: Probability: What are the Chances? Probability: What Are the Chances? 5.1 Randomness, Probability, and Simulation

Chapter 5: Probability: What are the Chances? Section 5.1 Randomness, Probability, and Simulation The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 5 Probability: What Are

### 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

### Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all

### MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions )

MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions ) Last modified: May 1, 2012 Reading Dineen, pages 57-67 Proof

### 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

### Chapter 1 Axioms of Probability. Wen-Guey Tzeng Computer Science Department National Chiao University

Chapter 1 Axioms of Probability Wen-Guey Tzeng Computer Science Department National Chiao University Introduction Luca Paccioli(1445-1514), Studies of chances of events Niccolo Tartaglia(1499-1557) Girolamo

### Discrete Mathematics Lecture 5. Harper Langston New York University

Discrete Mathematics Lecture 5 Harper Langston New York University Empty Set S = {x R, x 2 = -1} X = {1, 3}, Y = {2, 4}, C = X Y (X and Y are disjoint) Empty set has no elements Empty set is a subset of

### 4: Probability. What is probability? Random variables (RVs)

4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

### 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

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

### MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics

MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you should

### 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

### 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

### 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,

### 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

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

### 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

### 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

### Using Laws of Probability. Sloan Fellows/Management of Technology Summer 2003

Using Laws of Probability Sloan Fellows/Management of Technology Summer 2003 Uncertain events Outline The laws of probability Random variables (discrete and continuous) Probability distribution Histogram

### 1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.

1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first

### 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

### Chapter 5 Section 2 day 1 2014f.notebook. November 17, 2014. Honors Statistics

Chapter 5 Section 2 day 1 2014f.notebook November 17, 2014 Honors Statistics Monday November 17, 2014 1 1. Welcome to class Daily Agenda 2. Please find folder and take your seat. 3. Review Homework C5#3

Chapter 3 Probability 2012 Pearson Education, Inc. All rights reserved. 1 of 20 Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition

### Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

### 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)

### Chapter 5 - Probability

Chapter 5 - Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set

### Basic Probability Theory I

A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population

### 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

### 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

### Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.

Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111

### 4.1 4.2 Probability Distribution for Discrete Random Variables

4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.

### 1 Limiting distribution for a Markov chain

Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

### Probability Models.S1 Introduction to Probability

Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are

### COMMON CORE STATE STANDARDS FOR

COMMON CORE STATE STANDARDS FOR Mathematics (CCSSM) High School Statistics and Probability Mathematics High School Statistics and Probability Decisions or predictions are often based on data numbers in

### A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.

Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome

### Elements of probability theory

2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted

### Notes 3 Autumn Independence. Two events A and B are said to be independent if

MAS 108 Probability I Notes 3 Autumn 2005 Independence Two events A and B are said to be independent if P(A B) = P(A) P(B). This is the definition of independence of events. If you are asked in an exam