FUNCTIONS OF ONE RANDOM VARIABLE
|
|
- Arthur Barton
- 6 years ago
- Views:
Transcription
1 FUNCTIONS OF ONE RANDOM VARIABLE Constantin VERTAN April 9th, 05 As shown before, any random variable is a function, ξ : Ω R; this function can be composed with any real-valued, real argument function g : R R, resulting another random variable, denoted, for instance as η, which is: η g ξ gξ, η : Ω R. The following problem occurs?: how to characterize the new random variable η more specifically, how to find the probability density function of the random variable η knowing the statistical properties of the random variable ξ. Let s consider a given, fixed value x. The probability that the value of a particular realization of the random variable ξ s a is equal to x equals the probability that the value of that particular realization of the random variable ξ is within the infinitely small interval [x, x + dx] dx 0. But this is also: Probξ [x; x + dx]} F ξ x + dx F ξ x x+dx x w ξ tdt w ξ x dx. If the transform function g is bijective, the value x is uniquely mapped to the value y gx. in the same as we deduced, we can write that: Probη [y; y + dy]} w η y dy. But since y is uniquely obtained out of x, it follows that the probabilities given by and are equal, thus: w ξ x dx w η y dy. We are interested in w η y, so we will write: w η y w ξ x dx dy w ξx g x xg y w ξ g y g g y. 3 This formula 3 is valed only in the case of a transform function g which is bijective over its entire domain, or, in some other words, only if the equation y gx in which x is unknown and y is a parameter, has an unique solution. If the function g is not bijective, its domain must be decomposed in intervals of bijectivity. Within each such interval, the equation y gx, in which x is unknown and y is a parameter, has an unique solution denoted by x k. In this case, the formula 3 becomes: w η y k w ξ x k g x k xk g y. 4 The essential condition for the above relation to hold is that the number of intervals of bijectivity is finite or countable. This document is intended to be used in the framework of the course Decision and estimation for information processing, held for the students of the English stream at the Faculty of Electronics, Telecommunications and Information Technology of the Politehnica University of Bucharest. This material discusses applications of the fundamental theoretical definitions regarding the concept of random variable, is intended for use at the seminars and does not replaces the course notes.
2 For the pair of random variables ξ and η one can verify the theorem of the mean: η yw η ydy yw ξ xdx gxw ξ xdx. 5 Ex. Let ξ be a random variable uniformly distributed within π, π and consider the function g : π, π ;, with gx sinx. The random variable η is given by η gξ. Compute the pdf of the random variable η. First, one has to check the bijectivity of the transform function gx within its domain, and, if this does not hold, one has to divide this interval into subintervals for which the function is bijective. The bijectivity can be studied in a very simple manner, by solving the equation y gx, with the unknown x and the parameter y. In this case, the equation y sinx has one unique solution for x π, π, which is x arcsiny. Thus g y arcsiny, g : ; π, π. The derivative of the function gx is g x cosx. According to the formula of computing the new pdf 3 we have: f η y w ξ g y g g y w ξarcsiny cosarcsiny f ξarcsiny. y The random variable ξ is uniformly distributed within π, π ; then it follows that w ξ x π, if x π, π, Then: w η y π, if arcsiny π, π y 0, otherwise y π, if y, 0, otherwise Ex. A random variable ξ is transformed via a linear function gx x + β 0, into the random variable η. Compute the pdf, the mean and the variance of the random variable η, considering that ξ is distributed according to a normal distribution, and respectively according to an uniform distribution. Any linear function is bijective; the equation y gx with the unknown x has the solution x y β, and thus g y y β. The derivative of the linear function is g x. Under these circumstances, the pdf of the random variable η is given by the formula 3, being: w η y w ξ g y w y β ξ g g y y β w ξ. 6 If ξ is normally distributed with mean µ and variance σ then: w ξ x Nµ, σ exp πσ By replacing in 6 we get: w η y exp πσ y β x µ σ µ σ. π σ exp y µ + β π σ N µ + β, σ. This means that the distribution of the random variable obtained by the linear transformation is still normal, having a mean transformed via the same linear function and a variance scaled times.
3 If the random variable ξ is uniformly distributed within [a; b], for instance, its pdf is: w ξ x b a, if x [a, b], Then: w η y w ξ y β b a y β, if [a, b], One has to consider two cases, according to the sign of ; case, > 0: w η y b a, if y [a + β, b + β], case, < 0: w η y b a, if y [b + β, a + β], In both cases one can notice that the distribution is still uniform after the transform and according to the previous proof, the mean is the middle of the definition interval, and the variance is / from the squared interval length: η a + b + β ξ + β and ση b a σ ξ. The new mean is the original mean transformed via the same function as the random variable and the new variance is the scaled version of the original variance. Ex. 3 One considers the random variable ξ, uniformly distributed in [ c, c]. Compute the pdf and the cumulative density function of the random variable η /ξ. The transform function is gx /x. The derivative is g x /x 3. The function is not bijective over the entire real axis, but it is bijective within the intervals, 0 and 0,. The solutions of the equation y gx are: x / y and x / y, for y > 0. If y < 0 the equation has no solutions and f η y 0. Then we can apply the formula 4 for y > 0: w η y w ξ x k g x k xk g y w ξ x g x x g y + k +w ξ x g x xg y w η y w ξ/ y / / y 3 + w ξ / y / / y 3 y 3 wξ / y + w ξ / y. The random variable ξ is uniformly distributed; then: w ξ x c, if x [ c, c], It follows that: w ξ / y c, if y, /c] [/c,, c, if y [/c,, w ξ / y c, if y, /c] [/c,, c, if y [/c,, 3
4 Then: w η y c y 3, if y [/c,, The cumulative density function of random variable η is: y 0, if y < /c, F η y w η tdt y, if y [/c,. Ex. 4 A signal with values distributed normally according to N0, σ zero mean and variance σ is rectified by a ideal diode. Compute the pdf of the values of the resulting signal. The transform function of the rectifier circuit containing a single diode half-wave rectifier is: x, if x 0, gx 0, if x < 0. The function gx is not bijective; in this case one cannot simply apply formula 4 sice the solution set for the equation y gx with x < 0 is not countable; more precisely x R gx 0}, 0]. The problem will be solved by computing the cumulative density function F η y Probη y}. If y < 0, F η y Probη y} Probη < 0} 0. If y 0, F η y Probη 0} Probη 0} Probξ 0} F ξ If y > 0, F η y Probη y} Probξ x} F ξ x. In concluzie, Fξ y, if y 0, F η y The desired pdf is the derivative of F η y, that is: w η y df ηy dy 0, if y < 0, 0.5δy + N0, σ Uy, for y 0., where U is the unitary step function. One has to notice that the new random variable η has a concentrated probabiliy at value zero, that is the probability of obtaining exactly the value zero is non-zero: P η 0} lim ε 0 ε ε w η ydy lim ε 0 ε ε δydy + lim ε 0 ε 0 w ξ xdx. Ex. 5 Find the transform function that maps an uniform distribution within [0; ] into a Rayleigh distribution. Let ξ be the uniformly distributed random variable and η be the Rayleigh distributed random variable. Then:, if x [0, ], w ξ x, y w η y e y, if y 0, The supports of the two pdf s are [0, ], respectivly [0,, and then the unknown transform function must satisfy g : [0, ] [0,. Let assume that the function g is bijective; then, according to 3 we have: w η y w ξ g y g g y. 4
5 The inverse of the transform function exists and it is g : [0, [0, ]. This means that f ξ g y and thus: g g y w η y. But, since g is bijective, then it is strictly monotonic. By imposing g0 0, it follows that the function cannot be otherwise than increasing, and its derivative is positive. g y wη y, g y y w η tdt y 0 t e t From here one evaluates y as a function of x and then: Thus gx ln x. y ln x. dt e y x. Ex. 6 Prove that in the case when ξ is a random variable, its cumulative density function will transform it into an uniformly distributed random variable within [0, ]. If the pdf of random variable ξ is w ξ x, then its associated cumulative density function is F ξ x x w ξ tdt. If this is also the transform function of the random variable, then gx F ξ x, with g : R [0, ]. The derivative of the transform function is: g x df ξx dx w ξ x, and g x w ξ x w ξ x since the pdf is positive. then, according to 3 we have: w η y w ξ x g x xg y w ξ x w ξ x xf ξ y for y [0; ]. This is indeed an uniform distribution within [0, ]. Ex. 7 The dissipated electrical power into a resistor R kω is modelled as a random variable having an uniform distribution within [P min, P max ] [W, 0W ]. Find the distribution of the current values within the resistor. Ex. 8 A constant valued resistor R is conected at a current generator. The current generated by the generator can be considered a random variable uniformly distributed within [I min, I max ]. Compute the mean power dispersed by the resistor and the pdf of that power. Ex. 9 Prove using the theorem of the mean 5 that for a linear transform function gx x + β the variance of the resulting random variable is the original variance scaled times, and the resulting mean is original mean transformed via gx. Ex. 0 Compute the mean information obtained following the realization of an event which probability is distributed according to a /x function within [, ]. Ex. By measuring the anodic current of a vacuum diode one notices a linear distribution within 0 and ma. The anodic voltage of the vacuum diode is generated by a voltage source that has to be tested the dispersion of the voltage must not exceed 0 V. If A /000 ma/v 3/ is the tested generator fit? Ex. The transfer function of an ideal bi-phase rectifier is described by the function gx x. Compute the pdf, the mean voltage and the mean power of the random signal ξt obtained after rectification, if ξt is a normally distributed N0, σ, b uniformly distributed in [ ; ] and [0; ]. Ex. 3 A non-linear limiter is defined by the transfer function: 0, if x 0, y gx x, if x 0, ],, if x >. 5
6 At the input of the circuit is applied a signal with values following the pdf w X x e x + δx. Compute the pdf of the output signal. Acknowledgement This work has been funded by the Sectoral Operational Programme Human Resources Development of the Romanian Ministry of Education and Scientific Research through the Financial Agreement POSDRU/74/.3/S/4955. References [] M. Ciuc, C. Vertan: Statistical signal processing, in Romanian: Prelucrarea statistică a semnalelor, Ed. MatrixROM, Bucharest, 005. [] A. T. Murgan, I. Spânu, I. Gavăt, I. Sztojanov, V. E. Neagoe, A. Vlad: Exercises for the Theory of Information Transmission in Romanian: Teoria Transmisiunii Informaţiei - probleme, Ed. Didactică şi Pedagogică, Bucharest, 983. [3] C. Vertan, I. Gavăt, R. Stoian: Random variables and processes: principles and applications in Romanian: Variabile şi procese aleatoare: principii şi aplicaţii, Ed. Printech, Bucharest, 999. [4] A. Papoulis: Probability, random variables and stochastic processes, McGraw Hill Inc., 99. 6
p. 6-1 Continuous Random Variables p. 6-2
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability (>). Often, there is interest in random variables
More informationChapter 2: Random Variables
ECE54: Stochastic Signals and Systems Fall 28 Lecture 2 - September 3, 28 Dr. Salim El Rouayheb Scribe: Peiwen Tian, Lu Liu, Ghadir Ayache Chapter 2: Random Variables Example. Tossing a fair coin twice:
More informationProbability and Distributions
Probability and Distributions What is a statistical model? A statistical model is a set of assumptions by which the hypothetical population distribution of data is inferred. It is typically postulated
More informationNorthwestern University Department of Electrical Engineering and Computer Science
Northwestern University Department of Electrical Engineering and Computer Science EECS 454: Modeling and Analysis of Communication Networks Spring 2008 Probability Review As discussed in Lecture 1, probability
More informationSolution to Assignment 3
The Chinese University of Hong Kong ENGG3D: Probability and Statistics for Engineers 5-6 Term Solution to Assignment 3 Hongyang Li, Francis Due: 3:pm, March Release Date: March 8, 6 Dear students, The
More informationChapter 2. Random Variable. Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance.
Chapter 2 Random Variable CLO2 Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance. 1 1. Introduction In Chapter 1, we introduced the concept
More informationChapter 3: Random Variables 1
Chapter 3: Random Variables 1 Yunghsiang S. Han Graduate Institute of Communication Engineering, National Taipei University Taiwan E-mail: yshan@mail.ntpu.edu.tw 1 Modified from the lecture notes by Prof.
More informationBrief Review of Probability
Maura Department of Economics and Finance Università Tor Vergata Outline 1 Distribution Functions Quantiles and Modes of a Distribution 2 Example 3 Example 4 Distributions Outline Distribution Functions
More information2 (Statistics) Random variables
2 (Statistics) Random variables References: DeGroot and Schervish, chapters 3, 4 and 5; Stirzaker, chapters 4, 5 and 6 We will now study the main tools use for modeling experiments with unknown outcomes
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 14: Continuous random variables Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin www.cs.cmu.edu/
More information1 Joint and marginal distributions
DECEMBER 7, 204 LECTURE 2 JOINT (BIVARIATE) DISTRIBUTIONS, MARGINAL DISTRIBUTIONS, INDEPENDENCE So far we have considered one random variable at a time. However, in economics we are typically interested
More informationMultiple Random Variables
Multiple Random Variables Joint Probability Density Let X and Y be two random variables. Their joint distribution function is F ( XY x, y) P X x Y y. F XY ( ) 1, < x
More informationLecture 3: Random variables, distributions, and transformations
Lecture 3: Random variables, distributions, and transformations Definition 1.4.1. A random variable X is a function from S into a subset of R such that for any Borel set B R {X B} = {ω S : X(ω) B} is an
More information2 Functions of random variables
2 Functions of random variables A basic statistical model for sample data is a collection of random variables X 1,..., X n. The data are summarised in terms of certain sample statistics, calculated as
More informationMultivariate Distribution Models
Multivariate Distribution Models Model Description While the probability distribution for an individual random variable is called marginal, the probability distribution for multiple random variables is
More informationChapter 3, 4 Random Variables ENCS Probability and Stochastic Processes. Concordia University
Chapter 3, 4 Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University ENCS6161 p.1/47 The Notion of a Random Variable A random variable X is a function that assigns a real
More informationContinuous random variables
Continuous random variables CE 311S What was the difference between discrete and continuous random variables? The possible outcomes of a discrete random variable (finite or infinite) can be listed out;
More informationWill Landau. Feb 21, 2013
Iowa State University Feb 21, 2013 Iowa State University Feb 21, 2013 1 / 31 Outline Iowa State University Feb 21, 2013 2 / 31 random variables Two types of random variables: Discrete random variable:
More informationMath 480 The Vector Space of Differentiable Functions
Math 480 The Vector Space of Differentiable Functions The vector space of differentiable functions. Let C (R) denote the set of all infinitely differentiable functions f : R R. Then C (R) is a vector space,
More informationwhere r n = dn+1 x(t)
Random Variables Overview Probability Random variables Transforms of pdfs Moments and cumulants Useful distributions Random vectors Linear transformations of random vectors The multivariate normal distribution
More informationRandom Variables. Random variables. A numerically valued map X of an outcome ω from a sample space Ω to the real line R
In probabilistic models, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. As a function or a map, it maps from an element (or an outcome) of a sample
More informationSystem Identification
System Identification Arun K. Tangirala Department of Chemical Engineering IIT Madras July 27, 2013 Module 3 Lecture 1 Arun K. Tangirala System Identification July 27, 2013 1 Objectives of this Module
More informationMTH 202 : Probability and Statistics
MTH 202 : Probability and Statistics Lecture 9 - : 27, 28, 29 January, 203 4. Functions of a Random Variables 4. : Borel measurable functions Similar to continuous functions which lies to the heart of
More informationReview: mostly probability and some statistics
Review: mostly probability and some statistics C2 1 Content robability (should know already) Axioms and properties Conditional probability and independence Law of Total probability and Bayes theorem Random
More informationMATH4210 Financial Mathematics ( ) Tutorial 7
MATH40 Financial Mathematics (05-06) Tutorial 7 Review of some basic Probability: The triple (Ω, F, P) is called a probability space, where Ω denotes the sample space and F is the set of event (σ algebra
More informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More information3 Operations on One Random Variable - Expectation
3 Operations on One Random Variable - Expectation 3.0 INTRODUCTION operations on a random variable Most of these operations are based on a single concept expectation. Even a probability of an event can
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2017 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Exercise 4.1 Let X be a random variable with p(x)
More informationChapter 3: Random Variables 1
Chapter 3: Random Variables 1 Yunghsiang S. Han Graduate Institute of Communication Engineering, National Taipei University Taiwan E-mail: yshan@mail.ntpu.edu.tw 1 Modified from the lecture notes by Prof.
More informationTransformation of Probability Densities
Transformation of Probability Densities This Wikibook shows how to transform the probability density of a continuous random variable in both the one-dimensional and multidimensional case. In other words,
More informationf (x) = k=0 f (0) = k=0 k=0 a k k(0) k 1 = a 1 a 1 = f (0). a k k(k 1)x k 2, k=2 a k k(k 1)(0) k 2 = 2a 2 a 2 = f (0) 2 a k k(k 1)(k 2)x k 3, k=3
1 M 13-Lecture Contents: 1) Taylor Polynomials 2) Taylor Series Centered at x a 3) Applications of Taylor Polynomials Taylor Series The previous section served as motivation and gave some useful expansion.
More informationChap 2.1 : Random Variables
Chap 2.1 : Random Variables Let Ω be sample space of a probability model, and X a function that maps every ξ Ω, toa unique point x R, the set of real numbers. Since the outcome ξ is not certain, so is
More informationContinuous Random Variables and Continuous Distributions
Continuous Random Variables and Continuous Distributions Continuous Random Variables and Continuous Distributions Expectation & Variance of Continuous Random Variables ( 5.2) The Uniform Random Variable
More informationRandom Variables. P(x) = P[X(e)] = P(e). (1)
Random Variables Random variable (discrete or continuous) is used to derive the output statistical properties of a system whose input is a random variable or random in nature. Definition Consider an experiment
More informationContinuous distributions
CHAPTER 7 Continuous distributions 7.. Introduction A r.v. X is said to have a continuous distribution if there exists a nonnegative function f such that P(a X b) = ˆ b a f(x)dx for every a and b. distribution.)
More informationconditional cdf, conditional pdf, total probability theorem?
6 Multiple Random Variables 6.0 INTRODUCTION scalar vs. random variable cdf, pdf transformation of a random variable conditional cdf, conditional pdf, total probability theorem expectation of a random
More informationECE353: Probability and Random Processes. Lecture 7 -Continuous Random Variable
ECE353: Probability and Random Processes Lecture 7 -Continuous Random Variable Xiao Fu School of Electrical Engineering and Computer Science Oregon State University E-mail: xiao.fu@oregonstate.edu Continuous
More informationLecture 5: Integrals and Applications
Lecture 5: Integrals and Applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde 1 1 / 21 Outline The
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More information1 Presessional Probability
1 Presessional Probability Probability theory is essential for the development of mathematical models in finance, because of the randomness nature of price fluctuations in the markets. This presessional
More informationMetric Spaces. Exercises Fall 2017 Lecturer: Viveka Erlandsson. Written by M.van den Berg
Metric Spaces Exercises Fall 2017 Lecturer: Viveka Erlandsson Written by M.van den Berg School of Mathematics University of Bristol BS8 1TW Bristol, UK 1 Exercises. 1. Let X be a non-empty set, and suppose
More informationGrade: The remainder of this page has been left blank for your workings. VERSION D. Midterm D: Page 1 of 12
First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm D: Page of 2 Indefinite Integrals. 9 marks Each part is worth marks. Please
More informationParallel Circuits. Chapter
Chapter 5 Parallel Circuits Topics Covered in Chapter 5 5-1: The Applied Voltage V A Is the Same Across Parallel Branches 5-2: Each Branch I Equals V A / R 5-3: Kirchhoff s Current Law (KCL) 5-4: Resistance
More information4 Pairs of Random Variables
B.Sc./Cert./M.Sc. Qualif. - Statistical Theory 4 Pairs of Random Variables 4.1 Introduction In this section, we consider a pair of r.v. s X, Y on (Ω, F, P), i.e. X, Y : Ω R. More precisely, we define a
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationBasics of Stochastic Modeling: Part II
Basics of Stochastic Modeling: Part II Continuous Random Variables 1 Sandip Chakraborty Department of Computer Science and Engineering, INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR August 10, 2016 1 Reference
More informationELEMENTS OF PROBABILITY THEORY
ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable
More informationSTAT Chapter 5 Continuous Distributions
STAT 270 - Chapter 5 Continuous Distributions June 27, 2012 Shirin Golchi () STAT270 June 27, 2012 1 / 59 Continuous rv s Definition: X is a continuous rv if it takes values in an interval, i.e., range
More informationECE 302 Solution to Homework Assignment 5
ECE 2 Solution to Assignment 5 March 7, 27 1 ECE 2 Solution to Homework Assignment 5 Note: To obtain credit for an answer, you must provide adequate justification. Also, if it is possible to obtain a numeric
More informationContinuous Random Variables
1 / 24 Continuous Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 27, 2013 2 / 24 Continuous Random Variables
More informationContinuous Random Variables
MATH 38 Continuous Random Variables Dr. Neal, WKU Throughout, let Ω be a sample space with a defined probability measure P. Definition. A continuous random variable is a real-valued function X defined
More informationLecture 4: Fourier Transforms.
1 Definition. Lecture 4: Fourier Transforms. We now come to Fourier transforms, which we give in the form of a definition. First we define the spaces L 1 () and L 2 (). Definition 1.1 The space L 1 ()
More informationMathematical MCQ for international students admitted to École polytechnique
Mathematical MCQ for international students admitted to École polytechnique This multiple-choice questionnaire is intended for international students admitted to the first year of the engineering program
More informationSection 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures College of Science MATHS : Calculus I (University of Bahrain) Integrals / 7 The Substitution Method Idea: To replace a relatively
More informationLecture 4. Continuous Random Variables and Transformations of Random Variables
Math 408 - Mathematical Statistics Lecture 4. Continuous Random Variables and Transformations of Random Variables January 25, 2013 Konstantin Zuev (USC) Math 408, Lecture 4 January 25, 2013 1 / 13 Agenda
More informationGeneral Random Variables
1/65 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Probability A general random variable is discrete, continuous, or mixed. A discrete random variable
More informationF X (x) = P [X x] = x f X (t)dt. 42 Lebesgue-a.e, to be exact 43 More specifically, if g = f Lebesgue-a.e., then g is also a pdf for X.
10.2 Properties of PDF and CDF for Continuous Random Variables 10.18. The pdf f X is determined only almost everywhere 42. That is, given a pdf f for a random variable X, if we construct a function g by
More informationStatistics, Data Analysis, and Simulation SS 2015
Statistics, Data Analysis, and Simulation SS 2015 08.128.730 Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, 27. April 2015 Dr. Michael O. Distler
More informationGaussian Random Fields
Gaussian Random Fields March 22, 2007 Random Fields A N dimensional random field is a set of random variables Y (x), x R N, which has a collection of distribution functions F (Y (x ) y,..., Y (x n ) y
More informationLecture 11. Probability Theory: an Overveiw
Math 408 - Mathematical Statistics Lecture 11. Probability Theory: an Overveiw February 11, 2013 Konstantin Zuev (USC) Math 408, Lecture 11 February 11, 2013 1 / 24 The starting point in developing the
More information3 Continuous Random Variables
Jinguo Lian Math437 Notes January 15, 016 3 Continuous Random Variables Remember that discrete random variables can take only a countable number of possible values. On the other hand, a continuous random
More informationMathematics 1. (Integration)
Mthemtics 1. (Integrtion) University of Debrecen 2018-2019 fll Definition Let I R be n open, non-empty intervl, f : I R be function. F : I R is primitive function of f if F is differentible nd F = f on
More informationSystem Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models
System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models Fatih Cavdur fatihcavdur@uludag.edu.tr March 20, 2012 Introduction Introduction The world of the model-builder
More informationLecture 4: Integrals and applications
Lecture 4: Integrals and applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: autumn 2013 Lejla Batina Version: autumn 2013 Calculus en Kansrekenen 1 / 18
More information2.1 Random Events and Probability
2 Random Variables The idea of a random variable or random event involves no assumption about the intrinsic nature of the phenomenon under investigation. Indeed, it may be a perfectly deterministic phenomenon,
More informationMTH739U/P: Topics in Scientific Computing Autumn 2016 Week 6
MTH739U/P: Topics in Scientific Computing Autumn 16 Week 6 4.5 Generic algorithms for non-uniform variates We have seen that sampling from a uniform distribution in [, 1] is a relatively straightforward
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional
More information1 Introduction. 2 Measure theoretic definitions
1 Introduction These notes aim to recall some basic definitions needed for dealing with random variables. Sections to 5 follow mostly the presentation given in chapter two of [1]. Measure theoretic definitions
More informationIntegration by Substitution
November 22, 2013 Introduction 7x 2 cos(3x 3 )dx =? 2xe x2 +5 dx =? Chain rule The chain rule: d dx (f (g(x))) = f (g(x)) g (x). Use the chain rule to find f (x) and then write the corresponding anti-differentiation
More informationOrder Statistics and Distributions
Order Statistics and Distributions 1 Some Preliminary Comments and Ideas In this section we consider a random sample X 1, X 2,..., X n common continuous distribution function F and probability density
More informationIowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions
Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined
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: autumn 2013 Lejla Batina Version: autumn 2013 Wiskunde
More informationPoisson random measure: motivation
: motivation The Lévy measure provides the expected number of jumps by time unit, i.e. in a time interval of the form: [t, t + 1], and of a certain size Example: ν([1, )) is the expected number of jumps
More information1 Random Variable: Topics
Note: Handouts DO NOT replace the book. In most cases, they only provide a guideline on topics and an intuitive feel. 1 Random Variable: Topics Chap 2, 2.1-2.4 and Chap 3, 3.1-3.3 What is a random variable?
More informationFundamental Tools - Probability Theory II
Fundamental Tools - Probability Theory II MSc Financial Mathematics The University of Warwick September 29, 2015 MSc Financial Mathematics Fundamental Tools - Probability Theory II 1 / 22 Measurable random
More informationNormal Random Variables and Probability
Normal Random Variables and Probability An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2015 Discrete vs. Continuous Random Variables Think about the probability of selecting
More informationECE 302 Division 2 Exam 2 Solutions, 11/4/2009.
NAME: ECE 32 Division 2 Exam 2 Solutions, /4/29. You will be required to show your student ID during the exam. This is a closed-book exam. A formula sheet is provided. No calculators are allowed. Total
More information1.1 Review of Probability Theory
1.1 Review of Probability Theory Angela Peace Biomathemtics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology. CRC Press,
More informationDUBLIN CITY UNIVERSITY
DUBLIN CITY UNIVERSITY SAMPLE EXAMINATIONS 2017/2018 MODULE: QUALIFICATIONS: Simulation for Finance MS455 B.Sc. Actuarial Mathematics ACM B.Sc. Financial Mathematics FIM YEAR OF STUDY: 4 EXAMINERS: Mr
More informationProduct measure and Fubini s theorem
Chapter 7 Product measure and Fubini s theorem This is based on [Billingsley, Section 18]. 1. Product spaces Suppose (Ω 1, F 1 ) and (Ω 2, F 2 ) are two probability spaces. In a product space Ω = Ω 1 Ω
More informationSingular Integrals. 1 Calderon-Zygmund decomposition
Singular Integrals Analysis III Calderon-Zygmund decomposition Let f be an integrable function f dx 0, f = g + b with g Cα almost everywhere, with b
More informationExam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationProving the central limit theorem
SOR3012: Stochastic Processes Proving the central limit theorem Gareth Tribello March 3, 2019 1 Purpose In the lectures and exercises we have learnt about the law of large numbers and the central limit
More informationStat410 Probability and Statistics II (F16)
Stat4 Probability and Statistics II (F6 Exponential, Poisson and Gamma Suppose on average every /λ hours, a Stochastic train arrives at the Random station. Further we assume the waiting time between two
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions
More informationF (x) = P [X x[. DF1 F is nondecreasing. DF2 F is right-continuous
7: /4/ TOPIC Distribution functions their inverses This section develops properties of probability distribution functions their inverses Two main topics are the so-called probability integral transformation
More informationBounded uniformly continuous functions
Bounded uniformly continuous functions Objectives. To study the basic properties of the C -algebra of the bounded uniformly continuous functions on some metric space. Requirements. Basic concepts of analysis:
More informationE[X n ]= dn dt n M X(t). ). What is the mgf? Solution. Found this the other day in the Kernel matching exercise: 1 M X (t) =
Chapter 7 Generating functions Definition 7.. Let X be a random variable. The moment generating function is given by M X (t) =E[e tx ], provided that the expectation exists for t in some neighborhood of
More informationCh3 Operations on one random variable-expectation
Ch3 Operations on one random variable-expectation Previously we define a random variable as a mapping from the sample space to the real line We will now introduce some operations on the random variable.
More informationSTAT 450: Statistical Theory. Distribution Theory. Reading in Casella and Berger: Ch 2 Sec 1, Ch 4 Sec 1, Ch 4 Sec 6.
STAT 45: Statistical Theory Distribution Theory Reading in Casella and Berger: Ch 2 Sec 1, Ch 4 Sec 1, Ch 4 Sec 6. Basic Problem: Start with assumptions about f or CDF of random vector X (X 1,..., X p
More informationChapter 4. Continuous Random Variables 4.1 PDF
Chapter 4 Continuous Random Variables In this chapter we study continuous random variables. The linkage between continuous and discrete random variables is the cumulative distribution (CDF) which we will
More informationProbability Theory and Statistics. Peter Jochumzen
Probability Theory and Statistics Peter Jochumzen April 18, 2016 Contents 1 Probability Theory And Statistics 3 1.1 Experiment, Outcome and Event................................ 3 1.2 Probability............................................
More informationMATH115. Infinite Series. Paolo Lorenzo Bautista. July 17, De La Salle University. PLBautista (DLSU) MATH115 July 17, / 43
MATH115 Infinite Series Paolo Lorenzo Bautista De La Salle University July 17, 2014 PLBautista (DLSU) MATH115 July 17, 2014 1 / 43 Infinite Series Definition If {u n } is a sequence and s n = u 1 + u 2
More informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics and Engineering Lecture notes for PDEs Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 The integration theory
More information1.1. BASIC ANTI-DIFFERENTIATION 21 + C.
.. BASIC ANTI-DIFFERENTIATION and so e x cos xdx = ex sin x + e x cos x + C. We end this section with a possibly surprising complication that exists for anti-di erentiation; a type of complication which
More informationSTA 4321/5325 Solution to Homework 5 March 3, 2017
STA 4/55 Solution to Homework 5 March, 7. Suppose X is a RV with E(X and V (X 4. Find E(X +. By the formula, V (X E(X E (X E(X V (X + E (X. Therefore, in the current setting, E(X V (X + E (X 4 + 4 8. Therefore,
More informationUnsteady State Heat Conduction in a Bounded Solid
Unsteady State Heat Conduction in a Bounded Solid Consider a sphere of radius R. Initially the sphere is at a uniform temperature T. It is cooled by convection to an air stream at temperature T a. What
More informationSTAT 450: Statistical Theory. Distribution Theory. Reading in Casella and Berger: Ch 2 Sec 1, Ch 4 Sec 1, Ch 4 Sec 6.
STAT 450: Statistical Theory Distribution Theory Reading in Casella and Berger: Ch 2 Sec 1, Ch 4 Sec 1, Ch 4 Sec 6. Example: Why does t-statistic have t distribution? Ingredients: Sample X 1,...,X n from
More informationIntroduction to Probability Theory
Introduction to Probability Theory Ping Yu Department of Economics University of Hong Kong Ping Yu (HKU) Probability 1 / 39 Foundations 1 Foundations 2 Random Variables 3 Expectation 4 Multivariate Random
More information