Random variable X is a mapping that maps each outcome s in the sample space to a unique real number x, < x <. ( ) X s. Real Line
|
|
- Angel Mason
- 5 years ago
- Views:
Transcription
1 Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number, < <. ( ) s s : outcome Sample Space Real Line Eample Toss a coin. Define random variable as follows: = 0 if heads if tails Roll a dice. Define random variable Y as follows: if,3,or 5 Y = 0 if 2,4, or 6 Consider a packet router. Eamine its queue at a random time. Define random variable N as the number of packets waiting in the queue. Go to the bus stop at a random time. Define random variable W as the amount of time you wait until the net bus arrives. Each time we repeat the eperiment, the outcome may change in the sample space and the value of the random variable changes according to the rule of the mapping, Notes on the Mapping Range of random variable : R = { = ( s) for some s S} In general, is a many-to-one mapping, but never one-to-many
2 2 Eample- Bernoulli Toss a biased coin. The coin falls down heads with probability p. Define random variable as = 0 for heads for tails Range: R =? P= [ ] =? P [ = 0.5] =?, P [ 0.] =?, P [ > 0] =?, P [ 2] =?, P [ > 3] =?. is referred to as a Bernoulli rv. Eample - Geometric Toss a biased coin. is the number of times we toss the coin until we see the first head. Range: R = {,2,3, } P [ = j] =? is a referred to as a geometric rv. Eample Eponential Consider a packet router that passes arriving packets to net destination routers. Measure the time between successive packets arrivals, referred to as the packet inter-arrival time. Let denote the packet inter-arrival time. Range: R = { r 0 r< } Eperiments show frequently [ ] λ P [ ] e for 0, where λ is the packet arrival rate, e.g., packets/sec. is referred to as an eponential rv.
3 3 Use of Random Variables With rvs, we can define functions of rvs. For eample, Find Pe [ ] U = + Y. Find P U. [ ] W = Rsin Θ. Find P W w. Notation P[ = ] : is a random variable, and is a constant or a simple variable. Types of Random Variables Discrete rv Continuous rv Discrete Random Variables The range consists of finite, countable real numbers such as{4,6,8}, or countably infinite real numbers such as {0,,2, } or {, 2,,0,,2, }. Continuous Random Variables When the range is not countable, the random variable is a continuous one.
4 4 Probability Distributions cumulative distribution function (cdf) probability density function probability mass function cumulative distribution function (cdf) The cdf of a random variable F ( ) = P [ ] is defined as Eample Uniform Distribution Throw a dart at a spinning wheel. is the phase where the dart hits the wheel. The range of the random variable is R = {0 < 2 π}. Within the range, 2 P [ 2] = for any phases π. 2π The cdf is 2π F ( ) = P[ ] = 0 < 2π 2π 0 < 0 F ( ) 0 2π is a continuous random variable. is referred to as a uniform random variable, or is said to have a uniform probability distribution.
5 5 Eample Uniform Distribution Buses arrive periodically with a period of T. You arrive at the bus stop at a random time. Define random variable W as the time you wait until the net bus arrives. Find the cdf of W. Ans. > T FW ( ) = P[ W ] = 0 T T 0 < 0 Uniform Distribution ( ) In short, U a, b, a< b. b a F ( ) = P[ ] = a < b b a 0 < a Eample - Bernoulli Toss a coin. Define as for a head with prob p = 0 for a tail with prob p. is referred to as a Bernoulli random variable. F ( ) = P[ ] = p 0 < 0 < 0 F ( ) p 0 All discrete rvs have discontinuities in their cdf. The value of the cdf is taken approaching from the right.
6 6 Properties of cdf 0 F ( ) 2 F ( ) = 3 F ( ) = 0 4 F ( ) is a non-decreasing function of 5 F ( ) is continuous from the right: that is, F ( b) = lim F ( b+ h). h 0 6 P[ a< b] = F ( b) F ( a) However, for a continuous rv, feeel safe to say Pa [ b] = F ( b) F ( a)
7 7 probability density function (pdf) The pdf of a rv is the derivative of its cdf: d f( ) F( ) d Properties of pdf The pdf is the rate at which the cdf increases. 2 Integrate the pdf to get the cdf. F( ) = f( u) du 3 To find the probability b P[ a b] = F ( b) F ( a) = f ( u) du a Be careful of the = sign for discrete random variables. 4 f ( ) 0 for any, and f ( u) du=.
8 8 Eample Y is an eponential random variable with thee cdf FY ( y) = e [ ] i) Find P Y 2. 2 y for y 0. ii) Find the pdf of Y. iii) Plot the cdf and pdf. For any pdf, i) non - negative ii) area sums to. For any cdf, i) nonn - decreasing from 0 to
9 9 probability mass function (pmf) The cdf of a discrete random variable has discontinuities. F ( ) p 0 The pdf consists of delta functions. f ( ) p p 0 When the range is {,,, }, we often use the notation p P[ = ]. 0 2 { p k } of a discrete random variable is referred to as the probability mass function (pmf). k k For a discrete random variable, it is easier to find the pmf first and then the cdf from the following relation F ( ) = P[ ] = p. k k
10 0 Notes on Continuous Distributions For a continuous random variable, it is easier to find the cdf first and then find the pdf by differentiating the cdf. d f ( ) = F( ) d 2 It is wrong to say P[ ] f ( ) = = for a continuous random variable. For a continuous random variable, P[ = ] = 0 for any. However f ( ) may not be zero. Then how are P[ = ] and f ( ) are related? Ans. Note for a continuous rv, For a small Δ, [ ] = ( ) Pa b f ( d ) for any inerval ab,. [ ] P +Δ f ( ) Δ. ( c) Eq. c is often used for finding the pdf directly. b a
11 Roots of Popular Probability Distributions
12 2
Random variable X is a mapping that maps each outcome s in the sample space to a unique real number x, x. X s. Real Line
Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number,. s s : outcome Sample Space Real Line Eamples Toss a coin. Define the random variable
More informationChapter 3 Single Random Variables and Probability Distributions (Part 1)
Chapter 3 Single Random Variables and Probability Distributions (Part 1) Contents What is a Random Variable? Probability Distribution Functions Cumulative Distribution Function Probability Density Function
More informationRVs and their probability distributions
RVs and their probability distributions RVs and their probability distributions In these notes, I will use the following notation: The probability distribution (function) on a sample space will be denoted
More informationChapter 1 Probability Theory
Review for the previous lecture Eample: how to calculate probabilities of events (especially for sampling with replacement) and the conditional probability Definition: conditional probability, statistically
More informationTransform Techniques - CF
Transform Techniques - CF [eview] Moment Generating Function For a real t, the MGF of the random variable is t t M () t E[ e ] e Characteristic Function (CF) k t k For a real ω, the characteristic function
More informationChapter 2: The Random Variable
Chapter : The Random Variable The outcome of a random eperiment need not be a number, for eample tossing a coin or selecting a color ball from a bo. However we are usually interested not in the outcome
More informationRandom Variables. Statistics 110. Summer Copyright c 2006 by Mark E. Irwin
Random Variables Statistics 110 Summer 2006 Copyright c 2006 by Mark E. Irwin Random Variables A Random Variable (RV) is a response of a random phenomenon which is numeric. Examples: 1. Roll a die twice
More informationTransform Techniques - CF
Transform Techniques - CF [eview] Moment Generating Function For a real t, the MGF of the random variable is t t M () t E[ e ] e Characteristic Function (CF) k t k For a real ω, the characteristic function
More informationRecitation 2: Probability
Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions
More informationTransform Techniques - CF
Transform Techniques - CF [eview] Moment Generating Function For a real t, the MGF of the random variable is t e k p ( k) discrete t t k M () t E[ e ] e t e f d continuous Characteristic Function (CF)
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 29, 2014 Introduction Introduction The world of the model-builder
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 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 informationRandom variables. DS GA 1002 Probability and Statistics for Data Science.
Random variables DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Motivation Random variables model numerical quantities
More information(Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3)
3 Probability Distributions (Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3) Probability Distribution Functions Probability distribution function (pdf): Function for mapping random variables to real numbers. Discrete
More informationLecture Notes 2 Random Variables. Discrete Random Variables: Probability mass function (pmf)
Lecture Notes 2 Random Variables Definition Discrete Random Variables: Probability mass function (pmf) Continuous Random Variables: Probability density function (pdf) Mean and Variance Cumulative Distribution
More informationRandom Variable. Discrete Random Variable. Continuous Random Variable. Discrete Random Variable. Discrete Probability Distribution
Random Variable Theoretical Probability Distribution Random Variable Discrete Probability Distributions A variable that assumes a numerical description for the outcome of a random eperiment (by chance).
More information(Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3)
3 Probability Distributions (Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3) Probability Distribution Functions Probability distribution function (pdf): Function for mapping random variables to real numbers. Discrete
More informationChapter 2 Random Variables
Stochastic Processes Chapter 2 Random Variables Prof. Jernan Juang Dept. of Engineering Science National Cheng Kung University Prof. Chun-Hung Liu Dept. of Electrical and Computer Eng. National Chiao Tung
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 informationCHAPTER 3 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. 3.1 Concept of a Random Variable. 3.2 Discrete Probability Distributions
CHAPTER 3 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 3.1 Concept of a Random Variable Random Variable A random variable is a function that associates a real number with each element in the sample space.
More informationLecture Notes 2 Random Variables. Random Variable
Lecture Notes 2 Random Variables Definition Discrete Random Variables: Probability mass function (pmf) Continuous Random Variables: Probability density function (pdf) Mean and Variance Cumulative Distribution
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 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 informationStochastic processes Lecture 1: Multiple Random Variables Ch. 5
Stochastic processes Lecture : Multiple Random Variables Ch. 5 Dr. Ir. Richard C. Hendriks 26/04/8 Delft University of Technology Challenge the future Organization Plenary Lectures Book: R.D. Yates and
More informationProbability and Statisitcs
Probability and Statistics Random Variables De La Salle University Francis Joseph Campena, Ph.D. January 25, 2017 Francis Joseph Campena, Ph.D. () Probability and Statisitcs January 25, 2017 1 / 17 Outline
More informationExpected value of r.v. s
10 Epected value of r.v. s CDF or PDF are complete (probabilistic) descriptions of the behavior of a random variable. Sometimes we are interested in less information; in a partial characterization. 8 i
More informationContinuous Random Variables
Contents IV Continuous Random Variables 1 13 Introduction 1 13.1 Probability Mass Function Does Not Exist........................... 1 13.2 Probability Distribution.....................................
More informationProbability and Statistics Concepts
University of Central Florida Computer Science Division COT 5611 - Operating Systems. Spring 014 - dcm Probability and Statistics Concepts Random Variable: a rule that assigns a numerical value to each
More informationChapter 4 Multiple Random Variables
Chapter 4 Multiple Random Variables Chapter 41 Joint and Marginal Distributions Definition 411: An n -dimensional random vector is a function from a sample space S into Euclidean space n R, n -dimensional
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 informationChapter 5. Statistical Models in Simulations 5.1. Prof. Dr. Mesut Güneş Ch. 5 Statistical Models in Simulations
Chapter 5 Statistical Models in Simulations 5.1 Contents Basic Probability Theory Concepts Discrete Distributions Continuous Distributions Poisson Process Empirical Distributions Useful Statistical Models
More informationExample A. Define X = number of heads in ten tosses of a coin. What are the values that X may assume?
Stat 400, section.1-.2 Random Variables & Probability Distributions notes by Tim Pilachowski For a given situation, or experiment, observations are made and data is recorded. A sample space S must contain
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 informationRandom Variables. Definition: A random variable (r.v.) X on the probability space (Ω, F, P) is a mapping
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 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 informationn px p x (1 p) n x. p x n(n 1)... (n x + 1) x!
Lectures 3-4 jacques@ucsd.edu 7. Classical discrete distributions D. The Poisson Distribution. If a coin with heads probability p is flipped independently n times, then the number of heads is Bin(n, p)
More informationContinuous Random Variables
1 Continuous Random Variables Example 1 Roll a fair die. Denote by X the random variable taking the value shown by the die, X {1, 2, 3, 4, 5, 6}. Obviously the probability mass function is given by (since
More informationSTAT 516: Basic Probability and its Applications
Lecture 4: Random variables Prof. Michael September 15, 2015 What is a random variable? Often, it is hard and/or impossible to enumerate the entire sample space For a coin flip experiment, the sample space
More informationChapter 3. Chapter 3 sections
sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional Distributions 3.7 Multivariate Distributions
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 informationSTAT 430/510 Probability Lecture 7: Random Variable and Expectation
STAT 430/510 Probability Lecture 7: Random Variable and Expectation Pengyuan (Penelope) Wang June 2, 2011 Review Properties of Probability Conditional Probability The Law of Total Probability Bayes Formula
More informationRecap of Basic Probability Theory
02407 Stochastic Processes Recap of Basic Probability Theory Uffe Høgsbro Thygesen Informatics and Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: uht@imm.dtu.dk
More informationRecap of Basic Probability Theory
02407 Stochastic Processes? Recap of Basic Probability Theory Uffe Høgsbro Thygesen Informatics and Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: uht@imm.dtu.dk
More informationRS Chapter 2 Random Variables 9/28/2017. Chapter 2. Random Variables
RS Chapter Random Variables 9/8/017 Chapter Random Variables Random Variables A random variable is a convenient way to epress the elements of Ω as numbers rather than abstract elements of sets. Definition:
More informationRelationship between probability set function and random variable - 2 -
2.0 Random Variables A rat is selected at random from a cage and its sex is determined. The set of possible outcomes is female and male. Thus outcome space is S = {female, male} = {F, M}. If we let X be
More informationClass 26: review for final exam 18.05, Spring 2014
Probability Class 26: review for final eam 8.05, Spring 204 Counting Sets Inclusion-eclusion principle Rule of product (multiplication rule) Permutation and combinations Basics Outcome, sample space, event
More informationWhy study probability? Set theory. ECE 6010 Lecture 1 Introduction; Review of Random Variables
ECE 6010 Lecture 1 Introduction; Review of Random Variables Readings from G&S: Chapter 1. Section 2.1, Section 2.3, Section 2.4, Section 3.1, Section 3.2, Section 3.5, Section 4.1, Section 4.2, Section
More informationConditional Probability
Conditional Probability Idea have performed a chance experiment but don t know the outcome (ω), but have some partial information (event A) about ω. Question: given this partial information what s the
More informationTheorem 1.7 [Bayes' Law]: Assume that,,, are mutually disjoint events in the sample space s.t.. Then Pr( )
Theorem 1.7 [Bayes' Law]: Assume that,,, are mutually disjoint events in the sample space s.t.. Then Pr Pr = Pr Pr Pr() Pr Pr. We are given three coins and are told that two of the coins are fair and the
More informationDiscrete Random Variables. Discrete Random Variables
Random Variables In many situations, we are interested in numbers associated with the outcomes of a random experiment. For example: Testing cars from a production line, we are interested in variables such
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 informationn(1 p i ) n 1 p i = 1 3 i=1 E(X i p = p i )P(p = p i ) = 1 3 p i = n 3 (p 1 + p 2 + p 3 ). p i i=1 P(X i = 1 p = p i )P(p = p i ) = p1+p2+p3
Introduction to Probability Due:August 8th, 211 Solutions of Final Exam Solve all the problems 1. (15 points) You have three coins, showing Head with probabilities p 1, p 2 and p 3. You perform two different
More informationB.N.Bandodkar College of Science, Thane. Subject : Computer Simulation and Modeling.
B.N.Bandodkar College of Science, Thane Subject : Computer Simulation and Modeling. Simulation is a powerful technique for solving a wide variety of problems. To simulate is to copy the behaviors of a
More informationStatistics and Econometrics I
Statistics and Econometrics I Random Variables Shiu-Sheng Chen Department of Economics National Taiwan University October 5, 2016 Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, 2016
More informationQueueing Theory and Simulation. Introduction
Queueing Theory and Simulation Based on the slides of Dr. Dharma P. Agrawal, University of Cincinnati and Dr. Hiroyuki Ohsaki Graduate School of Information Science & Technology, Osaka University, Japan
More informationMath 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential.
Math 8A Lecture 6 Friday May 7 th Epectation Recall the three main probability density functions so far () Uniform () Eponential (3) Power Law e, ( ), Math 8A Lecture 6 Friday May 7 th Epectation Eample
More informationNotes 6 Autumn Example (One die: part 1) One fair six-sided die is thrown. X is the number showing.
MAS 08 Probability I Notes Autumn 005 Random variables A probability space is a sample space S together with a probability function P which satisfies Kolmogorov s aioms. The Holy Roman Empire was, in the
More informationWhat is a random variable
OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE MATH 256 Probability and Random Processes 04 Random Variables Fall 20 Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr
More informationComputer Science, Informatik 4 Communication and Distributed Systems. Simulation. Discrete-Event System Simulation. Dr.
Simulation Discrete-Event System Simulation Chapter 4 Statistical Models in Simulation Purpose & Overview The world the model-builder sees is probabilistic rather than deterministic. Some statistical model
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 informationIntroduction to Probability Theory for Graduate Economics Fall 2008
Introduction to Probability Theory for Graduate Economics Fall 008 Yiğit Sağlam October 10, 008 CHAPTER - RANDOM VARIABLES AND EXPECTATION 1 1 Random Variables A random variable (RV) is a real-valued function
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 informationDiscrete Random Variable
Discrete Random Variable Outcome of a random experiment need not to be a number. We are generally interested in some measurement or numerical attribute of the outcome, rather than the outcome itself. n
More informationThe random variable 1
The random variable 1 Contents 1. Definition 2. Distribution and density function 3. Specific random variables 4. Functions of one random variable 5. Mean and variance 2 The random variable A random variable
More informationChapter 3 Discrete Random Variables
MICHIGAN STATE UNIVERSITY STT 351 SECTION 2 FALL 2008 LECTURE NOTES Chapter 3 Discrete Random Variables Nao Mimoto Contents 1 Random Variables 2 2 Probability Distributions for Discrete Variables 3 3 Expected
More informationECE Lecture 4. Overview Simulation & MATLAB
ECE 450 - Lecture 4 Overview Simulation & MATLAB Random Variables: Concept and Definition Cumulative Distribution Functions (CDF s) Eamples & Properties Probability Distribution Functions (pdf s) 1 Random
More informationDISCRETE RANDOM VARIABLES: PMF s & CDF s [DEVORE 3.2]
DISCRETE RANDOM VARIABLES: PMF s & CDF s [DEVORE 3.2] PROBABILITY MASS FUNCTION (PMF) DEFINITION): Let X be a discrete random variable. Then, its pmf, denoted as p X(k), is defined as follows: p X(k) :=
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 informationChapter 2 Queueing Theory and Simulation
Chapter 2 Queueing Theory and Simulation Based on the slides of Dr. Dharma P. Agrawal, University of Cincinnati and Dr. Hiroyuki Ohsaki Graduate School of Information Science & Technology, Osaka University,
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 informationII. Probability. II.A General Definitions
II. Probability II.A General Definitions The laws of thermodynamics are based on observations of macroscopic bodies, and encapsulate their thermal properties. On the other hand, matter is composed of atoms
More informationStochastic Processes. Review of Elementary Probability Lecture I. Hamid R. Rabiee Ali Jalali
Stochastic Processes Review o Elementary Probability bili Lecture I Hamid R. Rabiee Ali Jalali Outline History/Philosophy Random Variables Density/Distribution Functions Joint/Conditional Distributions
More informationPart 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park August 19, 2008 1 Introduction There are three main objectives to this section: 1. To introduce the concepts of probability and random variables.
More informationBayesian Updating with Continuous Priors Class 13, Jeremy Orloff and Jonathan Bloom
Bayesian Updating with Continuous Priors Class 3, 8.05 Jeremy Orloff and Jonathan Bloom Learning Goals. Understand a parameterized family of distributions as representing a continuous range of hypotheses
More informationComputer Applications for Engineers ET 601
Computer Applications for Engineers ET 601 Asst. Prof. Dr. Prapun Suksompong prapun@siit.tu.ac.th Random Variables (Con t) 1 Office Hours: (BKD 3601-7) Wednesday 9:30-11:30 Wednesday 16:00-17:00 Thursday
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 informationLecture notes for probability. Math 124
Lecture notes for probability Math 124 What is probability? Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments whose result
More informationPreliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix
More informationNotes 12 Autumn 2005
MAS 08 Probability I Notes Autumn 005 Conditional random variables Remember that the conditional probability of event A given event B is P(A B) P(A B)/P(B). Suppose that X is a discrete random variable.
More informationStatistical Concepts. Distributions of Data
Module : Review of Basic Statistical Concepts. Understanding Probability Distributions, Parameters and Statistics A variable that can take on any value in a range is called a continuous variable. Example:
More informationRecap. The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY INFERENTIAL STATISTICS
Recap. Probability (section 1.1) The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY Population Sample INFERENTIAL STATISTICS Today. Formulation
More informationDept. of Linguistics, Indiana University Fall 2015
L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 34 To start out the course, we need to know something about statistics and This is only an introduction; for a fuller understanding, you would
More informationMA 250 Probability and Statistics. Nazar Khan PUCIT Lecture 15
MA 250 Probability and Statistics Nazar Khan PUCIT Lecture 15 RANDOM VARIABLES Random Variables Random variables come in 2 types 1. Discrete set of outputs is real valued, countable set 2. Continuous set
More informationSTAT2201. Analysis of Engineering & Scientific Data. Unit 3
STAT2201 Analysis of Engineering & Scientific Data Unit 3 Slava Vaisman The University of Queensland School of Mathematics and Physics What we learned in Unit 2 (1) We defined a sample space of a random
More informationApplied Statistics I
Applied Statistics I Liang Zhang Department of Mathematics, University of Utah June 17, 2008 Liang Zhang (UofU) Applied Statistics I June 17, 2008 1 / 22 Random Variables Definition A dicrete random variable
More informationName: Firas Rassoul-Agha
Midterm 1 - Math 5010 - Spring 016 Name: Firas Rassoul-Agha Solve the following 4 problems. You have to clearly explain your solution. The answer carries no points. Only the work does. CALCULATORS ARE
More informationReview of Elementary Probability Lecture I Hamid R. Rabiee
Stochastic Processes Review o Elementar Probabilit Lecture I Hamid R. Rabiee Outline Histor/Philosoph Random Variables Densit/Distribution Functions Joint/Conditional Distributions Correlation Important
More informationLecture 2: CDF and EDF
STAT 425: Introduction to Nonparametric Statistics Winter 2018 Instructor: Yen-Chi Chen Lecture 2: CDF and EDF 2.1 CDF: Cumulative Distribution Function For a random variable X, its CDF F () contains all
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 informationP (A) = P (B) = P (C) = P (D) =
STAT 145 CHAPTER 12 - PROBABILITY - STUDENT VERSION The probability of a random event, is the proportion of times the event will occur in a large number of repititions. For example, when flipping a coin,
More informationChapter 4. Probability-The Study of Randomness
Chapter 4. Probability-The Study of Randomness 4.1.Randomness Random: A phenomenon- individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.
More informationp. 4-1 Random Variables
Random Variables A Motivating Example Experiment: Sample k students without replacement from the population of all n students (labeled as 1, 2,, n, respectively) in our class. = {all combinations} = {{i
More informationM378K In-Class Assignment #1
The following problems are a review of M6K. M7K In-Class Assignment # Problem.. Complete the definition of mutual exclusivity of events below: Events A, B Ω are said to be mutually exclusive if A B =.
More informationBandits, Experts, and Games
Bandits, Experts, and Games CMSC 858G Fall 2016 University of Maryland Intro to Probability* Alex Slivkins Microsoft Research NYC * Many of the slides adopted from Ron Jin and Mohammad Hajiaghayi Outline
More informationStatistics 100A Homework 5 Solutions
Chapter 5 Statistics 1A Homework 5 Solutions Ryan Rosario 1. Let X be a random variable with probability density function a What is the value of c? fx { c1 x 1 < x < 1 otherwise We know that for fx to
More informationIAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES VARIABLE Studying the behavior of random variables, and more importantly functions of random variables is essential for both the
More informationECE 302: Probabilistic Methods in Electrical Engineering
ECE 302: Probabilistic Methods in Electrical Engineering Test I : Chapters 1 3 3/22/04, 7:30 PM Print Name: Read every question carefully and solve each problem in a legible and ordered manner. Make sure
More informationMore on Distribution Function
More on Distribution Function The distribution of a random variable X can be determined directly from its cumulative distribution function F X. Theorem: Let X be any random variable, with cumulative distribution
More informationMAT 271E Probability and Statistics
MAT 271E Probability and Statistics Spring 2011 Instructor : Class Meets : Office Hours : Textbook : Supp. Text : İlker Bayram EEB 1103 ibayram@itu.edu.tr 13.30 16.30, Wednesday EEB? 10.00 12.00, Wednesday
More information