Jointly Distributed Random Variables


 Natalie Eunice Maxwell
 2 years ago
 Views:
Transcription
1 Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables Definition Joint cdfs Joint Probability Mass Functions Joint Probability Density Functions Independence and Expectation Independence Expectation Conditional Expectation Covariance and Correlation Examples 6 1 Jointly Distributed Random Variables 1.1 Definition Joint Probability Distribution Suppose we have two random variables X and Y defined on a sample space S with probability measure P(E), E S. Jointly, they form a map (X, Y) : S R 2 where s (X(s), Y(s)). From before, we know to define the marginal probability distributions P X and P Y by, for example, P X (B) = P(X 1 (B)), B R. We now define the joint probability distribution P XY by P XY (B X, B Y ) = P{X 1 (B X ) Y 1 (B Y )}, B X, B Y R. So P XY (B X, B Y ), the probability that X B X and Y B Y, is given by the probability P of the set of all points in the sample space that get mapped both into B X by X and into B Y by Y. More generally, for a single region B XY R 2, find the collection of sample space elements and define S BXY = {s S (X(s), Y(s)) B XY } P XY (B XY ) = P(S BXY ). 1
2 1.2 Joint cdfs Joint Cumulative Distribution Function We thus define the joint cumulative distribution function as F XY (x, y) = P XY ((, x], (, y]), x, y R. It is easy to check that the marginal cdfs for X and Y can be recovered by and that the two definitions will agree. F X (x) = F XY (x, ), x R, F Y (y) = F XY (, y), y R, Properties of a Joint cdf For F XY to be a valid cdf, we need to make sure the following conditions hold F XY (x, y) 1, x, y R; 2. Monotonicity: x 1, x 2, y 1, y 2 R, x 1 < x 2 F XY (x 1, y 1 ) F XY (x 2, y 1 ) and y 1 < y 2 F XY (x 1, y 1 ) F XY (x 1, y 2 ); 3. x, y R, F XY (x, ) = 0, F XY (, y) = 0 and F XY (, ) = 1. Interval Probabilities Suppose we are interested in whether the random variable pair (X, Y) lie in the interval cross product (x 1, x 2 ] (y 1, y 2 ]; that is, if x 1 < X x 2 and y 1 < Y y 2. Y y 2 y 1 x 1 x 2 X First note that P XY ((x 1, x 2 ], (, y]) = F XY (x 2, y) F XY (x 1, y). It is then easy to see that P XY ((x 1, x 2 ], (y 1, y 2 ]) is given by F XY (x 2, y 2 ) F XY (x 1, y 2 ) F XY (x 2, y 1 ) + F XY (x 1, y 1 ). 1.3 Joint Probability Mass Functions If X and Y are both discrete random variables, then we can define the joint probability mass function as p XY (x, y) = P XY ({x}, {y}), x, y R. We can recover the marginal pmfs p X and p Y since, by the law of total probability, x, y R p X (x) = y p XY (x, y), p Y (y) = x p XY (x, y). 2
3 Properties of a Joint pmf For p XY to be a valid pmf, we need to make sure the following conditions hold p XY (x, y) 1, x, y R; 2. p XY (x, y) = 1. y x 1.4 Joint Probability Density Functions On the other hand, if f XY : R R R s.t. P XY (B XY ) = f XY (x, y)dxdy, B XY R R, (x,y) B XY then we say X and Y are jointly continuous and we refer to f XY as the joint probability density function of X and Y. In this case, we have F XY (x, y) = y t= x s= f XY (s, t)dsdt, x, y R, By the Fundamental Theorem of Calculus we can identify the joint pdf as f XY (x, y) = 2 x y F XY(x, y). Furthermore, we can recover the marginal densities f X and f Y : f X (x) = d dx F X(x) = d dx F XY(x, ) = d x f XY (s, y)dsdy. dx s= By the Fundamental Theorem of Calculus, and through a symmetric argument for Y, we thus get f X (x) = f XY (x, y)dy, f Y (y) = x= f XY (x, y)dx. Properties of a Joint pdf For f XY to be a valid pdf, we need to make sure the following conditions hold. 1. f XY (x, y) 0, x, y R; 2. x= f XY (x, y)dxdy = 1. 3
4 2 Independence and Expectation 2.1 Independence Independence of Random Variables Two random variables X and Y are independent iff B X, B Y R., P XY (B X, B Y ) = P X (B X )P Y (B Y ). More specifically, two discrete random variables X and Y are independent iff p XY (x, y) = p X (x)p Y (y), x, y R; and two continuous random variables X and Y are independent iff f XY (x, y) = f X (x) f Y (y), x, y R. Conditional Distributions For two r.v.s X, Y we define the conditional probability distribution P Y X by P Y X (B Y B X ) = P XY(B X, B Y ), B X, B P X (B X ) Y R, P X (B X ) = 0. This is the revised probability of Y falling inside B Y given that we now know X B X. Then we have X and Y are independent P Y X (B Y B X ) = P Y (B Y ), B X, B Y R. For discrete r.v.s X, Y we define the conditional probability mass function p Y X by p Y X (y x) = p XY(x, y), x, y R, p X (x) = 0, p X (x) and for continuous r.v.s X, Y we define the conditional probability density function f Y X by f Y X (y x) = f XY(x, y), x, y R. f X (x) [Justification: P(Y y X [x, x + dx)) = P XY([x, x + dx), (, y]) P X ([x, x + dx)) F(y, x + dx) F(y, x) = F(x + dx) F(x) {F(y, x + dx) F(y, x)}/dx = {F(x + dx) F(x)}/dx d{f(y, x + dx) F(y, x)}/dxdy = f (y X [x, x + dx)) = {F(x + dx) F(x)}/dx f (y, x) = f (y x) = lim f (y X [x, x + dx)) = dx >0 f (x).] R. In either case, X and Y are independent p Y X (y x) = p Y (y) or f Y X (y x) = f Y (y), x, y 4
5 2.2 Expectation E{g(X, Y)} Suppose we have a bivariate function of interest of the random variables X and Y, g : R R R. If X and Y are discrete, we define E{g(X, Y)} by E XY {g(x, Y)} = y g(x, y)p XY (x, y). x If X and Y are jointly continuous, we define E{g(X, Y)} by E XY {g(x, Y)} = x= g(x, y) f XY (x, y)dxdy. Immediately from these definitions we have the following: If g(x, Y) = g 1 (X) + g 2 (Y), E XY {g 1 (X) + g 2 (Y)} = E X {g 1 (X)} + E Y {g 2 (Y)}. If g(x, Y) = g 1 (X)g 2 (Y) and X and Y are independent, E XY {g 1 (X)g 2 (Y)} = E X {g 1 (X)}E Y {g 2 (Y)}. In particular, considering g(x, Y) = XY for independent X, Y we have E XY (XY) = E X (X)E Y (Y). 2.3 Conditional Expectation E Y X (Y X = x) In general E XY (XY) = E X (X)E Y (Y). Suppose X and Y are discrete r.v.s with joint pmf p(x, y). If we are given the value x of the r.v. X, our revised pmf for Y is the conditional pmf p(y x), for y R. The conditional expectation of Y given X = x is therefore Similarly, if X and Y were continuous, E Y X (Y X = x) = y p(y x). y E Y X (Y X = x) = y f (y x)dy. In either case, the conditional expectation is a function of x but not the unknown Y. 5
6 2.4 Covariance and Correlation Covariance For a single variable X we considered the expectation of g(x) = (X µ X )(X µ X ), called the variance and denoted σ 2 X. The bivariate extension of this is the expectation of g(x, Y) = (X µ X )(Y µ Y ). We define the covariance of X and Y by σ XY = Cov(X, Y) = E XY [(X µ X )(Y µ Y )]. Correlation Covariance measures how the random variables move in tandem with one another, and so is closely related to the idea of correlation. We define the correlation of X and Y by ρ XY = Cor(X, Y) = σ XY σ X σ Y. Unlike the covariance, the correlation is invariant to the scale of the r.v.s X and Y. It is easily shown that if X and Y are independent random variables, then σ XY = ρ XY = 0. 3 Examples Example 1 Suppose that the lifetime, X, and brightness, Y of a light bulb are modelled as continuous random variables. Let their joint pdf be given by Are lifetime and brightness independent? The marginal pdf for X is f (x) = f (x, y) = λ 1 λ 2 e λ 1x λ 2 y, x, y > 0. = λ 1 e λ 1x. f (x, y)dy = y=0 λ 1 λ 2 e λ 1x λ 2 y dy Similarly f (y) = λ 2 e λ 2y. Hence f (x, y) = f (x) f (y) and X and Y are independent. Example 2 Suppose continuous r.v.s (X, Y) R 2 have joint pdf { 1 f (x, y) = π, x2 + y 2 1 0, otherwise. Determine the marginal pdfs for X and Y. Well x 2 + y 2 1 y 1 x 2. So 1 x 2 1 f (x) = f (x, y)dy = y= 1 x 2 π dy = 2 1 x π 2. Similarly f (y) = 2 1 y π 2. Hence f (x, y) = f (x) f (y) and X and Y are not independent. 6
Random Variables, Expectation, Distributions
Random Variables, Expectation, Distributions CS 5960/6960: Nonparametric Methods Tom Fletcher January 21, 2009 Review Random Variables Definition A random variable is a function defined on a probability
More informationBivariate Distributions
Chapter 4 Bivariate Distributions 4.1 Distributions of Two Random Variables In many practical cases it is desirable to take more than one measurement of a random observation: (brief examples) 1. What is
More informationSTAT 430/510 Probability Lecture 14: Joint Probability Distribution, Continuous Case
STAT 430/510 Probability Lecture 14: Joint Probability Distribution, Continuous Case Pengyuan (Penelope) Wang June 20, 2011 Joint density function of continuous Random Variable When X and Y are two continuous
More informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 101634501 Probability and Statistics for Engineers Winter 20102011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete
More informationTopic 4: Multivariate random variables. Multiple random variables
Topic 4: Multivariate random variables Joint, marginal, and conditional pmf Joint, marginal, and conditional pdf and cdf Independence Expectation, covariance, correlation Conditional expectation Two jointly
More information4. Joint Distributions of Two Random Variables
4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint
More informationContinuous 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
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationSection 6.1 Joint Distribution Functions
Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function
More informationJoint Distributions. Tieming Ji. Fall 2012
Joint Distributions Tieming Ji Fall 2012 1 / 33 X : univariate random variable. (X, Y ): bivariate random variable. In this chapter, we are going to study the distributions of bivariate random variables
More informationST 371 (VIII): Theory of Joint Distributions
ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or
More informationJoint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single
More informationProbability & 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 informationContinuous Random Variables
Probability 2  Notes 7 Continuous Random Variables Definition. A random variable X is said to be a continuous random variable if there is a function f X (x) (the probability density function or p.d.f.)
More informationChapter 4 Expected Values
Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges
More informationContinuous Distributions
MAT 2379 3X (Summer 2012) Continuous Distributions Up to now we have been working with discrete random variables whose R X is finite or countable. However we will have to allow for variables that can take
More informationLecture Notes 1. Brief Review of Basic Probability
Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters 3 are a review. I will assume you have read and understood Chapters 3. Here is a very
More informationM2S1 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 informationChapter 4. Multivariate Distributions
1 Chapter 4. Multivariate Distributions Joint p.m.f. (p.d.f.) Independent Random Variables Covariance and Correlation Coefficient Expectation and Covariance Matrix Multivariate (Normal) Distributions Matlab
More informationDefinition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).
Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The
More information3.6: General Hypothesis Tests
3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.
More informationLecture 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 informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationContinuous 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 socalled continuous random variables. A random
More information6. Jointly Distributed Random Variables
6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.
More informationChapters 5. Multivariate Probability Distributions
Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are
More informationContinuous 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
More informationMATH 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 6sided dice, five 8sided dice, and six 0sided dice. A die is drawn from the bag and then rolled.
More informationCovariance and Correlation. Consider the joint probability distribution f XY (x, y).
Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Section 52 Covariance and Correlation Consider the joint probability distribution f XY (x, y). Is there a relationship between X and Y? If so, what kind?
More informationExpectation Discrete RV  weighted average Continuous RV  use integral to take the weighted average
PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV  weighted average Continuous RV  use integral to take the weighted average Variance Variance is the average of (X µ) 2
More informationL10: Probability, statistics, and estimation theory
L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian
More informationChange of Continuous Random Variable
Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make
More informationMath 432 HW 2.5 Solutions
Math 432 HW 2.5 Solutions Assigned: 110, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/
More information5. 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 informationChapter 4  Lecture 1 Probability Density Functions and Cumul. Distribution Functions
Chapter 4  Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the
More informationExamination 110 Probability and Statistics Examination
Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiplechoice test questions. The test is a threehour examination
More informationLesson 5 Chapter 4: Jointly Distributed Random Variables
Lesson 5 Chapter 4: Jointly Distributed Random Variables Department of Statistics The Pennsylvania State University 1 Marginal and Conditional Probability Mass Functions The Regression Function Independence
More informationStatistiek (WISB361)
Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17
More informationECE302 Spring 2006 HW7 Solutions March 11, 2006 1
ECE32 Spring 26 HW7 Solutions March, 26 Solutions to HW7 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where
More informationJoint 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
More informationWorked examples Multiple Random Variables
Worked eamples Multiple Random Variables Eample Let X and Y be random variables that take on values from the set,, } (a) Find a joint probability mass assignment for which X and Y are independent, and
More informationExample. Two fair dice are tossed and the two outcomes recorded. As is familiar, we have
Lectures 910 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 informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2014 1 2 3 4 5 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Experiment, outcome, sample space, and
More informationMULTIVARIATE 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 informationJoint 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 informationContinuous Probability Distributions (120 Points)
Economics : Statistics for Economics Problem Set 5  Suggested Solutions University of Notre Dame Instructor: Julio Garín Spring 22 Continuous Probability Distributions 2 Points. A management consulting
More informationMath 576: Quantitative Risk Management
Math 576: Quantitative Risk Management Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 4 Haijun Li Math 576: Quantitative Risk Management Week 4 1 / 22 Outline 1 Basics
More informationP(a X b) = f X (x)dx. A p.d.f. must integrate to one: f X (x)dx = 1. Z b
Continuous Random Variables The probability that a continuous random variable, X, has a value between a and b is computed by integrating its probability density function (p.d.f.) over the interval [a,b]:
More informationJoint Continuous Distributions
Joint Continuous Distributions Statistics 11 Summer 6 Copright c 6 b Mark E. Irwin Joint Continuous Distributions Not surprisingl we can look at the joint distribution of or more continuous RVs. For eample,
More informationPrinciple of Data Reduction
Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then
More informationLecture Notes 3 Random Vectors. Specifying a Random Vector. Mean and Covariance Matrix. Coloring and Whitening. Gaussian Random Vectors
Lecture Notes 3 Random Vectors Specifying a Random Vector Mean and Covariance Matrix Coloring and Whitening Gaussian Random Vectors EE 278B: Random Vectors 3 Specifying a Random Vector Let X,X 2,...,X
More informationFeb 28 Homework Solutions Math 151, Winter 2012. Chapter 6 Problems (pages 287291)
Feb 8 Homework Solutions Math 5, Winter Chapter 6 Problems (pages 879) Problem 6 bin of 5 transistors is known to contain that are defective. The transistors are to be tested, one at a time, until the
More informationRecitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere
Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x
More informationRenewal Theory. (iv) For s < t, N(t) N(s) equals the number of events in (s, t].
Renewal Theory Def. A stochastic process {N(t), t 0} is said to be a counting process if N(t) represents the total number of events that have occurred up to time t. X 1, X 2,... times between the events
More information10 BIVARIATE DISTRIBUTIONS
BIVARIATE DISTRIBUTIONS After some discussion of the Normal distribution, consideration is given to handling two continuous random variables. The Normal Distribution The probability density function f(x)
More informationSection 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 informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationChapter 5: Joint Probability Distributions. Chapter Learning Objectives. The Joint Probability Distribution for a Pair of Discrete Random
Chapter 5: Joint Probability Distributions 51 Two or More Random Variables 51.1 Joint Probability Distributions 51.2 Marginal Probability Distributions 51.3 Conditional Probability Distributions 51.4
More informationConditional expectation
A Conditional expectation A.1 Review of conditional densities, expectations We start with the continuous case. This is sections 6.6 and 6.8 in the book. Let X, Y be continuous random variables. We defined
More informationRandom Variables. Chapter 2. Random Variables 1
Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets
More informationDiscrete and Continuous Random Variables. Summer 2003
Discrete and Continuous Random Variables Summer 003 Random Variables A random variable is a rule that assigns a numerical value to each possible outcome of a probabilistic experiment. We denote a random
More information3. 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
More informationSECONDORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS
L SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECONDORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A secondorder linear differential equation is one of the form d
More informationLecture 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 informationSolution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute
Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the
More informationChap 3 : Two Random Variables
Chap 3 : Two Random Variables Chap 3.1: Distribution Functions of Two RVs In many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to record
More informationSet 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 informationChapter 5: Multivariate Distributions
Chapter 5: Multivariate Distributions Professor Ron Fricker Naval Postgraduate School Monterey, California 3/15/15 Reading Assignment: Sections 5.1 5.12 1 Goals for this Chapter Bivariate and multivariate
More informationST 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 informationMASSACHUSETTS 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( ) = 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 informationMT426 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 20042012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................
More informationMC3: Econometric Theory and Methods
University College London Department of Economics M.Sc. in Economics MC3: Econometric Theory and Methods Course notes: Econometric models, random variables, probability distributions and regression Andrew
More informationTRANSFORMATIONS OF RANDOM VARIABLES
TRANSFORMATIONS OF RANDOM VARIABLES 1. INTRODUCTION 1.1. Definition. We are often interested in the probability distributions or densities of functions of one or more random variables. Suppose we have
More informationThe random variable X  the no. of defective items when three electronic components are tested would be
RANDOM VARIABLES and PROBABILITY DISTRIBUTIONS Example: Give the sample space giving a detailed description of each possible outcome when three electronic components are tested, where N  denotes nondefective
More informationMath 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 informationLecture 8: Continuous random variables, expectation and variance
Lecture 8: Continuous random variables, expectation and variance Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde
More informationMath 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions
Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,
More informationAB2.14: Heat Equation: Solution by Fourier Series
AB2.14: Heat Equation: Solution by Fourier Series Consider the boundary value problem for the onedimensional heat equation describing the temperature variation in a bar with the zerotemperature ends:
More information3 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 informationStatistical Inference
Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this
More informationIntroduction 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 informationLecture 1: Entropy and mutual information
Lecture 1: Entropy and mutual information 1 Introduction Imagine two people Alice and Bob living in Toronto and Boston respectively. Alice (Toronto) goes jogging whenever it is not snowing heavily. Bob
More informationExercises with solutions (1)
Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal
More informationLecture 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 informationRandom 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 informationTopic 8 The Expected Value
Topic 8 The Expected Value Functions of Random Variables 1 / 12 Outline Names for Eg(X ) Variance and Standard Deviation Independence Covariance and Correlation 2 / 12 Names for Eg(X ) If g(x) = x, then
More informationLecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. LogNormal Distribution
Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Logormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last
More informationMultivariable Calculus Practice Midterm 2 Solutions Prof. Fedorchuk
Multivariable Calculus Practice Midterm Solutions Prof. Fedorchuk. ( points) Let f(x, y, z) xz + e y x. a. (4 pts) Compute the gradient f. b. ( pts) Find the directional derivative D,, f(,, ). c. ( pts)
More informationQuiz 1(1): Experimental Physics LabI Solution
Quiz 1(1): Experimental Physics LabI Solution 1. Suppose you measure three independent variables as, x = 20.2 ± 1.5, y = 8.5 ± 0.5, θ = (30 ± 2). Compute the following quantity, q = x(y sin θ). Find the
More informationContents. 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
More informationDepartment of Mathematics, Indian Institute of Technology, Kharagpur Assignment 23, Probability and Statistics, March 2015. Due:March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 3, Probability and Statistics, March 05. Due:March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
More informationLesson 3. Conditional Probability Review
SA402 Dynamic and Stochastic Models Fall 2016 Assoc. Prof. Nelson Uhan Lesson 3. Conditional Probability Review Course standards covered in this lesson: B3 Joint probabilities, B4 Conditional probabilities,
More informationStatistics 100A Homework 7 Solutions
Chapter 6 Statistics A Homework 7 Solutions Ryan Rosario. A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 5 percent will purchase
More informationLecture 10: Other Continuous Distributions and Probability Plots
Lecture 10: Other Continuous Distributions and Probability Plots Devore: Section 4.44.6 Page 1 Gamma Distribution Gamma function is a natural extension of the factorial For any α > 0, Γ(α) = 0 x α 1 e
More informationFor 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 (19031987) 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 informationWorked examples Random Processes
Worked examples Random Processes Example 1 Consider patients coming to a doctor s office at random points in time. Let X n denote the time (in hrs) that the n th patient has to wait before being admitted
More informationECE302 Spring 2006 HW5 Solutions February 21, 2006 1
ECE3 Spring 6 HW5 Solutions February 1, 6 1 Solutions to HW5 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics
More informationData Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
More informationP (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