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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 Probability Theory A random variable is neither random nor variable. Gian-Carlo Rota, M.I.T.. Florian Herzog 2013

2 Probability space Probability space A probability space W is a unique triple W = {Ω, F, P }: Ω is its sample space F its σ-algebra of events P its probability measure Remarks: (1) The sample space Ω is the set of all possible samples or elementary events ω: Ω = {ω ω Ω}. (2)The σ-algebra F is the set of all of the considered events A, i.e., subsets of Ω: F = {A A Ω, A F}. (3) The probability measure P assigns a probability P (A) to every event A F: P : F [0, 1]. Stochastic Systems,

3 Sample space The sample space Ω is sometimes called the universe of all samples or possible outcomes ω. Example 1. Sample space Toss of a coin (with head and tail): Ω = {H, T }. Two tosses of a coin: Ω = {HH, HT, T H, T T }. A cubic die: Ω = {ω 1, ω 2, ω 3, ω 4, ω 5, ω 6 }. The positive integers: Ω = {1, 2, 3,... }. The reals: Ω = {ω ω R}. Note that the ωs are a mathematical construct and have per se no real or scientific meaning. The ωs in the die example refer to the numbers of dots observed when the die is thrown. Stochastic Systems,

4 Event An event A is a subset of Ω. If the outcome ω of the experiment is in the subset A, then the event A is said to have occurred. The set of all subsets of the sample space are denoted by 2 Ω. Example 2. Events Head in the coin toss: A = {H}. Odd number in the roll of a die: A = {ω 1, ω 3, ω 5 }. An integer smaller than 5: A = {1, 2, 3, 4}, where Ω = {1, 2, 3,... }. A real number between 0 and 1: A = [0, 1], where Ω = {ω ω R}. We denote the complementary event of A by A c = Ω\A. When it is possible to determine whether an event A has occurred or not, we must also be able to determine whether A c has occurred or not. Stochastic Systems,

5 Probability Measure I Definition 1. Probability measure A probability measure P on the countable sample space Ω is a set function P : F [0, 1], satisfying the following conditions P(Ω) = 1. P(ω i ) = p i. If A 1, A 2, A 3,... F are mutually disjoint, then ( ) P A i = i=1 P (A i ). i=1 Stochastic Systems,

6 Probability The story so far: Sample space: Ω = {ω 1,..., ω n }, finite! Events: F = 2 Ω : All subsets of Ω Probability: P (ω i ) = p i P (A Ω) = w i A p i Probability axioms of Kolmogorov (1931) for elementary probability: P(Ω) = 1. If A Ω then P (A) 0. If A 1, A 2, A 3,... Ω are mutually disjoint, then ( ) P A i = i=1 P (A i ). i=1 Stochastic Systems,

7 Uncountable sample spaces Most important uncountable sample space for engineering: R, resp. R n. Consider the example Ω = [0, 1], every ω is equally likely. Obviously, P(ω) = 0. Intuitively, P([0, a]) = a, basic concept: length! Question: Has every subset of [0, 1] a determinable length? Answer: No! (e.g. Vitali sets, Banach-Tarski paradox) Question: Is this of importance in practice? Answer: No! Question: Does it matter for the underlying theory? Answer: A lot! Stochastic Systems,

8 Fundamental mathematical tools Not every subset of [0, 1] has a determinable length collect the ones with a determinable length in F. Such a mathematical construct, which has additional, desirable properties, is called σ-algebra. Definition 2. σ-algebra A collection F of subsets of Ω is called a σ-algebra on Ω if Ω F and F ( denotes the empty set) If A F then Ω\A = A c F: The complementary subset of A is also in Ω For all A i F: i=1 A i F The intuition behind it: collect all events in the σ-algebra F, make sure that by performing countably many elementary set operation (,, c ) on elements of F yields again an element in F (closeness). The pair {Ω, F} is called measure space. Stochastic Systems,

9 Example of σ-algebra Example 3. σ-algebra of two coin-tosses Ω = {HH, HT, T H, T T } = {ω 1, ω 2, ω 3, ω 4 } F min = {, Ω} = {, {ω 1, ω 2, ω 3, ω 4 }}. F 1 = {, {ω 1, ω 2 }, {ω 3, ω 4 }, {ω 1, ω 2, ω 3, ω 4 }}. F max = {, {ω 1 }, {ω 2 }, {ω 3 }, {ω 4 }, {ω 1, ω 2 }, {ω 1, ω 3 }, {ω 1, ω 4 }, {ω 2, ω 3 }, {ω 2, ω 4 }, {ω 3, ω 4 }, {ω 1, ω 2, ω 3 }, {ω 1, ω 2, ω 4 }, {ω 1, ω 3, ω 4 }, {ω 2, ω 3, ω 4 }, Ω}. Stochastic Systems,

10 Generated σ-algebras The concept of generated σ-algebras is important in probability theory. Definition 3. σ(c): σ-algebra generated by a class C of subsets Let C be a class of subsets of Ω. The σ-algebra generated by C, denoted by σ(c), is the smallest σ-algebra F which includes all elements of C, i.e., C F. Identify the different events we can measure of an experiment (denoted by A), we then just work with the σ-algebra generated by A and have avoided all the measure theoretic technicalities. Stochastic Systems,

11 Borel σ-algebra The Borel σ-algebra includes all subsets of R which are of interest in practical applications (scientific or engineering). Definition 4. Borel σ-algebra B(R)) The Borel σ-algebra B(R) is the smallest σ-algebra containing all open intervals in R. The sets in B(R) are called Borel sets. The extension to the multi-dimensional case, B(R n ), is straightforward. (, a), (b, ), (, a) (b, ) [a, b] = (, a) (b, ), (, a] = n=1 [a n, a] and [b, ) = n=1 [b, b + n], (a, b] = (, b] (a, ), {a} = n=1 (a 1 n, a + 1 n ), {a 1,, a n } = n k=1 a k. Stochastic Systems,

12 Measure Definition 5. Measure Let F be the σ-algebra of Ω and therefore (Ω, F) be a measurable space. The map µ : F [0, ] is called a measure on (Ω, F) if µ is countably additive. The measure µ is countably additive (or σ-additive) if µ( ) = 0 and for every sequence of disjoint sets (F i : i N) in F with F = i N F i we have µ(f ) = i N µ(f i ). If µ is countably additive, it is also additive, meaning for every F, G F we have µ(f G) = µ(f ) + µ(g) if and only if F G = The triple (Ω, F, µ) is called a measure space. Stochastic Systems,

13 Lebesgue Measure The measure of length on the straight line is known as the Lebesgue measure. Definition 6. Lebesgue measure on B(R) The Lebesgue measure on B(R), denoted by λ, is defined as the measure on (R, B(R)) which assigns the measure of each interval to be its length. Examples: Lebesgue measure of one point: λ({a}) = 0. Lebesgue measure of countably many points: λ(a) = i=1 λ({a i}) = 0. The Lebesgue measure of a set containing uncountably many points: zero positive and finite infinite Stochastic Systems,

14 Probability Measure Definition 7. Probability measure A probability measure P on the sample space Ω with σ-algebra F is a set function P : F [0, 1], satisfying the following conditions P(Ω) = 1. If A F then P (A) 0. If A 1, A 2, A 3,... F are mutually disjoint, then ( ) P A i = i=1 P (A i ). i=1 The triple (Ω, F, P ) is called a probability space. Stochastic Systems,

15 F-measurable functions Definition 8. F-measurable function The function f : Ω R defined on (Ω, F, P ) is called F-measurable if f 1 (B) = {ω Ω : f(ω) B} F for all B B(R), i.e., the inverse f 1 maps all of the Borel sets B R to F. Sometimes it is easier to work with following equivalent condition: y R {ω Ω : f(ω) y} F This means that once we know the (random) value X(ω) we know which of the events in F have happened. F = {, Ω}: only constant functions are measurable F = 2 Ω : all functions are measurable Stochastic Systems,

16 F-measurable functions Ω: Sample space, A i : Event, F = σ(a 1,..., A n ): σ-algebra of events X(ω) : Ω R R ω A 1 A 2 X(ω) R (, a] B A 3 A 4 X 1 : B F Ω X: random variable, B: Borel σ-algebra Stochastic Systems,

17 F-measurable functions - Example Roll of a die: Ω = {ω 1, ω 2, ω 3, ω 4, ω 5, ω 6 } We only know, whether an even or and odd number has shown up: F = {, {ω 1, ω 3, ω 5 }, {ω 2, ω 4, ω 6 }, Ω} = σ({ω 1, ω 3, ω 5 }) Consider the following random variable: f(ω) = { 1, if ω = ω1, ω 2, ω 3 ; 1, if ω = ω 4, ω 5, ω 6. Check measurability with the condition y R {ω Ω : f(ω) y} F {ω Ω : f(ω) 0} = {ω 4, ω 5, ω 6 } / F f is not F-measurable. Stochastic Systems,

18 Lebesgue integral I Definition 9. Lebesgue Integral (Ω, F) a measure space, µ : Ω R a measure, f : Ω R is F-measurable. If f is a simple function, i.e., f(x) = c i, for all x A i, c i R Ω fdµ = n c i µ(a i ). If f is nonnegative, we can always construct a sequence of simple functions f n with f n (x) f n+1 (x) which converges to f: lim n f n (x) = f(x). With this sequence, the Lebesgue integral is defined by fdµ = lim f n dµ. Ω n Ω i=1 Stochastic Systems,

19 Lebesgue integral II Definition 10. Lebesgue Integral (Ω, F) a measure space, µ : Ω R a measure, f : Ω R is F-measurable. If f is an arbitrary, measurable function, we have f = f + f with f + (x) = max(f(x), 0) and f (x) = max( f(x), 0), and then define Ω fdµ = Ω f + dp Ω f dp. The integral above may be finite or infinite. It is not defined if Ω f + dp and Ω f dp are both infinite. Stochastic Systems,

20 Riemann vs. Lebesgue The most important concept of the Lebesgue integral is that the limit of approximate sums (as the Riemann integral): for Ω = R: f(x) f(x) f x x x Stochastic Systems,

21 Riemann vs. Lebesgue integral Theorem 1. Riemann-Lebesgue integral equivalence Let f be a bounded and continuous function on [x 1, x 2 ] except at a countable number of points in [x 1, x 2 ]. Then both the Riemann and the Lebesgue integral with Lebesgue measure µ exist and are the same: x2 x 1 f(x) dx = [x 1,x 2 ] fdµ. There are more functions which are Lebesgue integrable than Riemann integrable. Stochastic Systems,

22 Popular example for Riemann vs. Lebesgue Consider the function f(x) = { 0, x Q; 1, x R\Q. The Riemann integral 1 0 f(x)dx does not exist, since lower and upper sum do not converge to the same value. However, the Lebesgue integral fdλ = 1 [0,1] does exist, since f(x) is the indicator function of x R\Q. Stochastic Systems,

23 Random Variable Definition 11. Random variable A real-valued random variable X is a F-measurable function defined on a probability space (Ω, F, P ) mapping its sample space Ω into the real line R: X : Ω R. Since X is F-measurable we have X 1 : B F. Stochastic Systems,

24 Distribution function Definition 12. Distribution function The distribution function of a random variable X, defined on a probability space (Ω, F, P ), is defined by: F (x) = P (X(ω) x) = P ({ω : X(ω) x}). From this the probability measure of the half-open sets in R is P (a < X b) = P ({ω : a < X(ω) b}) = F (b) F (a). Stochastic Systems,

25 Density function Closely related to the distribution function is the density function. Let f : R R be a nonnegative function, satisfying fdλ = 1. The function f is called R a density function (with respect to the Lebesgue measure) and the associated probability measure for a random variable X, defined on (Ω, F, P ), is for all A F. P ({ω : ω A}) = A fdλ. Stochastic Systems,

26 Important Densities I Poisson density or probability mass function (λ > 0): Univariate Normal density f(x) = λx x! e λ, x = 0, 1, 2,.... f(x) = 1 2πσ e 1 2 ( ) x µ σ. The normal variable is abbreviated as N (µ, σ). Multivariate normal density (x, µ R n ; Σ R n n ): f(x) = 1 (2π)n det(σ) e 1 2 (x µ)t Σ 1 (x µ). Stochastic Systems,

27 Important Densities II Univariate student t-density ν degrees of freedom (x, µ R 1 ; σ R 1 f(x) = Γ(ν+1 2 ) ( Γ( ν 2 ) (x µ) 2 ) 1 2 (ν+1) πνσ ν σ 2 Multivariate student t-density with ν degrees of freedom (x, µ R n ; Σ R n n ): f(x) = Γ( ν+n 2 ) ( Γ( ν 2 ) (πν) n det(σ) ν (x µ)t Σ (ν+n) (x µ)). Stochastic Systems,

28 Important Densities II The chi square distribution with degree-of-freedom (dof) n has the following density )n 2 2 ( f(x) = e x 2 x 2 2Γ( n 2 ) which is abreviated as Z χ 2 (n) and where Γ denotes the gamma function. A chi square distributed random variable Y is created by Y = n i=1 X 2 i where X are independent standard normal distributed random variables N (0, 1). Stochastic Systems,

29 Important Densities III A standard student-t distributed random variable Y is generated by Y = X Z ν, where X N (0, 1) and Z χ 2 (ν). Another important density is the Laplace distribution: p(x) = 1 x µ 2σ e σ with mean µ and diffusion σ. The variance of this distribution is given as 2σ 2. Stochastic Systems,

30 Expectation & Variance Definition 13. Expectation of a random variable The expectation of a random variable X, defined on a probability space (Ω, F, P ), is defined by: E[X] = Ω XdP = Ω xfdλ. With this definition at hand, it does not matter what the sample Ω is. The calculations for the two familiar cases of a finite Ω and Ω R with continuous random variables remain the same. Definition 14. Variance of a random variable The variance of a random variable X, defined on a probability space (Ω, F, P ), is defined by: var(x) = E[(X E[X]) 2 ] = (X E[X]) 2 dp = E[X 2 ] E[X] 2. Ω Stochastic Systems,

31 Normally distributed random variables The shorthand notation X N (µ, σ 2 ) for normally distributed random variables with parameters µ and σ is often found in the literature. The following properties are useful when dealing with normally distributed random variables: If X N (µ, σ 2 )) and Y = ax + b, then Y N (aµ + b, a 2 σ 2 )). If X 1 N (µ 1, σ 2 1 )) and X 2 N (µ 2, σ 2 2 )) then X 1 + X 2 N (µ 1 + µ 2, σ σ2 2 ) (if X 1 and X 2 are independent) Stochastic Systems,

32 Conditional Expectation I From elementary probability theory (Bayes rule): P (A B) = P (A B) P (B) = P (B A)P (A) P (B), P (B) > 0. A A B B Ω E(X B) = E(XI B) P (B), P (B) > 0. Stochastic Systems,

33 Conditional Expectation II General case: (Ω, F, P ) Definition 15. Conditional expectation Let X be a random variable defined on the probability space (Ω, F, P ) with E[ X ] <. Furthermore let G be a sub-σ-algebra of F (G F). Then there exists a random variable Y with the following properties: 1. Y is G-measurable. 2. E[ Y ] <. 3. For all sets G in G we have Y dp = G G XdP, for all G G. The random variable Y = E[X G] is called conditional expectation. Stochastic Systems,

34 Conditional Expectation: Example Y is a piecewise linear approximation of X. Y = E[X G] X(ω) Y X G 1 G 2 G 3 G 4 G 5 ω For the trivial σ-algebra {, Ω}: Y = E[X {, Ω}] = XdP = E[X]. Ω Stochastic Systems,

35 Conditional Expectation: Properties E(E(X F)) = E(X). If X is F-measurable, then E(X F) = X. Linearity: E(αX 1 + βx 2 F) = αe(x 1 F) + βe(x 2 F). Positivity: If X 0 almost surely, then E(X F) 0. Tower property: If G is a sub-σ-algebra of F, then E(E(X F) G) = E(X G). Taking out what is known: If Z is G-measurable, then E(ZX G) = Z E(X G). Stochastic Systems,

36 Summary σ-algebra: collection of the events of interest, closed under elementary set operations Borel σ-algebra: all the events of practical importance in R Lebesgue measure: defined as the length of an interval Density: transforms Lebesgue measure in a probability measure Measurable function: the σ-algebra of the probability space is rich enough Random variable X: a measurable function X : Ω R Expectation, Variance Conditional expectation is a piecewise linear approximation of the underlying random variable. Stochastic Systems,

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 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

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].

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

More information

Probabilities and Random Variables

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

More information

Chapter 1 - σ-algebras

Chapter 1 - σ-algebras Page 1 of 17 PROBABILITY 3 - Lecture Notes Chapter 1 - σ-algebras Let Ω be a set of outcomes. We denote by P(Ω) its power set, i.e. the collection of all the subsets of Ω. If the cardinality Ω of Ω is

More information

The Lebesgue Measure and Integral

The Lebesgue Measure and Integral The Lebesgue Measure and Integral Mike Klaas April 12, 2003 Introduction The Riemann integral, defined as the limit of upper and lower sums, is the first example of an integral, and still holds the honour

More information

Chapter 4 Lecture Notes

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,

More information

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

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

More information

RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis

RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied

More information

Lecture Notes on Measure Theory and Functional Analysis

Lecture Notes on Measure Theory and Functional Analysis Lecture Notes on Measure Theory and Functional Analysis P. Cannarsa & T. D Aprile Dipartimento di Matematica Università di Roma Tor Vergata cannarsa@mat.uniroma2.it daprile@mat.uniroma2.it aa 2006/07 Contents

More information

PROBABILITY AND MEASURE

PROBABILITY AND MEASURE PROBABILITY AND MEASURE J. R. NORRIS Contents 1. Measures 3 2. Measurable functions and random variables 11 3. Integration 18 4. Norms and inequalities 28 5. Completeness of L p and orthogonal projection

More information

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 ) 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

More information

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

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

More information

E3: PROBABILITY AND STATISTICS lecture notes

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............................

More information

ST 371 (IV): Discrete Random Variables

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

More information

Probability and Statistics

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

More information

Lecture 4: Probability Spaces

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

More information

4. Joint Distributions

4. Joint Distributions Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

More information

A Measure Theory Tutorial (Measure Theory for Dummies)

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

More information

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

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

More information

Notes on Continuous Random Variables

Notes on Continuous Random Variables Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

More information

1 Definition of Conditional Expectation

1 Definition of Conditional Expectation 1 DEFINITION OF CONDITIONAL EXPECTATION 4 1 Definition of Conditional Expectation 1.1 General definition Recall the definition of conditional probability associated with Bayes Rule P(A B) P(A B) P(B) For

More information

Our goal first will be to define a product measure on A 1 A 2.

Our goal first will be to define a product measure on A 1 A 2. 1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

More information

Lebesgue Measure on R n

Lebesgue Measure on R n 8 CHAPTER 2 Lebesgue Measure on R n Our goal is to construct a notion of the volume, or Lebesgue measure, of rather general subsets of R n that reduces to the usual volume of elementary geometrical sets

More information

Measure Theory and Probability

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

More information

4.1 4.2 Probability Distribution for Discrete Random Variables

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.

More information

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

More information

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 )

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

More information

Elements of probability theory

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

More information

In general, we will not be concerned with the probability of individual outcomes of an experiment, but with collections of outcomes, called events. Fo

In general, we will not be concerned with the probability of individual outcomes of an experiment, but with collections of outcomes, called events. Fo 1 Introduction A Probability Refresher The word probability evokes (in most people) nebulous concepts related to uncertainty, randomness", etc. Probability is also a concept which is hard to characterize

More information

Probability for Estimation (review)

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

More information

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

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

More information

MATH41112/61112 Ergodic Theory Lecture Measure spaces

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

More information

Probability distributions

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,

More information

PROBABILITY THEORY - PART 1 MEASURE THEORETICAL FRAMEWORK CONTENTS

PROBABILITY THEORY - PART 1 MEASURE THEORETICAL FRAMEWORK CONTENTS PROBABILITY THEORY - PART 1 MEASURE THEORETICAL FRAMEWORK MANJUNATH KRISHNAPUR CONTENTS 1. Discrete probability space 2 2. Uncountable probability spaces? 3 3. Sigma algebras and the axioms of probability

More information

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8. Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

More information

MULTIVARIATE PROBABILITY DISTRIBUTIONS

MULTIVARIATE PROBABILITY DISTRIBUTIONS MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined

More information

Section 5.1 Continuous Random Variables: Introduction

Section 5.1 Continuous Random Variables: Introduction Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,

More information

Set theory, and set operations

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

More information

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

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

More information

Lecture 13: Martingales

Lecture 13: Martingales Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

More information

and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space

and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space RAL ANALYSIS A survey of MA 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets

More information

Lecture 4: Random Variables

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

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

More information

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites

More information

Mathematical Background

Mathematical Background Appendix A Mathematical Background A.1 Joint, Marginal and Conditional Probability Let the n (discrete or continuous) random variables y 1,..., y n have a joint joint probability probability p(y 1,...,

More information

Joint Exam 1/P Sample Exam 1

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

More information

RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. DISCRETE RANDOM VARIABLES.. Definition of a Discrete Random Variable. A random variable X is said to be discrete if it can assume only a finite or countable

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

More information

Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

More information

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

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

More information

SOME APPLICATIONS OF MARTINGALES TO PROBABILITY THEORY

SOME APPLICATIONS OF MARTINGALES TO PROBABILITY THEORY SOME APPLICATIONS OF MARTINGALES TO PROBABILITY THEORY WATSON LADD Abstract. Martingales are a very simple concept with wide application in probability. We introduce the concept of a martingale, develop

More information

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE

REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE REAL ANALYSIS LECTURE NOTES: 1.4 OUTER MEASURE CHRISTOPHER HEIL 1.4.1 Introduction We will expand on Section 1.4 of Folland s text, which covers abstract outer measures also called exterior measures).

More information

6. Distribution and Quantile Functions

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

More information

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 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

More information

Introduction to probability theory in the Discrete Mathematics course

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

More information

Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

More information

( ) = P Z > = P( Z > 1) = 1 Φ(1) = 1 0.8413 = 0.1587 P X > 17

( ) = P Z > = P( Z > 1) = 1 Φ(1) = 1 0.8413 = 0.1587 P X > 17 4.6 I company that manufactures and bottles of apple juice uses a machine that automatically fills 6 ounce bottles. There is some variation, however, in the amounts of liquid dispensed into the bottles

More information

Random Variables and Measurable Functions.

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

More information

5. Continuous Random Variables

5. Continuous Random Variables 5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

More information

Lecture 7: Continuous Random Variables

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

More information

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

More information

Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22

Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22 Math 151. Rumbos Spring 2014 1 Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability

More information

Topic 8: The Expected Value

Topic 8: The Expected Value Topic 8: September 27 and 29, 2 Among the simplest summary of quantitative data is the sample mean. Given a random variable, the corresponding concept is given a variety of names, the distributional mean,

More information

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 Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

More information

Markov random fields and Gibbs measures

Markov random fields and Gibbs measures Chapter Markov random fields and Gibbs measures 1. Conditional independence Suppose X i is a random element of (X i, B i ), for i = 1, 2, 3, with all X i defined on the same probability space (.F, P).

More information

Combinatorics: The Fine Art of Counting

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

More information

Introduction to Probability

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

More information

Stochastic Processes and Advanced Mathematical Finance. Laws of Large Numbers

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

More information

Notes on Probability Theory

Notes on Probability Theory Notes on Probability Theory Christopher King Department of Mathematics Northeastern University July 31, 2009 Abstract These notes are intended to give a solid introduction to Probability Theory with a

More information

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 Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

More information

Measure Theory. 1 Measurable Spaces

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,

More information

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution 8 October 2007 In this lecture we ll learn the following: 1. how continuous probability distributions differ

More information

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures. Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

More information

Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

More information

Probability and statistics; Rehearsal for pattern recognition

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

More information

Estimating the Average Value of a Function

Estimating the Average Value of a Function Estimating the Average Value of a Function Problem: Determine the average value of the function f(x) over the interval [a, b]. Strategy: Choose sample points a = x 0 < x 1 < x 2 < < x n 1 < x n = b and

More information

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 )

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 =

More information

An Introduction to Basic Statistics and Probability

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

More information

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 Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

More information

Exploratory Data Analysis

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

More information

Basics of 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.

More information

3 Multiple Discrete Random Variables

3 Multiple Discrete Random Variables 3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f

More information

The Ergodic Theorem and randomness

The Ergodic Theorem and randomness The Ergodic Theorem and randomness Peter Gács Department of Computer Science Boston University March 19, 2008 Peter Gács (Boston University) Ergodic theorem March 19, 2008 1 / 27 Introduction Introduction

More information

Measure Theory. Jesus Fernandez-Villaverde University of Pennsylvania

Measure Theory. Jesus Fernandez-Villaverde University of Pennsylvania Measure Theory Jesus Fernandez-Villaverde University of Pennsylvania 1 Why Bother with Measure Theory? Kolmogorov (1933). Foundation of modern probability. Deals easily with: 1. Continuous versus discrete

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

More information

3. The Multivariate Normal Distribution

3. The Multivariate Normal Distribution 3. The Multivariate Normal Distribution 3.1 Introduction A generalization of the familiar bell shaped normal density to several dimensions plays a fundamental role in multivariate analysis While real data

More information

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 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

More information

Ri and. i=1. S i N. and. R R i

Ri and. i=1. S i N. and. R R i The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

More information

ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS 1. INTRODUCTION ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

More information

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course On STRUCTURAL RELIABILITY Module # 02 Lecture 6 Course Format: Web Instructor: Dr. Arunasis Chakraborty Department of Civil Engineering Indian Institute of Technology Guwahati 6. Lecture 06:

More information

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 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

More information

MATHEMATICAL METHODS OF STATISTICS

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

More information

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 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.........................

More information

Finite and discrete probability distributions

Finite and discrete probability distributions 8 Finite and discrete probability distributions To understand the algorithmic aspects of number theory and algebra, and applications such as cryptography, a firm grasp of the basics of probability theory

More information

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails 12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

More information

Math 431 An Introduction to Probability. Final Exam Solutions

Math 431 An Introduction to Probability. Final Exam Solutions Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

More information

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 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

More information

Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

More information

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is. Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

More information