# Probabilities and Random Variables

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 that we can draw inferences about them. The fundamental mathematical object is a triple (Ω, F, P ) called the probability space. A probability space is needed for each experiment or collection of experiments that we wish to describe mathematically. The ingredients of a probability space are a sample space Ω, a collection F of events, and a probability measure P. Let us examine each of these in turn. 1.1 The sample space Ω This is the set of all the possible outcomes of the experiment. Elements of Ω are called sample points and typically denoted by ω. These examples should clarify its meaning: Example 1. If the experiment is a roll of a six-sided die, then the natural sample space is Ω = {1, 2, 3, 4, 5, 6}. Each sample point is a natural number between 1 and 6. Example 2. Suppose the experiment consists of tossing a coin three times. Let us write 0 for heads and 1 for tails. The sample space must contain all the possible outcomes of the 3 successive tosses, in other words, all triples of 0 s and 1 s: Ω = {0, 1} 3 = {0, 1} {0, 1} {0, 1} = {(x 1, x 2, x 3 ) : x i {0, 1} for i = 1, 2, 3} = {(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}. The four formulas are examples of different ways of writing down the set Ω. A sample point ω = (0, 1, 0) means that the first and third tosses come out heads (0) and the second toss comes out tails (1). Example 3. Suppose the experiment consists of tossing a coin infinitely many times. Even though such an experiment cannot be physically arranged, it is of central importance to the theory to be able to handle idealized situations of this kind. A sample point ω is now an infinite sequence ω = (x 1, x 2, x 3,..., x i,...) = (x i ) whose terms x i are each 0 or 1. The interpretation is that x i = 0 (1) if the ith toss came out heads (tails). In this model the infinite sequence of coin tosses is regarded as a single experiment, although we may choose to observe the individual tosses one at a time. (If you need a mental picture, think of the goddess of chance simultaneously tossing all infinitely many coins.) The sample space Ω is the space of all infinite sequences of 0 s and 1 s: Ω = {ω = (x i ) : each x i is either 0 or 1 } = {0, 1} N. 1

2 In the last formula N = {1, 2, 3,...} stands for the set of natural numbers, and the notation {0, 1} N stands for the infinite product set {0, 1} N = {0, 1} {0, 1} {0, 1} {0, 1}. Example 4. If our experiment consists in observing the number of customers that arrive at a service desk during a fixed time period, the sample space should be the set of nonnegative integers: Ω = Z + = {0, 1, 2, 3,...}. Example 5. If the experiment consists in observing the lifetime of a light bulb, then a natural sample space would be the set of nonnegative real numbers: Ω = R + = [0, ). These sample spaces are mathematically very different. Examples 1 and 2 have finite sample spaces, while the rest are infinite. The sample space in Example 4 is countably infinite, which means that its elements can be arranged in a sequence. Finite and countably infinite spaces are also called discrete. Probability theory on discrete sample spaces requires no advanced mathematics beyond calculus and linear algebra, and that is the main focus of this course. A precise treatment of probability models on uncountable spaces requires measure theory. 1.2 The collection of events, F Events are simply subsets of the sample space. They can often be described both in words and in set-theoretic notation. Events are typically denoted by upper case letters: A, B, C, etc. Here are some possible events on the sample spaces described above: Example 1. A = {the outcome of the die is even} = {2, 4, 6}. Example 2. A = {exactly two tosses come out tails} = {(0, 1, 1), (1, 0, 1), (1, 1, 0)}. Example 3. A = {the second toss comes out tails and the fifth heads} = {ω = (x i ) {0, 1} N : x 2 = 1 and x 5 = 0 }. Example 4. Example 5. A = {at least 6 customers arrived} = {6, 7, 8, 9,...}. A = {the bulb lasted less than 4 hours} = [0, 4). In discrete sample spaces the class F of events contains all subsets of the sample space. But in more complicated sample spaces this cannot be the case. In the general theory F is a σ-algebra in the sample space. This means that F satisfies the following axioms: (i) Ω F and F (The sample space Ω and the empty set are events.) 2

3 (ii) If A F then A c F. (A c is the complement of A, this is the set of points of Ω that do not belong in A.) (iii) If A 1, A 2, A 3,... F then also also an event.) A i F. (In words: the union of a sequence of events is 1.3 The probability measure P This is a function on events that gives the probability P (A) of each event A in F. It must satisfy these axioms: (i) 0 P (A) 1 for all A F. (ii) P ( ) = 0 and P (Ω) = 1. (iii) If A 1, A 2, A 3,... are pairwise disjoint events, meaning that A i A j = whenever i j, then ( P A i ) = P (A i ). The necessity to restrict F in uncountable sample spaces arises from the fact that we cannot always consistently define probabilities for all subsets of uncountable sample spaces. The way around this difficulty is to admit as events only members of certain judiciously defined σ-algebras. These complications cannot be appreciated without some study of measure theory. The good news is that in practice one never encounters these bad events that the theory cannot handle. It is a hard exercise in real analysis to construct a nonmeasurable set on R which lies outside the natural σ-algebra. In discrete sample spaces these difficulties can be completely avoided. Suppose our sample space Ω is either finite or countably infinite. To define a probability measure on Ω we only need an assigment P (ω) of probabilities for all sample points ω that together satisfy and 0 P (ω) 1 for all ω Ω P (ω) = 1. ω Ω Then for any subset A of Ω we can define a probability by P (A) = ω A P (ω). (1) This concrete construction shows that there is no problem in giving each subset A Ω a probability, and it is easy to check that axioms (i) (iii) for P above are satisfied. This approach does not work for uncountable spaces such as the real line. The interesting probability measures there typically give individual sample points zero probability, and the summation in (1) has to be replaced by integration. 3

4 Here are examples of natural probability measures on the previously defined sample spaces: Example 1. If we assume that the die is fair, then it is reasonable to regard each outcome as equally likely, and the appropriate probability measure is P (ω) = 1/6 for each ω Ω. Example 2. If the coin is fair, then each outcome should be equally likely: P (ω) = 1/8 for each ω Ω. If the coin is not fair but gives tails with probability p and heads with probability 1 p for some fixed number p (0, 1), then the appropriate P is defined for sample points ω = (x 1, x 2, x 3 ) by P (x 1, x 2, x 3 ) = p x 1+x 2 +x 3 (1 p) 3 (x 1+x 2 +x 3 ). The uniform measure P (ω) = 1/8 is recovered in the case p = 1/2. Example 3. Defining a probability measure on the space of infinite sequences requires some measuretheoretic machinery. We will not concern ourselves with these technical issues. We simply take for granted that it is possible to construct the probability measures on Ω that we need. For example, there is a probability measure P on Ω such that, for any choice of numbers a 1, a 2,..., a n from {0, 1}, P {ω Ω : ω 1 = a 1,..., ω n = a n } = 2 n. (2) The number 2 n corresponds to a fair coin (p = 1/2 in Example 2), and the construction can be performed for any p just as well. The result from measure theory that makes all this work is called Kolmogorov s extension theorem. Example 4. A natural choice of probability measure here could be a Poisson distribution: P (ω) = e λ λ ω ω! for ω = 0, 1, 2, 3,... (3) The number λ > 0 is a parameter of the model that in an actual modeling task would be chosen to fit the observed data. Example 5. A possible choice here would be an exponential distribution with parameter λ > 0. This can be defined by saying that the probability of an interval [a, b] with 0 a < b < is given by P [a, b] = b a λe λx dx. Here f(x) = λe λx is the density function of the exponential distribution with parameter λ. Equivalently, the same probability measure can be defined by giving the (cumulative) distribution function F (x) = P [0, x] = 1 e λx, x 0. 2 Random Variables and Their Expectations From now on the discussion always takes place in the context of some fixed probability space (Ω, F, P ). You may assume that Ω is discrete, that F is the collection of all subsets of Ω, and that P is defined by (1) from some given numbers {P (ω) : ω Ω} that give the probabilities of individual sample points. 4

5 A random variable is a function defined on the sample space Ω. While functions in calculus are typically denoted by letters such as f or g, random variables are often denoted by capital letters such as X, Y and Z. Thus a random variable X associates a number X(ω) to each sample point ω. For each event A there is an indicator random variable I A defined by { 1, if ω A I A (ω) = 0, if ω / A. Other common notations for an indicator random variable are 1 A and χ A. If X 1, X 2,..., X n are random variables defined on some common probability space, then X = (X 1, X 2,..., X n ) defines an R n -valued random variable, also called a random vector. Most interesting events are of the type {ω Ω : X(ω) B}, where X is a random variable and B is a subset of the space into which X maps (so typically the real line R or, in the case of a random vector, a Euclidean space R n ). The above notation is usually abbreviated as {X B}, and it reads the event that the value of X lies in the set B. Examples: Example 2. Let the random variable Y be the outcome of the third toss. If we denote the elements of Ω by ω = (x 1, x 2, x 3 ), then Y is defined by Y (ω) = x 3. Suppose you receive \$5 each time the coin comes up heads. Let Z be the amount you won. Then you could express Z as a function on Ω by writing Z(ω) = 5(3 (x 1 + x 2 + x 3 )). The event {Z = 10} is the subset of sample points ω that satisfy the condition Z(ω) = 10. So {Z = 10} = {ω Ω : Z(ω) = 10} = {(0, 0, 1), (0, 1, 0), (1, 0, 0)}. The probability measure P on the sample space gives the probabilities of the values of a random variable. For example, in the case of a fair coin, P {Z = 10} = P {(0, 0, 1), (0, 1, 0), (1, 0, 0)} = 3/8. Example 3. In the sequence setting it is customary to use these so-called coordinate random variables X i : For a sample point ω = (x i ), define X i(ω) = x i. If our probability measure P is the one given by formula (2), then for any choice of numbers a 1, a 2,..., a n from {0, 1}, If S n is the number of tails in the first n tosses, then P {X 1 = a 1,..., X n = a n } = 2 n. (4) S n = n X i. (5) Here we are defining a new random variable S n in terms of the old ones X 1, X 2, X 3,... The sample point has been left out of formula (5) as is typical to simplify notation. The precise meaning of equation (5) is that for each ω Ω, the number S n (ω) is defined in terms of the numbers X 1 (ω), X 2 (ω), X 3 (ω),... by n S n (ω) = X i (ω). (6) 5

6 Again, probabilities of the values of random variables come from the probability measure on the sample space. From (2) or (4) it follows that P {S 5 = 3} = (the number of ways of getting 3 tails in 5 tosses) 2 5 ( ) 5 = 2 5 = Example 4. Let the random variable X be the number of customers. Then, as a function on Ω, X is somewhat trivial, namely the identity function: X(ω) = ω. Formula (3) from above can be expressed as P (X = k) = e λ λ k for k = 0, 1, 2, 3,... (7) k! Now we say that the random variable X is Poisson distributed with parameter λ. Examples 3 and 4 are instructive in the sense that we prefer to express things in terms of random variables. That is, we prefer formulas (4) and (7) over formulas (2) and (3). The probability space recedes to the background and sometimes vanishes entirely from the discussion. In example 4 we would simply say let X be a random variable with Poisson(λ) distribution by which mean that (6) holds. No probability space needs to be specified because it is understood that the simple probability space of Example 4 can be imagined in the background. No matter what the original probability space (Ω, F, P ) is, we can often translate all the relevant probabilistic information to a more familiar space such as R or Z + by using the distribution of a random variable. Suppose first that X is a Z + -valued random variable. (This means that X is a function from Ω into Z +.) The distribution (or probability distribution) of X is a sequence (p k ) k=0 of numbers defined by p k = P (X = k), k = 0, 1, 2, 3,... (8) Thus the distribution of X is actually a probability measure on Z +. All probabilistic statements about X can be expressed in terms of its distribution, so we can forget the original probability space. Sometimes we need to allow for the possibility that X takes on the value. Then an additional number p = P (X = ) = 1 needs to be added to the distribution. If we want to talk about more than one random variable simultaneously, we need joint distributions. Suppose X 1, X 2,..., X n are Z + -valued random variables defined on a common probability space. Their joint distribution is a collection k=0 (p k1,...,k n : k 1,..., k n Z + ) of numbers indexed by the n-tuples (k 1,..., k n ) of nonnegative integers, and defined by p k p k1,...,k n = P (X 1 = k 1,..., X n = k n ). An example: formula (4) above specifies the joint distribution of the first n coin tosses. Similar formulas are valid when the range of X is some other countable subset of R instead of Z +. But when the range of X is not countable, a sequence of numbers such as (8) cannot specify its 6

7 distribution. Then we need the (cumulative) distribution function (c.d.f.) of X, denoted by F and defined for all real numbers x by F (x) = P (X x). If F is differentiable, then f(x) = F (x) is the probability density of X. The expectation of a random variable X is a number defined by E[X] = x xp (X = x) (9) if X has only countably many possible values x, and by E[X] = + xf(x)dx if X has density f, provided the sum (integral) exists. Both formulas can be subsumed under the formula E[X] = + x df (x), where F is the c.d.f. of X, and the integral is interpreted as a Riemann-Stieltjes integral. (The Riemann-Stieltjes integral is a topic of a course on advanced calculus or beginning real analysis. Do not worry even if you are not familiar with it.) It follows from formula (9) that if X is Z + -valued, then E[X] = kp k, and furthermore, if f is a function defined on Z +, then the composition f(x) is a new random variable and it has expectation E[f(X)] = f(k)p k. Note that E[X] = if P (X = ) > 0. The expectation of an indicator random variable is the probability of the event in question, as an easy calculation shows: E[I A ] = 1 P (I A = 1) + 0 P (I A = 0) = 1 P (A) + 0 P (A c ) = P (A). k=0 k=0 Example 2. Let us calculate the expectation of the reward Z, assuming the coin is fair: E[Z] = 0 P (Z = 0) + 5 P (Z = 5) + 10 P (Z = 10) + 15 P (Z = 15) ( ) 3 = 5 1 ( ) ( ) = =

8 An important property of the expectation is its additivity: Suppose X 1,..., X n are random variables defined on a common probability space and c 1,..., c n are numbers. Then c 1 X c n X n is also a random variable and E[c 1 X c n X n ] = c 1 E[X 1 ] + + c n E[X n ]. 3 Independence and Conditioning Suppose X 1,..., X n are discrete random variables (they have countable ranges) defined on a common probability space. They are mutually independent if the following holds for all possible values y 1,..., y n of their range: P (X 1 = y 1, X 2 = y 2,..., X n = y n ) = P (X 1 = y 1 ) P (X 2 = y 2 ) P (X n = y n ). (10) A collection of events A 1,..., A n are mutually independent if their indicator random variables are mutually independent. Given an event B such that P (B) > 0, we can define a new probability measure P ( B) on Ω by conditioning on B: The conditional probability of an event A, given B, is defined by P (A B) = P (A B). P (B) You can think of conditioning as first restricting the sample space to the set B, and then renormalizing the probability measure by dividing by P (B) so that the total (new) space again has probability 1. Check that the following holds: Two events A and B are independent if and only if P (A B) = P (A) and P (B A) = P (B). The equality P (A B) = P (A) means that the event B gives no information about the event A, in the sense that the probability of A s occurrence is not influenced by whether or not B occurred. This then is the intuitive meaning of statistical independence: If X and Y are independent random variables, knowledge of the value of X should give us no information about the value of Y. Independence splits the expectation of a product of random variables into a product of expectations: Suppose X 1,..., X n are mutually independent random variables whose expectations are finite. Then the product X 1 X 2 X 3 X n is a random variable, and E[X 1 X 2 X 3 X n ] = E[X 1 ] E[X 2 ] E[X 3 ] E[X n ]. 8

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES Contents 1. Probabilistic experiments 2. Sample space 3. Discrete probability

### 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.

### 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

### 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

### 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

### 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

### 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

### Math 421: Probability and Statistics I Note Set 2

Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the

### 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,

### 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............................

### 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

### 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,

### Measure Theory and Probability

Chapter 2 Measure Theory and Probability 2.1 Introduction In advanced probability courses, a random variable (r.v.) is defined rigorously through measure theory, which is an in-depth mathematics discipline

### Chapter 1. Probability Theory: Introduction. Definitions Algebra. RS Chapter 1 Random Variables

Chapter 1 Probability Theory: Introduction Definitions Algebra Definitions: Semiring A collection of sets F is called a semiring if it satisfies: Φ F. If A, B F, then A B F. If A, B F, then there exists

### Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

### 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

### STAT 571 Assignment 1 solutions

STAT 571 Assignment 1 solutions 1. If Ω is a set and C a collection of subsets of Ω, let A be the intersection of all σ-algebras that contain C. rove that A is the σ-algebra generated by C. Solution: Let

### 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

### 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

### RENEWAL THEORY AND ITS APPLICATIONS

RENEWAL THEORY AND ITS APPLICATIONS PETER NEBRES Abstract. This article will delve into renewal theory. It will first look at what a random process is and then explain what renewal processes are. It will

### Topic 2: Scalar random variables. Definition of random variables

Topic 2: Scalar random variables Discrete and continuous random variables Probability distribution and densities (cdf, pmf, pdf) Important random variables Expectation, mean, variance, moments Markov and

### 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)

### 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

### Math/Stat 370: Engineering Statistics, Washington State University

Math/Stat 370: Engineering Statistics, Washington State University Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2 Haijun Li Math/Stat 370: Engineering 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 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

### 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

### 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

### MATH INTRODUCTION TO PROBABILITY FALL 2011

MATH 7550-01 INTRODUCTION TO PROBABILITY FALL 2011 Lecture 4. Measurable functions. Random variables. I have told you that we can work with σ-algebras generated by classes of sets in some indirect ways.

### Notes for Math 450 Lecture Notes 2

Notes for Math 450 Lecture Notes 2 Renato Feres 1 Probability Spaces We first explain the basic concept of a probability space, (Ω, F, P ). This may be interpreted as an experiment with random outcomes.

### Notes 2 for Honors Probability and Statistics

Notes 2 for Honors Probability and Statistics Ernie Croot August 24, 2010 1 Examples of σ-algebras and Probability Measures So far, the only examples of σ-algebras we have seen are ones where the sample

### Introduction to Random Variables (RVs)

ECE270: Handout 5 Introduction to Random Variables (RVs) Outline: 1. informal definition of a RV, 2. three types of a RV: a discrete RV, a continuous RV, and a mixed RV, 3. a general rule to find probability

### IEOR 4106: Introduction to Operations Research: Stochastic Models. SOLUTIONS to Homework Assignment 1

IEOR 4106: Introduction to Operations Research: Stochastic Models SOLUTIONS to Homework Assignment 1 Probability Review: Read Chapters 1 and 2 in the textbook, Introduction to Probability Models, by Sheldon

### MATH 201. Final ANSWERS August 12, 2016

MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6-sided dice, five 8-sided dice, and six 0-sided dice. A die is drawn from the bag and then rolled.

### 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

### MATH41112/61112 Ergodic Theory Lecture Measure spaces

8. Measure spaces 8.1 Background In Lecture 1 we remarked that ergodic theory is the study of the qualitative distributional properties of typical orbits of a dynamical system and that these properties

### 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

### 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.

### Stat 110. Unit 3: Random Variables Chapter 3 in the Text

Stat 110 Unit 3: Random Variables Chapter 3 in the Text 1 Unit 3 Outline Random Variables (RVs) and Distributions Discrete RVs and Probability Mass Functions (PMFs) Bernoulli and Binomial Distributions

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### 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

### 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

### 3. Continuous Random Variables

3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such

### NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

### 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

### 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

### Elements of Probability Distribution Theory

Chapter 1 Elements of Probability Distribution Theory 1.1 Introductory Definitions Statistics gives us methods to make inference about a population based on a random sample representing this population.

### 17.Convergence of Random Variables

17.Convergence of Random Variables In elementary mathematics courses (such as Calculus) one speaks of the convergence of functions: f n : R R, then n f n = f if n f n (x) = f(x) for all x in R. This is

### 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

### MATH 56A SPRING 2008 STOCHASTIC PROCESSES 31

MATH 56A SPRING 2008 STOCHASTIC PROCESSES 3.3. Invariant probability distribution. Definition.4. A probability distribution is a function π : S [0, ] from the set of states S to the closed unit interval

### Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014

STAT511 Spring 2014 Lecture Notes 1 Chapter 3: Discrete Random Variable and Probability Distribution January 28, 2014 3 Discrete Random Variables Chapter Overview Random Variable (r.v. Definition Discrete

### Probability II (MATH 2647)

Probability II (MATH 2647) Lecturer Dr. O. Hryniv email Ostap.Hryniv@durham.ac.uk office CM309 http://maths.dur.ac.uk/stats/courses/probmc2h/probability2h.html or via DUO This term we shall consider: Review

### 3. Examples of discrete probability spaces.. Here α ( j)

3. EXAMPLES OF DISCRETE PROBABILITY SPACES 13 3. Examples of discrete probability spaces Example 17. Toss n coins. We saw this before, but assumed that the coins are fair. Now we do not. The sample space

### 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.

### 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

### 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:

### CHAPTER III - MARKOV CHAINS

CHAPTER III - MARKOV CHAINS JOSEPH G. CONLON 1. General Theory of Markov Chains We have already discussed the standard random walk on the integers Z. A Markov Chain can be viewed as a generalization of

### f(x) is a singleton set for all x A. If f is a function and f(x) = {y}, we normally write

Math 525 Chapter 1 Stuff If A and B are sets, then A B = {(x,y) x A, y B} denotes the product set. If S A B, then S is called a relation from A to B or a relation between A and B. If B = A, S A A is called

### Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### 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,

### General theory of stochastic processes

CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result

### Introduction. Purpose of the Course. The purpose of this course is to study mathematically the behaviour of stochastic systems.

1 Introduction Purpose of the Course The purpose of this course is to study mathematically the behaviour of stochastic systems. Examples 1. A CPU with jobs arriving to it in a random fashion. 2. A communications

### Sums of Independent Random Variables

Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables

### Stochastic Processes and Advanced Mathematical Finance. Laws of Large Numbers

Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebraska-Lincoln Lincoln, NE 68588-0130 http://www.math.unl.edu Voice: 402-472-3731 Fax: 402-472-8466 Stochastic Processes and Advanced

### 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if. lim sup. j E j = E.

Real Analysis, course outline Denis Labutin 1 Measure theory I 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if (a), Ω A ; (b)

### Lecture 11: Probability models

Lecture 11: Probability models Probability is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model we need the following ingredients A sample

### 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

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### 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

### You flip a fair coin four times, what is the probability that you obtain three heads.

Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

### 3. Probability Measure

Virtual Laboratories > 1. Probability Spaces > 1 2 3 4 5 6 7 3. Probability Measure Definitions and Interpretations Suppose that we have a random experiment with sample space S. Intuitively, the probability

### 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

### Structure of Measurable Sets

Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations

### A Measure Theory Tutorial (Measure Theory for Dummies)

A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008

### 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............................................

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Mathematical foundations of Econometrics

Mathematical foundations of Econometrics G.Gioldasis, UniFe & prof. A.Musolesi, UniFe April 7, 2016.Gioldasis, UniFe & prof. A.Musolesi, UniFe Mathematical foundations of Econometrics April 7, 2016 1 /

### ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

### Finite Markov Chains and Algorithmic Applications. Matematisk statistik, Chalmers tekniska högskola och Göteborgs universitet

Finite Markov Chains and Algorithmic Applications Olle Häggström Matematisk statistik, Chalmers tekniska högskola och Göteborgs universitet PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

### Measure Theory. 1 Measurable Spaces

1 Measurable Spaces Measure Theory A measurable space is a set S, together with a nonempty collection, S, of subsets of S, satisfying the following two conditions: 1. For any A, B in the collection S,

### NOTES ON ELEMENTARY PROBABILITY

NOTES ON ELEMENTARY PROBABILITY KARL PETERSEN 1. Probability spaces Probability theory is an attempt to work mathematically with the relative uncertainties of random events. In order to get started, we

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### 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

### Fibonacci Numbers Modulo p

Fibonacci Numbers Modulo p Brian Lawrence April 14, 2014 Abstract The Fibonacci numbers are ubiquitous in nature and a basic object of study in number theory. A fundamental question about the Fibonacci

### Continuous random variables

Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the so-called continuous random variables. A random

### 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

### 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

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

### 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

### Random Variables and Measurable Functions.

Chapter 3 Random Variables and Measurable Functions. 3.1 Measurability Definition 42 (Measurable function) Let f be a function from a measurable space (Ω, F) into the real numbers. We say that the function

### Random Variables and Their Expected Values

Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution

### S = {1, 2,..., n}. P (1, 1) P (1, 2)... P (1, n) P (2, 1) P (2, 2)... P (2, n) P = . P (n, 1) P (n, 2)... P (n, n)

last revised: 26 January 2009 1 Markov Chains A Markov chain process is a simple type of stochastic process with many social science applications. We ll start with an abstract description before moving

### Random Variables. Consider a probability model (Ω, P ). Discrete Random Variables Chs. 2, 3, 4. Definition. A random variable is a function

Rom Variables Discrete Rom Variables Chs.,, 4 Rom Variables Probability Mass Functions Expectation: The Mean Variance Special Distributions Hypergeometric Binomial Poisson Joint Distributions Independence

### Continuous Random Variables

Continuous Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Continuous Random Variables 2 11 Introduction 2 12 Probability Density Functions 3 13 Transformations 5 2 Mean, Variance and Quantiles

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### 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

### 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