Probability and Statistics
|
|
- Gerald Barton
- 6 years ago
- Views:
Transcription
1 Probability and Statistics 1
2 Contents some stochastic processes Stationary Stochastic Processes 2
3 4. Some Stochastic Processes 4.1 Bernoulli process 4.2 Binomial process 4.3 Sine wave process 4.4 Random-telegraph process 3
4 4.1 Bernoulli Process Definition The infinite sequence of random variables is called a random sequence in particular 1 and 0 n are statistically independent P[ 1] p P[ 0] 1 p q n {,,, } {, n 1,2,3, } n n is a Bernoulli random variable, if n is fixed. { n, n 1, 2,3, } is called a Bernoulli random process 4
5 Probability Distribution Unending sequence of flips of a coin. Flip a coin at each positive-integer value of time (starting at time 1) and observe the result. To determine the joint probability flipping a coin, a finite number of times n. 1 If a head results from the coin flip at time n n 0 If not n are statistically independent the joint-probability distribution is the set of P[ x, x, x ] n P[ x ] P[ x ] P[ x ] 1 2 n n xi n 0,1 2 n probabilities n i1, 2,3, n 5
6 Example: two-dimensional case The joint-probability distribution is the set of probabilities and P[ 1, 1] p P[ 1, 0] pq P[ 0, 1] qp P[ 0, 0] q Exercise : Determine the probabilities of occurrence of the three particular Bernoulli sample sequences 6
7 Statistical Average E[ n] p var( ) pq p(1 p) n This expectation and variance are called the mean value and variance of the Bernoulli process 7
8 4.2 Binomial Process Definition A random process variable Y i1 random variables. n n i { Yn, n1, 2,3, } in which the counting random is defined to be a sum of independent Bernoulli Example: Unending sequence of flips of a coin by a zero-one Bernoulli process and count the number of heads in n flips. 8
9 Probability Distribution If the i can assume only the values +1 or 0,then n k P[ Yn k] P (1 P) k n n! k k!( n k)! k 0,1,2,, n nk Notice : while each counting random variable of independent random variable, the various independent. Y n Y n is the a sum are not 9
10 Probability Distribution E[ Yn ] np var( Yn ) npq The two difference counting random variables cov( Y, Y ) pq min( m, n) m n Yn and Ym var( Y Y ) m n pq m min( mn, ) : the smaller of the indices m and n n 10
11 4.3 Sine Wave Process Definition A random process { ( t), t, T} where the index set T is a continuous, and where ( t) V sin( t ) for all value of t in T,, and here are random variables V 11
12 Example in practice : The outputs of An electronics instrument manufacturer produces sine wave oscillators Output of a particular oscillator at any time can be characterized by the sample function x( t ) vsin( t ) 1 Outputs of various oscillator at a specified time can be characterized by the random variable Vsin( t ) t 1 1 Outputs of various oscillator at any time can be characterized by the sine wave process V sin( t ) {, t 0} t t 12
13 4.4 Random-telegraph process Random-telegraph process This is a real random process whose sample functions at any instant of time t may assume only the values zero or one. And it is assumed that 1 P[ ( t) 0] P[ ( t) 1] 2 The probability Pk [, ] that k transversals from one value to another occur in a time interval of length is given by the Poisson probability distribution ( ) k e P[ k, ] k 0,1,2 k! 13
14 The occurrence of k transversals in an interval of is statistically independent of the value assumed by any particular sample function at the start of the given interval 1 x A random-telegraph sample function t 14
15 Exercise-1: For the above random-telegraph process a. Show that b. Show that 1 E[ ( t)] 2 R ( t, t ) P[ ( t ) 1, ( t ) 1] c. Show further that 1 R ( t1, t2) P[ k even] 2 Where P[ k even] is the probability that the number of transversals which occur in an interval of duration even t t 1 2 is 15
16 5. Strict-Sense Stationary (S.S.S) Definition if f ( x1, x2, xn, t1, t2, tn ) f ( x1, x2, xn, t1 c, t2 c, tn c) for any c, where the left side represents the joint density function of the random variables and 1 ( t1), 2 ( t2),, n ( tn ) the right side corresponds to the joint density function of the random variables t c), ( t c),, ( t 1 ( n n c A process (t) is said to be strict-sense stationary if the above is true for all, i 1, 2,, n, n 1, 2, and any c. t i Stationary processes exhibit statistical properties that are invariant to shift in the time index. In strict terms, the statistical properties are governed by the joint probability density function. 16 ).
17 Strict-Sense Stationary (S.S.S) Probability Distribution s Properties For a first-order strict sense stationary process, ( x, t) f ( x, t c) for any c. In particular c = t gives f f ( x, t) f ( x) i.e., the first-order density of (t) is independent of t. In that case Similarly, for a second-order strict-sense stationary process we have from the previous page for any c. For c = t 2 we get f ( x, x ; t, t ) f ( x, x ; t t ) f ( x, x ; t, t ) f ( x, x ; t c, t c)
18 Strict-Sense Stationary (S.S.S) Statistical Average s Properties E[ ( t)] x f ( x) dx, a constant. the autocorrelation function is given by R ( t, t ) E{ ( t ) ( t )} * x x f ( x, x, t t ) dx dx * R t t R R * ( 1 2) ( ) ( ), i.e., the autocorrelation function of a second order strict-sense stationary process depends only on the difference of the time Indices. 18
19 Strict-Sense Stationary (S.S.S) Exercise-2 Consider the sine wave process {, t 0}. V cost t Show whether or not this random process is stationary in the strict sense. t 19
20 6. Wide-Sense Stationary (W.S.S), Definition a process (t) is said to be Wide-Sense Stationary if (i) (ii) (iii) 2 E ( t) E{ ( t)} E{ ( t ) ( t )} R ( t t ) R ( ) * for wide-sense stationary processes, the mean is a constant and the autocorrelation function depends only on the difference between the time indices. Notice that they do not say anything about the nature of the probability density functions, and instead deal with the average behavior of the process. 20
21 Relationship between S.S.S and W.S.S Since they follow the previous definition, strict-sense stationarity always implies wide-sense stationarity only when the process is limited in power. The converse is not true in general, the only exception being the Gaussian process. 21
22 Properties of stationary stochastic process Autocorrelation Function If real random process { ( t), t } is stationary, then R ( ) R ( ) R ( ) R (0) lim R ( ) R ( ) m 2 in practice if then lim R ( ) R ( ) m 2 2 R (0) m 2 2 R R (0) ( ) 22
23 7. Jointly Stationarity 1) Definition A pair of real random process { ( t), t } and { Y( t), t } are jointly stationary in wide sense, when E[ ( t)] m E[ Y ( t)] m R ( t, t ) R ( ) R ( t, t ) R ( ) R ( t, t ) R ( ) R ( t, t ) R ( ) for all values t Y Y Y Y Y Y Y 23
24 Exercise-3 Let the two random process { U( t), t } { V ( t), t } be such that U( t) cos t Y sin t V ( t) Y cost sint for all t, where and Y are independent real random variables for which E E Y E E Y 2 2 [ ] 0 [ ] [ ] 1 [ ] and a. Show that the two processes are individually stationary in the wide sense. b. Show that they are not jointly stationary in the wide sense. 24
25 2) Properties of Cross-correlation Function The two real random process { ( t), t } and { Y( t), t } are individually stationary in the wide sense and are jointly stationary in the wide sense. Then R Y ( ) R ( ) Y 2 RY ( ) R (0) RY (0) E Y E E Y This two processes are jointly stationary, then Z( t) ( t) Y( t) is stationary. 25
26 3) Cross-covariance Function The two random process { ( t), t } and { Y( t), t } are individually stationary in the wide sense and are jointly stationary in the wide sense. Then C Y ( ) C ( ) Y C ( ) Y Y 26
27 Homework 10.6,10.8,10.9, 10.10, 10.13,
Stationary 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 informationStochastic Processes. Monday, November 14, 11
Stochastic Processes 1 Definition and Classification X(, t): stochastic process: X : T! R (, t) X(, t) where is a sample space and T is time. {X(, t) is a family of r.v. defined on {, A, P and indexed
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 informationELEG 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 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 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 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 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 informationChapter 4 Random process. 4.1 Random process
Random processes - Chapter 4 Random process 1 Random processes Chapter 4 Random process 4.1 Random process 4.1 Random process Random processes - Chapter 4 Random process 2 Random process 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 informationChapter 1 Statistical Reasoning Why statistics? Section 1.1 Basics of Probability Theory
Chapter 1 Statistical Reasoning Why statistics? Uncertainty of nature (weather, earth movement, etc. ) Uncertainty in observation/sampling/measurement Variability of human operation/error imperfection
More informationSuppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8.
Suppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8. Coin A is flipped until a head appears, then coin B is flipped until
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 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 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 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 informationRandom Processes. DS GA 1002 Probability and Statistics for Data Science.
Random Processes DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Aim Modeling quantities that evolve in time (or space)
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 informationRandom Process. Random Process. Random Process. Introduction to Random Processes
Random Process A random variable is a function X(e) that maps the set of experiment outcomes to the set of numbers. A random process is a rule that maps every outcome e of an experiment to a function X(t,
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 05 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA6 MATERIAL NAME : University Questions MATERIAL CODE : JM08AM004 REGULATION : R008 UPDATED ON : Nov-Dec 04 (Scan
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 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 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 informationStatistics, Probability Distributions & Error Propagation. James R. Graham
Statistics, Probability Distributions & Error Propagation James R. Graham Sample & Parent Populations Make measurements x x In general do not expect x = x But as you take more and more measurements a pattern
More informationUCSD ECE250 Handout #24 Prof. Young-Han Kim Wednesday, June 6, Solutions to Exercise Set #7
UCSD ECE50 Handout #4 Prof Young-Han Kim Wednesday, June 6, 08 Solutions to Exercise Set #7 Polya s urn An urn initially has one red ball and one white ball Let X denote the name of the first ball drawn
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 information6 The normal distribution, the central limit theorem and random samples
6 The normal distribution, the central limit theorem and random samples 6.1 The normal distribution We mentioned the normal (or Gaussian) distribution in Chapter 4. It has density f X (x) = 1 σ 1 2π e
More informationTSKS01 Digital Communication Lecture 1
TSKS01 Digital Communication Lecture 1 Introduction, Repetition, and Noise Modeling Emil Björnson Department of Electrical Engineering (ISY) Division of Communication Systems Emil Björnson Course Director
More informationProblems on Discrete & Continuous R.Vs
013 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Probability & Random Process : MA 61 : University Questions : SKMA1004 Name of the Student: Branch: Unit I (Random Variables) Problems on Discrete
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 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 informationLecture - 30 Stationary Processes
Probability and Random Variables Prof. M. Chakraborty Department of Electronics and Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture - 30 Stationary Processes So,
More informationReview of Probability. CS1538: Introduction to Simulations
Review of Probability CS1538: Introduction to Simulations Probability and Statistics in Simulation Why do we need probability and statistics in simulation? Needed to validate the simulation model Needed
More informationSTOCHASTIC PROCESSES, DETECTION AND ESTIMATION Course Notes
STOCHASTIC PROCESSES, DETECTION AND ESTIMATION 6.432 Course Notes Alan S. Willsky, Gregory W. Wornell, and Jeffrey H. Shapiro Department of Electrical Engineering and Computer Science Massachusetts Institute
More informationRandom Variables Example:
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More informationChapter 5. Means and Variances
1 Chapter 5 Means and Variances Our discussion of probability has taken us from a simple classical view of counting successes relative to total outcomes and has brought us to the idea of a probability
More informationBinomial and Poisson Probability Distributions
Binomial and Poisson Probability Distributions Esra Akdeniz March 3, 2016 Bernoulli Random Variable Any random variable whose only possible values are 0 or 1 is called a Bernoulli random variable. What
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 08 SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : University Questions REGULATION : R03 UPDATED ON : November 07 (Upto N/D 07 Q.P) (Scan the
More informationProbability Space. J. McNames Portland State University ECE 538/638 Stochastic Signals Ver
Stochastic Signals Overview Definitions Second order statistics Stationarity and ergodicity Random signal variability Power spectral density Linear systems with stationary inputs Random signal memory Correlation
More informationBasics on Probability. Jingrui He 09/11/2007
Basics on Probability Jingrui He 09/11/2007 Coin Flips You flip a coin Head with probability 0.5 You flip 100 coins How many heads would you expect Coin Flips cont. You flip a coin Head with probability
More informationLecture 2: Repetition of probability theory and statistics
Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:
More informationECE 450 Homework #3. 1. Given the joint density function f XY (x,y) = 0.5 1<x<2, 2<y< <x<4, 2<y<3 0 else
ECE 450 Homework #3 0. Consider the random variables X and Y, whose values are a function of the number showing when a single die is tossed, as show below: Exp. Outcome 1 3 4 5 6 X 3 3 4 4 Y 0 1 3 4 5
More informationLecture 4 - Random walk, ruin problems and random processes
Lecture 4 - Random walk, ruin problems and random processes Jan Bouda FI MU April 19, 2009 Jan Bouda (FI MU) Lecture 4 - Random walk, ruin problems and random processesapril 19, 2009 1 / 30 Part I Random
More informationQuestion Paper Code : AEC11T03
Hall Ticket No Question Paper Code : AEC11T03 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)
More informationEE 121: Introduction to Digital Communication Systems. 1. Consider the following discrete-time communication system. There are two equallly likely
EE 11: Introduction to Digital Communication Systems Midterm Solutions 1. Consider the following discrete-time communication system. There are two equallly likely messages to be transmitted, and they are
More informationELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Random Variable Discrete Random
More informationReview of Mathematical Concepts. Hongwei Zhang
Review of Mathematical Concepts Hongwei Zhang http://www.cs.wayne.edu/~hzhang Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable
More informationEcon 424 Time Series Concepts
Econ 424 Time Series Concepts Eric Zivot January 20 2015 Time Series Processes Stochastic (Random) Process { 1 2 +1 } = { } = sequence of random variables indexed by time Observed time series of length
More information1 INFO Sep 05
Events A 1,...A n are said to be mutually independent if for all subsets S {1,..., n}, p( i S A i ) = p(a i ). (For example, flip a coin N times, then the events {A i = i th flip is heads} are mutually
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 information1: PROBABILITY REVIEW
1: PROBABILITY REVIEW Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) Slides 1: Probability Review 1 / 56 Outline We will review the following
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 informationFinal Examination Solutions (Total: 100 points)
Final Examination Solutions (Total: points) There are 4 problems, each problem with multiple parts, each worth 5 points. Make sure you answer all questions. Your answer should be as clear and readable
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 informationLecture 15. Theory of random processes Part III: Poisson random processes. Harrison H. Barrett University of Arizona
Lecture 15 Theory of random processes Part III: Poisson random processes Harrison H. Barrett University of Arizona 1 OUTLINE Poisson and independence Poisson and rarity; binomial selection Poisson point
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 informationExpectation. DS GA 1002 Statistical and Mathematical Models. Carlos Fernandez-Granda
Expectation DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos Fernandez-Granda Aim Describe random variables with a few numbers: mean, variance,
More informationECE Homework Set 3
ECE 450 1 Homework Set 3 0. Consider the random variables X and Y, whose values are a function of the number showing when a single die is tossed, as show below: Exp. Outcome 1 3 4 5 6 X 3 3 4 4 Y 0 1 3
More informationPart IA Probability. Definitions. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Definitions Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
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 informationLecture 3. Discrete Random Variables
Math 408 - Mathematical Statistics Lecture 3. Discrete Random Variables January 23, 2013 Konstantin Zuev (USC) Math 408, Lecture 3 January 23, 2013 1 / 14 Agenda Random Variable: Motivation and Definition
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 informationDiscrete Random Variables
Discrete Random Variables An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan Introduction The markets can be thought of as a complex interaction of a large number of random processes,
More informationSTAT/MATH 395 A - PROBABILITY II UW Winter Quarter Moment functions. x r p X (x) (1) E[X r ] = x r f X (x) dx (2) (x E[X]) r p X (x) (3)
STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 07 Néhémy Lim Moment functions Moments of a random variable Definition.. Let X be a rrv on probability space (Ω, A, P). For a given r N, E[X r ], if it
More informationDiscrete Random Variables
CPSC 53 Systems Modeling and Simulation Discrete Random Variables Dr. Anirban Mahanti Department of Computer Science University of Calgary mahanti@cpsc.ucalgary.ca Random Variables A random variable is
More informationIntroduction to Statistics and Error Analysis
Introduction to Statistics and Error Analysis Physics116C, 4/3/06 D. Pellett References: Data Reduction and Error Analysis for the Physical Sciences by Bevington and Robinson Particle Data Group notes
More informationECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen. D. van Alphen 1
ECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen D. van Alphen 1 Lecture 10 Overview Part 1 Review of Lecture 9 Continuing: Systems with Random Inputs More about Poisson RV s Intro. to Poisson Processes
More informationLycka till!
Avd. Matematisk statistik TENTAMEN I SF294 SANNOLIKHETSTEORI/EXAM IN SF294 PROBABILITY THE- ORY MONDAY THE 14 T H OF JANUARY 28 14. p.m. 19. p.m. Examinator : Timo Koski, tel. 79 71 34, e-post: timo@math.kth.se
More informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Tobias Neckel, Ionuț-Gabriel Farcaș Lehrstuhl Informatik V Summer Semester 2017 Lecture 2: Repetition of probability theory and statistics Example: coin flip Example
More informationEE/CpE 345. Modeling and Simulation. Fall Class 5 September 30, 2002
EE/CpE 345 Modeling and Simulation Class 5 September 30, 2002 Statistical Models in Simulation Real World phenomena of interest Sample phenomena select distribution Probabilistic, not deterministic Model
More informationRandom Process Review
Rando Process Review Consider a rando process t, and take k saples. For siplicity, we will set k. However it should ean any nuber of saples. t () t x t, t, t We have a rando vector t, t, t. If we find
More informationProbability reminders
CS246 Winter 204 Mining Massive Data Sets Probability reminders Sammy El Ghazzal selghazz@stanfordedu Disclaimer These notes may contain typos, mistakes or confusing points Please contact the author so
More informationP 1.5 X 4.5 / X 2 and (iii) The smallest value of n for
DHANALAKSHMI COLLEGE OF ENEINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING MA645 PROBABILITY AND RANDOM PROCESS UNIT I : RANDOM VARIABLES PART B (6 MARKS). A random variable X
More informationExpectation. DS GA 1002 Probability and Statistics for Data Science. Carlos Fernandez-Granda
Expectation DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Aim Describe random variables with a few numbers: mean,
More informationChapter 8. Some Approximations to Probability Distributions: Limit Theorems
Chapter 8. Some Approximations to Probability Distributions: Limit Theorems Sections 8.2 -- 8.3: Convergence in Probability and in Distribution Jiaping Wang Department of Mathematical Science 04/22/2013,
More informationMath489/889 Stochastic Processes and Advanced Mathematical Finance Solutions for Homework 7
Math489/889 Stochastic Processes and Advanced Mathematical Finance Solutions for Homework 7 Steve Dunbar Due Mon, November 2, 2009. Time to review all of the information we have about coin-tossing fortunes
More informationG.PULLAIAH COLLEGE OF ENGINEERING & TECHNOLOGY DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING PROBABILITY THEORY & STOCHASTIC PROCESSES
G.PULLAIAH COLLEGE OF ENGINEERING & TECHNOLOGY DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING PROBABILITY THEORY & STOCHASTIC PROCESSES LECTURE NOTES ON PTSP (15A04304) B.TECH ECE II YEAR I SEMESTER
More informationStat 134 Fall 2011: Notes on generating functions
Stat 3 Fall 0: Notes on generating functions Michael Lugo October, 0 Definitions Given a random variable X which always takes on a positive integer value, we define the probability generating function
More informationData, Estimation and Inference
Data, Estimation and Inference Pedro Piniés ppinies@robots.ox.ac.uk Michaelmas 2016 1 2 p(x) ( = ) = δ 0 ( < < + δ ) δ ( ) =1. x x+dx (, ) = ( ) ( ) = ( ) ( ) 3 ( ) ( ) 0 ( ) =1 ( = ) = ( ) ( < < ) = (
More informationLecture 4: Sampling, Tail Inequalities
Lecture 4: Sampling, Tail Inequalities Variance and Covariance Moment and Deviation Concentration and Tail Inequalities Sampling and Estimation c Hung Q. Ngo (SUNY at Buffalo) CSE 694 A Fun Course 1 /
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 information5.9 Power Spectral Density Gaussian Process 5.10 Noise 5.11 Narrowband Noise
Chapter 5 Random Variables and Processes Wireless Information Transmission System Lab. Institute of Communications Engineering g National Sun Yat-sen University Table of Contents 5.1 Introduction 5. Probability
More informationPHYS 114 Exam 1 Answer Key NAME:
PHYS 4 Exam Answer Key AME: Please answer all of the questions below. Each part of each question is worth points, except question 5, which is worth 0 points.. Explain what the following MatLAB commands
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 informationChapter 7: Theoretical Probability Distributions Variable - Measured/Categorized characteristic
BSTT523: Pagano & Gavreau, Chapter 7 1 Chapter 7: Theoretical Probability Distributions Variable - Measured/Categorized characteristic Random Variable (R.V.) X Assumes values (x) by chance Discrete R.V.
More informationThe distribution inherited by Y is called the Cauchy distribution. Using that. d dy ln(1 + y2 ) = 1 arctan(y)
Stochastic Processes - MM3 - Solutions MM3 - Review Exercise Let X N (0, ), i.e. X is a standard Gaussian/normal random variable, and denote by f X the pdf of X. Consider also a continuous random variable
More informationME 597: AUTONOMOUS MOBILE ROBOTICS SECTION 2 PROBABILITY. Prof. Steven Waslander
ME 597: AUTONOMOUS MOBILE ROBOTICS SECTION 2 Prof. Steven Waslander p(a): Probability that A is true 0 pa ( ) 1 p( True) 1, p( False) 0 p( A B) p( A) p( B) p( A B) A A B B 2 Discrete Random Variable X
More information04. Random Variables: Concepts
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 215 4. Random Variables: Concepts Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More informationPreliminary statistics
1 Preliminary statistics The solution of a geophysical inverse problem can be obtained by a combination of information from observed data, the theoretical relation between data and earth parameters (models),
More information1 Generating functions
1 Generating functions Even quite straightforward counting problems can lead to laborious and lengthy calculations. These are greatly simplified by using generating functions. 2 Definition 1.1. Given a
More informationA SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES
A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES ROY M. HOWARD Department of Electrical Engineering & Computing Curtin University of Technology Perth, Australia WILEY CONTENTS Preface xiii 1 A Signal
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 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 Distributions
A simplest example of random experiment is a coin-tossing, formally called Bernoulli trial. It happens to be the case that many useful distributions are built upon this simplest form of experiment, whose
More informationPhysics 403 Probability Distributions II: More Properties of PDFs and PMFs
Physics 403 Probability Distributions II: More Properties of PDFs and PMFs Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Last Time: Common Probability Distributions
More informationLecture 2 Binomial and Poisson Probability Distributions
Binomial Probability Distribution Lecture 2 Binomial and Poisson Probability Distributions Consider a situation where there are only two possible outcomes (a Bernoulli trial) Example: flipping a coin James
More information1 Random variables and distributions
Random variables and distributions In this chapter we consider real valued functions, called random variables, defined on the sample space. X : S R X The set of possible values of X is denoted by the set
More information6.041/6.431 Fall 2010 Quiz 2 Solutions
6.04/6.43: Probabilistic Systems Analysis (Fall 200) 6.04/6.43 Fall 200 Quiz 2 Solutions Problem. (80 points) In this problem: (i) X is a (continuous) uniform random variable on [0, 4]. (ii) Y is an exponential
More informationParametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes
Parametric Signal Modeling and Linear Prediction Theory 1. Discrete-time Stochastic Processes Electrical & Computer Engineering North Carolina State University Acknowledgment: ECE792-41 slides were adapted
More informationDennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa
Dennis Bricker Dept of Mechanical & Industrial Engineering The University of Iowa dennis-bricker@uiowa.edu Probability Theory Results page 1 D.Bricker, U. of Iowa, 2002 Probability of simultaneous occurrence
More information