Deterministic. Deterministic data are those can be described by an explicit mathematical relationship
|
|
- Maximillian Beasley
- 5 years ago
- Views:
Transcription
1 Random data
2 Deterministic Deterministic data are those can be described by an explicit mathematical relationship
3 Deterministic x(t) =X cos r! k m t
4 Non deterministic There is no way to predict an exact value at a future instant of time These data are random in character and must be described in terms of probability statements and statistical averages
5 In practical terms The decision of whether physical data are deterministic or random is usually based on the ability of reproduce the data by controlled experiments
6 In practical terms If the experiment can be repeated producing identical data, within the limits of experimental error -> deterministic If an experiment cannot be designed that will produce identical results when repeated -> non deterministic (random)
7 Terminology A single time history representing a random phenomena is called a sample function or a sample record The collection of all sample function that a random phenomenon might have produced is called random process or stochastic process A sample record of data may be thought of as one physical realization of a random process
8 Classification of random data
9 Random data classification Random Stationary Nonstationary Ergodic Nonergodic
10 Statistical properties Considering a collection of sample functions:
11 Statistical properties Considering a collection of sample functions: mean value of a random process at some time ti can be computed averaging all instantaneous values of each sample correlation between values at two different times is the average of the product of instantaneous values at time ti and ti+tau
12 Statistical properties µ x (t i )= lim N!+1 1 N NX k=1 x k (t i ) R xx (t i,t i + ) = lim N!+1 1 N NX x k (t i )x k (t i + ) k=1
13 Stationary vs Nonstationary When mean value and autocorrelation vary as time ti varies, the random process is said to be nonstationary When mean value and autocorrelation do not vary as time ti varies, the random process is said to be weakly stationary or stationary in a wide sense
14 Weakly stationary For weakly stationary random process, the mean value is constant and the autocorrelation function is dependent only on the time displacement tau. µ x (t i )=µ x R xx (t i,t i + ) =R xx ( )
15 Weakly stationary In most case it is possible to describe the properties of a stationary random process by computing time averages over specific sample function µ x (k) = lim T!1 R xx (,k)= lim T!1 1 T 1 T Z T 0 Z T 0 x k (t) dt x k (t)x k (t + ) dt
16 Ergodic random data If mean value and autocorrelation function do not differ over different sample functions the random process is said to be ergodic!! µ x (k) =µ x R xx (,k)=r xx ( ) Only stationary random process can be ergodic
17 Ergodic random data Ergodic random process are an important class of random processes All properties of ergodic random process can be determined by a single sample function Fortunately, in practice, random data representing stationary physical phenomena are generally ergodic
18 Nonstationary random data The properties of nonstationary random process are generally time-varying function In practice it is often not feasible to obtain a sufficient number of sample records to permit an accurate measurement of properties of the ensemble This has tend to impede the development of practical techniques for measuring and analyzing nonstationary random data
19 Stationary sample records Data in the form of sample records are referred to be stationary or nonstationary Z ti +T µ x (t i,k)= 1 T x k (t) dt t i Z ti +T R xx (t i,t i +,k)= 1 T x k (t)x k (t + ) dt t i
20 Stationary sample records A single time series is referred to be stationary if properties computed over short time intervals do not vary significantly from one interval to the next If the sample properties vary significantly as the starting time ti varies the individual sample record is said to be nonstationary
21 Stationary sample records A sample record obtained from an ergodic random process will be stationary Sample records from nonstationary random process will be nonstationary Hence if an ergodic assumption is justified verification of stationarity of a single sample records will justify an assumption of stationarity and ergodicity for the random process
22 Analysis of random data
23 Analysis of random data Since no explicit mathematical equation can be written, statistical procedures must be used to define the descriptive properties of the data
24 Basic descriptive properties Mean and mean square values Probability density functions Autocorrelation functions Power spectral density functions
25 Joint statistical properties Joint probability density functions Cross-correlation functions Cross-spectral density functions Frequency response functions Coherence functions
26 Probability density function Is a function that describes the relative likelihood for this random variable to take on a given value The probability of the random variable falling within a particular range of values is given by the integral of this variable s density over that range
27 Probability density function Z b Pr[a apple X apple b] = a f X (x) dx Z x CDF X (x) =P (X apple x) = 1 f X (u) du
28 Probability density function
29 Time domain analysis
30 Expected value E[X] = 1X x i p i,! i=1! E[X] = Z 1 xf(x) dx! 1 Arithmetic mean is an estimator of the expected value of a random process nx 2 x = 1 n i=1 x i Var( x) = n
31 Expected value E[X + c] =E[X]+c E[X + Y ]=E[X]+E[Y] E[aX] =a E[X] E[XY ]= Z Z xy j(x, y) dx dy Cov(X, Y )=E[XY ] E[X]E[Y ] E[XY ]= = Z Z Z Z xy j(x, y) dx dy = xyf(x)g(y) dy dx applez applez xf(x) dx yg(y) dy =E[X]E[Y ]
32 Variance Var(X) =E (X µ) 2 =E X 2 2X E[X]+(E[X]) 2 =E X 2 2E[X]E[X]+(E[X]) 2 =E X 2 (E[X]) 2 S 2 n(x) = 1 n nx (x i x) 2 s 2 n(x) = 1 i=1 n 1 nx (x i x) 2 i=1
33 Variance E[Sn]= 2 n 1 n Var(Sn)= 2 n 1 n n E[s 2 n]= 2 Var(s 2 n)= 2 4 n 1 biased unbiased smaller variance greater variance
34 Variance of biased and unbiased estimator σ =
35 Markov's inequality 8a >0 P(X a) apple 1 a E[X] more in general: P(g(X) a) apple 1 a E[g(X)] g(x) g : R! R 0, 8x 2 R
36 Chebyshev's inequality 8a >0 P( X µ a) apple 2 a 2 P( X µ a ) apple 1 a 2 no more than 1/a 2 of the distribution's values can be more than a standard deviations away from the mean
37 Autocorrelation C xx ( ) =E[(X t µ)(x t+ µ)] R xx ( ) = E[(X t µ)(x t+ µ)] 2 Often the autocovariance is called autocorrelation even if this normalization has not been performed and vice-versa
38 Frequency domain analysis
39 Fourier transform The Fourier transform is given by! and the inverse transform is given by! The Fourier transform is also a random variable X(f) = lim T!1 x(t) = lim F!1 Z T/2 Z F/2 F/2 T/2 e i2 ft x(t) dt e i2 ft X(f) df
40 Power spectral density Average value of the squared magnitude of the Fourier transform S(f) =h X(f) 2 i = hx(f)x (f)i = lim T!1 1 T Z T/2 T/2 e i2 ft x(t) dt Z T/2 T/2 e i2 ft x(t 0 ) dt 0
41 = Z 1 Z = lim T!1 = lim T!1 = lim T!1 Wiener-Khinchin S(f)e i2 f df hx(f)x (f)ie i2 f df 1 T 1 T 1 T 1 Z 1 1 Z 1 1 Z T/2 T/2 Z T/2 = lim T!1 T T/2 = hx(t)x(t )i Z T/2 T/2 Z T/2 Z T/2 T/2 Z T/2 T/2 x(t)x(t theorem e i2 ft x(t) dt T/2 Z T/2 T/2 e i2 ft0 x(t 0 ) dt 0 e i2 f df e i2 f(t t0 ) df x(t)x(t 0 ) dt dt 0 (t t 0 )x(t)x(t 0 ) dt dt 0 ) dt
42 Examples
43 Parseval s theorem Z 1 hx(t)x(t )i = Z 1 1 S(f)e i2 f df )hx 2 (t)i = 1 S(f) df The average value of the square of the signal (variance if the signal has zero mean) is equal to the integral of the power spectral density
44 Examples
45 Shot noise Generated by discrete arrival electrons in a wire rain on a roof Interactions can be ignored Arrival independent Poisson process
46 Shot noise <I>= qn/t I(t) =q NX n=1 (t t n ) Z T/2 NX NX I(f) = lim T!1 T/2 e i2 ft q n=1 (t t n ) dt = q n=1 e i2 ft n S I (f) =<I(f)I (f) >= lim T!1 = lim T!1 q 2 N T = q<i> q 2 T NX n=1 e i2 ft n NX m=1 e i2 ft m! <I 2 noise >= 2q<I> f
47 Johnson noise Relaxation of thermal fluctuation in a resistor Small voltage fluctuation associated with thermal motion of electrons <V 2 noise >= 4kTR f
ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random
More informationFig 1: Stationary and Non Stationary Time Series
Module 23 Independence and Stationarity Objective: To introduce the concepts of Statistical Independence, Stationarity and its types w.r.to random processes. This module also presents the concept of Ergodicity.
More informationBasic Descriptions and Properties
CHAPTER 1 Basic Descriptions and Properties This first chapter gives basic descriptions and properties of deterministic data and random data to provide a physical understanding for later material in this
More informationRandom Variables. Cumulative Distribution Function (CDF) Amappingthattransformstheeventstotherealline.
Random Variables Amappingthattransformstheeventstotherealline. Example 1. Toss a fair coin. Define a random variable X where X is 1 if head appears and X is if tail appears. P (X =)=1/2 P (X =1)=1/2 Example
More informationStochastic Processes
Elements of Lecture II Hamid R. Rabiee with thanks to Ali Jalali Overview Reading Assignment Chapter 9 of textbook Further Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic
More informationIV. Covariance Analysis
IV. Covariance Analysis Autocovariance Remember that when a stochastic process has time values that are interdependent, then we can characterize that interdependency by computing the autocovariance function.
More informationStochastic Processes. A stochastic process is a function of two variables:
Stochastic Processes Stochastic: from Greek stochastikos, proceeding by guesswork, literally, skillful in aiming. A stochastic process is simply a collection of random variables labelled by some parameter:
More informationFundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes
Fundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes Klaus Witrisal witrisal@tugraz.at Signal Processing and Speech Communication Laboratory www.spsc.tugraz.at Graz University of
More informationFor a stochastic process {Y t : t = 0, ±1, ±2, ±3, }, the mean function is defined by (2.2.1) ± 2..., γ t,
CHAPTER 2 FUNDAMENTAL CONCEPTS This chapter describes the fundamental concepts in the theory of time series models. In particular, we introduce the concepts of stochastic processes, mean and covariance
More informationCHAPTERl BASIC DESCRIPTIONS AND PROPERTIES
CHAPTERl BASIC DESCRIPTIONS AND PROPERTIES This first chapter gives basic descriptions and properties of deterministic data and random data to provide a physical understanding for later material in this
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 informationChapter 6 - Random Processes
EE385 Class Notes //04 John Stensby Chapter 6 - Random Processes Recall that a random variable X is a mapping between the sample space S and the extended real line R +. That is, X : S R +. A random process
More informationStochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.
More informationStatistical signal processing
Statistical signal processing Short overview of the fundamentals Outline Random variables Random processes Stationarity Ergodicity Spectral analysis Random variable and processes Intuition: A random variable
More informationLecture Notes 7 Stationary Random Processes. Strict-Sense and Wide-Sense Stationarity. Autocorrelation Function of a Stationary Process
Lecture Notes 7 Stationary Random Processes Strict-Sense and Wide-Sense Stationarity Autocorrelation Function of a Stationary Process Power Spectral Density Continuity and Integration of Random Processes
More informationENSC327 Communications Systems 19: Random Processes. Jie Liang School of Engineering Science Simon Fraser University
ENSC327 Communications Systems 19: Random Processes Jie Liang School of Engineering Science Simon Fraser University 1 Outline Random processes Stationary random processes Autocorrelation of random processes
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 informationSignals and Spectra - Review
Signals and Spectra - Review SIGNALS DETERMINISTIC No uncertainty w.r.t. the value of a signal at any time Modeled by mathematical epressions RANDOM some degree of uncertainty before the signal occurs
More informationSignals and Spectra (1A) Young Won Lim 11/26/12
Signals and Spectra (A) Copyright (c) 202 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later
More informationModule 9: Stationary Processes
Module 9: Stationary Processes Lecture 1 Stationary Processes 1 Introduction A stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space.
More informationFundamentals of Noise
Fundamentals of Noise V.Vasudevan, Department of Electrical Engineering, Indian Institute of Technology Madras Noise in resistors Random voltage fluctuations across a resistor Mean square value in a frequency
More informationRandom Processes Why we Care
Random Processes Why we Care I Random processes describe signals that change randomly over time. I Compare: deterministic signals can be described by a mathematical expression that describes the signal
More informationCS145: Probability & Computing
CS45: Probability & Computing Lecture 5: Concentration Inequalities, Law of Large Numbers, Central Limit Theorem Instructor: Eli Upfal Brown University Computer Science Figure credits: Bertsekas & Tsitsiklis,
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 informationExpectation of Random Variables
1 / 19 Expectation of Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 13, 2015 2 / 19 Expectation of Discrete
More informationNotes on Random Processes
otes on Random Processes Brian Borchers and Rick Aster October 27, 2008 A Brief Review of Probability In this section of the course, we will work with random variables which are denoted by capital letters,
More informationLimiting Distributions
We introduce the mode of convergence for a sequence of random variables, and discuss the convergence in probability and in distribution. The concept of convergence leads us to the two fundamental results
More information2. Suppose (X, Y ) is a pair of random variables uniformly distributed over the triangle with vertices (0, 0), (2, 0), (2, 1).
Name M362K Final Exam Instructions: Show all of your work. You do not have to simplify your answers. No calculators allowed. There is a table of formulae on the last page. 1. Suppose X 1,..., X 1 are independent
More informationLecture 25: Review. Statistics 104. April 23, Colin Rundel
Lecture 25: Review Statistics 104 Colin Rundel April 23, 2012 Joint CDF F (x, y) = P [X x, Y y] = P [(X, Y ) lies south-west of the point (x, y)] Y (x,y) X Statistics 104 (Colin Rundel) Lecture 25 April
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 informationStochastic Processes: I. consider bowl of worms model for oscilloscope experiment:
Stochastic Processes: I consider bowl of worms model for oscilloscope experiment: SAPAscope 2.0 / 0 1 RESET SAPA2e 22, 23 II 1 stochastic process is: Stochastic Processes: II informally: bowl + drawing
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 informationStat 5101 Notes: Algorithms (thru 2nd midterm)
Stat 5101 Notes: Algorithms (thru 2nd midterm) Charles J. Geyer October 18, 2012 Contents 1 Calculating an Expectation or a Probability 2 1.1 From a PMF........................... 2 1.2 From a PDF...........................
More informationJoint Distribution of Two or More Random Variables
Joint Distribution of Two or More Random Variables Sometimes more than one measurement in the form of random variable is taken on each member of the sample space. In cases like this there will be a few
More informationE X A M. Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours. Number of pages incl.
E X A M Course code: Course name: Number of pages incl. front page: 6 MA430-G Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours Resources allowed: Notes: Pocket calculator,
More informationChapter 6: Random Processes 1
Chapter 6: Random Processes 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 information2 Continuous Random Variables and their Distributions
Name: Discussion-5 1 Introduction - Continuous random variables have a range in the form of Interval on the real number line. Union of non-overlapping intervals on real line. - We also know that for any
More informationRandom Variables. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 13 Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 8, 2013 2 / 13 Random Variable Definition A real-valued
More informationMathematics 426 Robert Gross Homework 9 Answers
Mathematics 4 Robert Gross Homework 9 Answers. Suppose that X is a normal random variable with mean µ and standard deviation σ. Suppose that PX > 9 PX
More informationSpectral representations and ergodic theorems for stationary stochastic processes
AMS 263 Stochastic Processes (Fall 2005) Instructor: Athanasios Kottas Spectral representations and ergodic theorems for stationary stochastic processes Stationary stochastic processes Theory and methods
More information13. Power Spectrum. For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if.
For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if jt X ( ) = xte ( ) dt, (3-) then X ( ) represents its energy spectrum. his follows from Parseval
More informationIf we want to analyze experimental or simulated data we might encounter the following tasks:
Chapter 1 Introduction If we want to analyze experimental or simulated data we might encounter the following tasks: Characterization of the source of the signal and diagnosis Studying dependencies Prediction
More informationProperties of the Autocorrelation Function
Properties of the Autocorrelation Function I The autocorrelation function of a (real-valued) random process satisfies the following properties: 1. R X (t, t) 0 2. R X (t, u) =R X (u, t) (symmetry) 3. R
More informationLECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity.
LECTURES 2-3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series {X t } is a series of observations taken sequentially over time: x t is an observation
More informationProbability and Statistics for Final Year Engineering Students
Probability and Statistics for Final Year Engineering Students By Yoni Nazarathy, Last Updated: May 24, 2011. Lecture 6p: Spectral Density, Passing Random Processes through LTI Systems, Filtering Terms
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 informationChapter 5 Random Variables and Processes
Chapter 5 Random Variables and Processes Wireless Information Transmission System Lab. Institute of Communications Engineering National Sun Yat-sen University Table of Contents 5.1 Introduction 5. Probability
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 II. Basic definitions A time series is a set of observations X t, each
More informationLimiting Distributions
Limiting Distributions We introduce the mode of convergence for a sequence of random variables, and discuss the convergence in probability and in distribution. The concept of convergence leads us to the
More information4 Classical Coherence Theory
This chapter is based largely on Wolf, Introduction to the theory of coherence and polarization of light [? ]. Until now, we have not been concerned with the nature of the light field itself. Instead,
More informationECE 636: Systems identification
ECE 636: Systems identification Lectures 3 4 Random variables/signals (continued) Random/stochastic vectors Random signals and linear systems Random signals in the frequency domain υ ε x S z + y Experimental
More informationChapter 4. Chapter 4 sections
Chapter 4 sections 4.1 Expectation 4.2 Properties of Expectations 4.3 Variance 4.4 Moments 4.5 The Mean and the Median 4.6 Covariance and Correlation 4.7 Conditional Expectation SKIP: 4.8 Utility Expectation
More informationStationary independent increments. 1. Random changes of the form X t+h X t fixed h > 0 are called increments of the process.
Stationary independent increments 1. Random changes of the form X t+h X t fixed h > 0 are called increments of the process. 2. If each set of increments, corresponding to non-overlapping collection of
More informationSRI VIDYA COLLEGE OF ENGINEERING AND TECHNOLOGY UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS
UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS 1. Define random process? The sample space composed of functions of time is called a random process. 2. Define Stationary process? If a random process is divided
More informationExercises and Answers to Chapter 1
Exercises and Answers to Chapter The continuous type of random variable X has the following density function: a x, if < x < a, f (x), otherwise. Answer the following questions. () Find a. () Obtain mean
More informationLIST OF FORMULAS FOR STK1100 AND STK1110
LIST OF FORMULAS FOR STK1100 AND STK1110 (Version of 11. November 2015) 1. Probability Let A, B, A 1, A 2,..., B 1, B 2,... be events, that is, subsets of a sample space Ω. a) Axioms: A probability function
More informationLecture 3: Signal and Noise
Lecture 3: Signal and Noise J. M. D. Coey School of Physics and CRANN, Trinity College Dublin Ireland. 1. Detection techniques 2. Random processes 3. Noise mechanisms 4. Thermodynamics 5. Noise reduction
More information7.7 The Schottky Formula for Shot Noise
110CHAPTER 7. THE WIENER-KHINCHIN THEOREM AND APPLICATIONS 7.7 The Schottky Formula for Shot Noise On p. 51, we found that if one averages τ seconds of steady electron flow of constant current then the
More informationLecture 7: Chapter 7. Sums of Random Variables and Long-Term Averages
Lecture 7: Chapter 7. Sums of Random Variables and Long-Term Averages ELEC206 Probability and Random Processes, Fall 2014 Gil-Jin Jang gjang@knu.ac.kr School of EE, KNU page 1 / 15 Chapter 7. Sums of Random
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 informationIntroduction to Systems Analysis and Decision Making Prepared by: Jakub Tomczak
Introduction to Systems Analysis and Decision Making Prepared by: Jakub Tomczak 1 Introduction. Random variables During the course we are interested in reasoning about considered phenomenon. In other words,
More informationSTA 6857 Autocorrelation and Cross-Correlation & Stationary Time Series ( 1.4, 1.5)
STA 6857 Autocorrelation and Cross-Correlation & Stationary Time Series ( 1.4, 1.5) Outline 1 Announcements 2 Autocorrelation and Cross-Correlation 3 Stationary Time Series 4 Homework 1c Arthur Berg STA
More informationGaussian, Markov and stationary processes
Gaussian, Markov and stationary processes Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ November
More informationStat 5101 Notes: Algorithms
Stat 5101 Notes: Algorithms Charles J. Geyer January 22, 2016 Contents 1 Calculating an Expectation or a Probability 3 1.1 From a PMF........................... 3 1.2 From a PDF...........................
More informationIntroduction to Probability and Stochastic Processes I
Introduction to Probability and Stochastic Processes I Lecture 3 Henrik Vie Christensen vie@control.auc.dk Department of Control Engineering Institute of Electronic Systems Aalborg University Denmark Slides
More information16.584: Random (Stochastic) Processes
1 16.584: Random (Stochastic) Processes X(t): X : RV : Continuous function of the independent variable t (time, space etc.) Random process : Collection of X(t, ζ) : Indexed on another independent variable
More informationDiscrete Probability Refresher
ECE 1502 Information Theory Discrete Probability Refresher F. R. Kschischang Dept. of Electrical and Computer Engineering University of Toronto January 13, 1999 revised January 11, 2006 Probability theory
More informationChp 4. Expectation and Variance
Chp 4. Expectation and Variance 1 Expectation In this chapter, we will introduce two objectives to directly reflect the properties of a random variable or vector, which are the Expectation and Variance.
More informationChapter 6. Random Processes
Chapter 6 Random Processes Random Process A random process is a time-varying function that assigns the outcome of a random experiment to each time instant: X(t). For a fixed (sample path): a random process
More informationSTOCHASTIC PROBABILITY THEORY PROCESSES. Universities Press. Y Mallikarjuna Reddy EDITION
PROBABILITY THEORY STOCHASTIC PROCESSES FOURTH EDITION Y Mallikarjuna Reddy Department of Electronics and Communication Engineering Vasireddy Venkatadri Institute of Technology, Guntur, A.R < Universities
More informationSTAT 430/510: Lecture 16
STAT 430/510: Lecture 16 James Piette June 24, 2010 Updates HW4 is up on my website. It is due next Mon. (June 28th). Starting today back at section 6.7 and will begin Ch. 7. Joint Distribution of Functions
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 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 informationfor valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I
Code: 15A04304 R15 B.Tech II Year I Semester (R15) Regular Examinations November/December 016 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks:
More informationSTAT 248: EDA & Stationarity Handout 3
STAT 248: EDA & Stationarity Handout 3 GSI: Gido van de Ven September 17th, 2010 1 Introduction Today s section we will deal with the following topics: the mean function, the auto- and crosscovariance
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 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 informationInterest Rate Models:
1/17 Interest Rate Models: from Parametric Statistics to Infinite Dimensional Stochastic Analysis René Carmona Bendheim Center for Finance ORFE & PACM, Princeton University email: rcarmna@princeton.edu
More information2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit dwm/courses/2tf
Time-Frequency Analysis II (HT 20) 2AH 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 20 For hints and answers visit www.robots.ox.ac.uk/ dwm/courses/2tf David Murray. A periodic
More informationCommunication Theory II
Communication Theory II Lecture 8: Stochastic Processes Ahmed Elnakib, PhD Assistant Professor, Mansoura University, Egypt March 5 th, 2015 1 o Stochastic processes What is a stochastic process? Types:
More informationSUMMARY OF PROBABILITY CONCEPTS SO FAR (SUPPLEMENT FOR MA416)
SUMMARY OF PROBABILITY CONCEPTS SO FAR (SUPPLEMENT FOR MA416) D. ARAPURA This is a summary of the essential material covered so far. The final will be cumulative. I ve also included some review problems
More informationBASICS OF PROBABILITY
October 10, 2018 BASICS OF PROBABILITY Randomness, sample space and probability Probability is concerned with random experiments. That is, an experiment, the outcome of which cannot be predicted with certainty,
More informationProbability Models. 4. What is the definition of the expectation of a discrete random variable?
1 Probability Models The list of questions below is provided in order to help you to prepare for the test and exam. It reflects only the theoretical part of the course. You should expect the questions
More information3. Probability and Statistics
FE661 - Statistical Methods for Financial Engineering 3. Probability and Statistics Jitkomut Songsiri definitions, probability measures conditional expectations correlation and covariance some important
More informationProbability Models in Electrical and Computer Engineering Mathematical models as tools in analysis and design Deterministic models Probability models
Probability Models in Electrical and Computer Engineering Mathematical models as tools in analysis and design Deterministic models Probability models Statistical regularity Properties of relative frequency
More information7 The Waveform Channel
7 The Waveform Channel The waveform transmitted by the digital demodulator will be corrupted by the channel before it reaches the digital demodulator in the receiver. One important part of the channel
More informationFormulas for probability theory and linear models SF2941
Formulas for probability theory and linear models SF2941 These pages + Appendix 2 of Gut) are permitted as assistance at the exam. 11 maj 2008 Selected formulae of probability Bivariate probability Transforms
More informationStochastic Processes. Chapter Definitions
Chapter 4 Stochastic Processes Clearly data assimilation schemes such as Optimal Interpolation are crucially dependent on the estimates of background and observation error statistics. Yet, we don t know
More informationNotes for Math 324, Part 19
48 Notes for Math 324, Part 9 Chapter 9 Multivariate distributions, covariance Often, we need to consider several random variables at the same time. We have a sample space S and r.v. s X, Y,..., which
More informationP (x). all other X j =x j. If X is a continuous random vector (see p.172), then the marginal distributions of X i are: f(x)dx 1 dx n
JOINT DENSITIES - RANDOM VECTORS - REVIEW Joint densities describe probability distributions of a random vector X: an n-dimensional vector of random variables, ie, X = (X 1,, X n ), where all X is are
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 informationSpectral Analysis of Random Processes
Spectral Analysis of Random Processes Spectral Analysis of Random Processes Generally, all properties of a random process should be defined by averaging over the ensemble of realizations. Generally, all
More informationChapter 4. Continuous Random Variables
Chapter 4. Continuous Random Variables Review Continuous random variable: A random variable that can take any value on an interval of R. Distribution: A density function f : R R + such that 1. non-negative,
More information4. Distributions of Functions of Random Variables
4. Distributions of Functions of Random Variables Setup: Consider as given the joint distribution of X 1,..., X n (i.e. consider as given f X1,...,X n and F X1,...,X n ) Consider k functions g 1 : R n
More information5 Operations on Multiple Random Variables
EE360 Random Signal analysis Chapter 5: Operations on Multiple Random Variables 5 Operations on Multiple Random Variables Expected value of a function of r.v. s Two r.v. s: ḡ = E[g(X, Y )] = g(x, y)f X,Y
More informationThis is a Gaussian probability centered around m = 0 (the most probable and mean position is the origin) and the mean square displacement m 2 = n,or
Physics 7b: Statistical Mechanics Brownian Motion Brownian motion is the motion of a particle due to the buffeting by the molecules in a gas or liquid. The particle must be small enough that the effects
More informationReview of Probability Theory
Review of Probability Theory Arian Maleki and Tom Do Stanford University Probability theory is the study of uncertainty Through this class, we will be relying on concepts from probability theory for deriving
More informationMath 180C, Spring Supplement on the Renewal Equation
Math 18C Spring 218 Supplement on the Renewal Equation. These remarks supplement our text and set down some of the material discussed in my lectures. Unexplained notation is as in the text or in lecture.
More informationLecture 4 : Random variable and expectation
Lecture 4 : Random variable and expectation Study Objectives: to learn the concept of 1. Random variable (rv), including discrete rv and continuous rv; and the distribution functions (pmf, pdf and cdf).
More informationEAS 305 Random Processes Viewgraph 1 of 10. Random Processes
EAS 305 Random Processes Viewgraph 1 of 10 Definitions: Random Processes A random process is a family of random variables indexed by a parameter t T, where T is called the index set λ i Experiment outcome
More information