Independent Events. Two events are independent if knowing that one occurs does not change the probability of the other occurring
|
|
- Daniella George
- 5 years ago
- Views:
Transcription
1 Independent Events Two events are independent if knowing that one occurs does not change the probability of the other occurring Conditional probability is denoted P(A B), which is defined to be: P(A and B) / P(B) P(A B) = probability of A occurring given that B occurs A independent of B if P(A) = P(A B) Independence is equivalent to P(A and B) = P(A) * P(B) if A independent of B, then B is independent of A In terms of a Venn diagram, P(A) = P(A B) says that the ratio of the area of (A and B) to the area of B is the same as the area of A A A and B B 22
2 Independence of Random Variables Two random variables X and Y are independent if knowing the realized value of one of them (e.g.,y = y), doesn t provide any information on the probability of the other (X) taking on any particular value; notation: P(X = x) = P(X = x Y = y) for all possible realized values of X and Y This is equivalent to saying that the events A and B are independent, where A = {outcomes for which X = x} B = {outcomes for which Y = y} For all possible realized values of X and Y 23
3 Independence of Random Variables Observe 250 realizations of three random variables, X, Y, and Z, i.e., {x i, y i, z i } for i = 1 to 250 The R 2 (R = correlation) between the x i values and the y i values is a measure of what percentage of the deviations of the y i s from E(Y) can be predicted by knowing the deviations of the corresponding x i s from E(X) R 2 for these realizations: R 2 0: Top figure shows essentially no correlation between X and Y, i.e., knowing X s realizations tells you nothing about Y s realizations we can reasonably believe that X and Y are independent* R 2 = 1: Bottom figure shows perfect correlation between Z and Y, i.e., knowing Y s realizations tells you everything about Z s realizations we can reasonably believe that Y and Z are not independent Y realizations Zero Correlation Example R^2 close to 0% X realizations * Realizations from independent random variables will have zero correlation, but note that zero correlation does not necessarily imply independence in general 24
4 Properties of Random Numbers Random numbers are (independent) realizations of a random variable U(0,1) with uniform probability distribution on the closed interval [0, 1] (i.e., including 0 and 1) Can think of it as a series of realizations from U i, i = 1, 2, 3, where the U i are independent, identically distributed (i.i.d.) uniform random variables f ( x) = # 1, 0 $ x $ 1 "! 0, otherwise E( R) 1 =! xdx 0 = 2 x =
5 More General Uniform Probability Distribution A random variable U(a,b) is uniformly distributed on the interval (a,b) if its pdf and cdf are: $ 1!, f ( x) = # b & a!" 0, Properties P(x 1 < X < x 2 ) is proportional to the length of the interval, because F(x 2 ) F(x 1 ) = (x 2 -x 1 )/(b-a) E(X) = (a+b)/2 V(X) = (b-a) 2 /12 a % x % b otherwise $ 0,! x ' a F( x) = #,! b ' a " 1, f(x) x < a a & x < b x % b As described on the previous slide, U(0,1) is what we use to define random numbers 1 (b-a) a b 26
6 Review: Probability Density Functions and Cumulative Distributions Probability density functions on left Cumulative probability distributions on right Area under the left hand curve from 0 to a point x 0 = height of right hand curve at x 0 area = height Exponential (λ) x 0 x 0 For x 0 =4, area = (4-1)(0.2) = 0.6 Uniform (1, 6) 27
7 Generation of Pseudo-Random Numbers Pseudo, because generating numbers using a known method removes the potential for true randomness Goal: To produce a sequence of numbers falling into the closed interval [0,1] that simulates the random variable R, i.e., so that the realizations provide statistically significant evidence that the ideal properties of random numbers (RN) Important considerations in software that generates random variables Fast Portable to different computers Sufficiently long cycle Replicable Uniformity and independence One standard way to generate random numbers is using a linear congruential generator, or better yet a combination of them Random numbers are important in simulation Realizations of random variables are generated using random numbers These realizations are called random variates 28
8 Linear Congruential Random Number Generator Generate a sequence of integers Z 1, Z 2, Z 3, via the recursion Z i = (a Z i 1 + c) (mod m) for i = 0, 1, 2, m (at most) where a, c, and m are carefully chosen constants Specify a seed Z 0 to start off mod m means take the remainder after dividing (a Z i 1 + c) by m Because the Z i s are between 0 and m 1 Return the ith random number as U i = Z i / m Issues: Cycling (repeating values of the random numbers) Independence Uniformity The selection of the values for a, c, m, and Z 0 drastically affects the statistical properties and the cycle length Typical to overcome issues by combining several linear congruential generators 29
9 The Current (as of 2000) Arena Random Number Generator Uses some of the same ideas as LCG Modulo division, recursive on earlier values But is not an LCG Combines two separate component generators Recursion involves more than just the preceding value Combined multiple recursive generator (CMRG) A n = ( A n A n-3 ) mod B n = ( B n B n-3 ) mod Two simultaneous recursions Z n = (A n B n ) mod Combine the two Seed = a six-vector of first three A n s, B n s Cycle length = U n = Z n / if Z n > / if Z n = 0 The next random number 30
10 Tests for Random Numbers There are standard tests for uniformity and independence When to use tests: If a well-known simulation languages or random-number generators is used, it is probably unnecessary to test If the generator is not explicitly known or documented tests should be applied to many sample numbers. Types of tests: Theoretical tests: evaluate the mathematics of the random number generator without actually generating any numbers Empirical tests: use statistical tests on actual sequences of numbers produced Test of uniformity Two different methods that check to see if the observed relative frequencies are close to the expected relative frequencies Kolmogorov-Smirnov test Chi-square test Tests for autocorrelation Methods that check to see if there is correlation among the set of random numbers 31
11 Generating Realizations of a Discrete Random Variable Using inverse-transform technique Suppose the random variable X has a discrete probability distribution given by: x P(X =x) F(x) Given a random number, a realization of u = 0.73 of U(0,1), x, a realization of X, is given by: " 0, $ x = # 1, $ % 2, u & < u & < u &1.0 u 0.8 This is equivalent to: X = 0 X = 1 X = 2 x
12 Generating Exponentially Distributed Realizations X is an exponentially distributed random variable with λ = 1 Let u = P( X x) = F(x) = 1 e -λx for x 0 To generate realizations of X (random variates) U 1 Let x i = F -1 (u i ) = -(1/λ) ln(1-u i ) Where the u i are random numbers, i.e., realizations of U i (0, 1) U 2 We can use the inverse-transform technique for some continuous distributions U 1 33
13 Example Generation of Realizations Example: Generate 200 realizations from an exponentially distributed random variable with λ = 1 Examples of other distributions for which the inverse-transform technique works are: Uniform distribution Weibull distribution Triangular distribution All discrete distributions 34
14 Normal (Gaussian) Distribution A random variable X is normally distributed has the pdf: "! < µ <! Mean: Variance:! 2 > 0 Denoted as X ~ N(µ, σ) f (x) = - $ 1 " 2# e,- Special properties:. The pdf is symmetrical around the mean, i.e., f(µ - x) = f(µ + x) The maximum value of the pdf occurs at x = µ; i.e., the mean and mode are equal Unimodal, i.e., the pdf decreases as the distance from µ increases % x$µ (. ' * 0 2& " ) / 0, $ 1 < x < F(x) 0.5 µ 35
Systems Simulation Chapter 7: Random-Number Generation
Systems Simulation Chapter 7: Random-Number Generation Fatih Cavdur fatihcavdur@uludag.edu.tr April 22, 2014 Introduction Introduction Random Numbers (RNs) are a necessary basic ingredient in the simulation
More informationHow does the computer generate observations from various distributions specified after input analysis?
1 How does the computer generate observations from various distributions specified after input analysis? There are two main components to the generation of observations from probability distributions.
More informationUNIT 5:Random number generation And Variation Generation
UNIT 5:Random number generation And Variation Generation RANDOM-NUMBER GENERATION Random numbers are a necessary basic ingredient in the simulation of almost all discrete systems. Most computer languages
More informationIE 303 Discrete-Event Simulation L E C T U R E 6 : R A N D O M N U M B E R G E N E R A T I O N
IE 303 Discrete-Event Simulation L E C T U R E 6 : R A N D O M N U M B E R G E N E R A T I O N Review of the Last Lecture Continuous Distributions Uniform distributions Exponential distributions and memoryless
More informationHow does the computer generate observations from various distributions specified after input analysis?
1 How does the computer generate observations from various distributions specified after input analysis? There are two main components to the generation of observations from probability distributions.
More informationCPSC 531: Random Numbers. Jonathan Hudson Department of Computer Science University of Calgary
CPSC 531: Random Numbers Jonathan Hudson Department of Computer Science University of Calgary http://www.ucalgary.ca/~hudsonj/531f17 Introduction In simulations, we generate random values for variables
More informationRandom Number Generators
1/18 Random Number Generators Professor Karl Sigman Columbia University Department of IEOR New York City USA 2/18 Introduction Your computer generates" numbers U 1, U 2, U 3,... that are considered independent
More informationB.N.Bandodkar College of Science, Thane. Random-Number Generation. Mrs M.J.Gholba
B.N.Bandodkar College of Science, Thane Random-Number Generation Mrs M.J.Gholba Properties of Random Numbers A sequence of random numbers, R, R,., must have two important statistical properties, uniformity
More informationSlides 3: Random Numbers
Slides 3: Random Numbers We previously considered a few examples of simulating real processes. In order to mimic real randomness of events such as arrival times we considered the use of random numbers
More informationSources of randomness
Random Number Generator Chapter 7 In simulations, we generate random values for variables with a specified distribution Ex., model service times using the exponential distribution Generation of random
More information2 Random Variable Generation
2 Random Variable Generation Most Monte Carlo computations require, as a starting point, a sequence of i.i.d. random variables with given marginal distribution. We describe here some of the basic methods
More informationRandom Number Generation. CS1538: Introduction to simulations
Random Number Generation CS1538: Introduction to simulations Random Numbers Stochastic simulations require random data True random data cannot come from an algorithm We must obtain it from some process
More information2008 Winton. Review of Statistical Terminology
1 Review of Statistical Terminology 2 Formal Terminology An experiment is a process whose outcome is not known with certainty The experiment s sample space S is the set of all possible outcomes. A random
More informationQualifying Exam CS 661: System Simulation Summer 2013 Prof. Marvin K. Nakayama
Qualifying Exam CS 661: System Simulation Summer 2013 Prof. Marvin K. Nakayama Instructions This exam has 7 pages in total, numbered 1 to 7. Make sure your exam has all the pages. This exam will be 2 hours
More informationReview of Statistical Terminology
Review of Statistical Terminology An experiment is a process whose outcome is not known with certainty. The experiment s sample space S is the set of all possible outcomes. A random variable is a function
More informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More informationUniform random numbers generators
Uniform random numbers generators Lecturer: Dmitri A. Moltchanov E-mail: moltchan@cs.tut.fi http://www.cs.tut.fi/kurssit/tlt-2707/ OUTLINE: The need for random numbers; Basic steps in generation; Uniformly
More informationCIVL Continuous Distributions
CIVL 3103 Continuous Distributions Learning Objectives - Continuous Distributions Define continuous distributions, and identify common distributions applicable to engineering problems. Identify the appropriate
More informationIE 581 Introduction to Stochastic Simulation
1. List criteria for choosing the majorizing density r (x) when creating an acceptance/rejection random-variate generator for a specified density function f (x). 2. Suppose the rate function of a nonhomogeneous
More informationSlides 8: Statistical Models in Simulation
Slides 8: Statistical Models in Simulation Purpose and Overview The world the model-builder sees is probabilistic rather than deterministic: Some statistical model might well describe the variations. An
More informationH 2 : otherwise. that is simply the proportion of the sample points below level x. For any fixed point x the law of large numbers gives that
Lecture 28 28.1 Kolmogorov-Smirnov test. Suppose that we have an i.i.d. sample X 1,..., X n with some unknown distribution and we would like to test the hypothesis that is equal to a particular distribution
More informationSTAT Chapter 5 Continuous Distributions
STAT 270 - Chapter 5 Continuous Distributions June 27, 2012 Shirin Golchi () STAT270 June 27, 2012 1 / 59 Continuous rv s Definition: X is a continuous rv if it takes values in an interval, i.e., range
More informationStatistics 3657 : Moment Approximations
Statistics 3657 : Moment Approximations Preliminaries Suppose that we have a r.v. and that we wish to calculate the expectation of g) for some function g. Of course we could calculate it as Eg)) by the
More informationUniform Random Number Generators
JHU 553.633/433: Monte Carlo Methods J. C. Spall 25 September 2017 CHAPTER 2 RANDOM NUMBER GENERATION Motivation and criteria for generators Linear generators (e.g., linear congruential generators) Multiple
More informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01: Probability and Statistics for Engineers Spring 2013 Contents 1 Joint Probability Distributions 2 1.1 Two Discrete
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
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 informationContinuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.
UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Christopher Barr University of California, Los Angeles,
More informationExperimental Design and Statistics - AGA47A
Experimental Design and Statistics - AGA47A Czech University of Life Sciences in Prague Department of Genetics and Breeding Fall/Winter 2014/2015 Matúš Maciak (@ A 211) Office Hours: M 14:00 15:30 W 15:30
More informationProbability Density Functions
Statistical Methods in Particle Physics / WS 13 Lecture II Probability Density Functions Niklaus Berger Physics Institute, University of Heidelberg Recap of Lecture I: Kolmogorov Axioms Ingredients: Set
More informationContinuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( )
UCLA STAT 35 Applied Computational and Interactive Probability Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Chris Barr Continuous Random Variables and Probability
More informationMonte Carlo Simulation
Monte Carlo Simulation 198 Introduction Reliability indices of an actual physical system could be estimated by collecting data on the occurrence of failures and durations of repair. The Monte Carlo method
More informationfunctions Poisson distribution Normal distribution Arbitrary functions
Physics 433: Computational Physics Lecture 6 Random number distributions Generation of random numbers of various distribuition functions Normal distribution Poisson distribution Arbitrary functions Random
More informationContinuous random variables
Continuous random variables Can take on an uncountably infinite number of values Any value within an interval over which the variable is definied has some probability of occuring This is different from
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 informationECE 302 Division 2 Exam 2 Solutions, 11/4/2009.
NAME: ECE 32 Division 2 Exam 2 Solutions, /4/29. You will be required to show your student ID during the exam. This is a closed-book exam. A formula sheet is provided. No calculators are allowed. Total
More informationProbability Distributions Columns (a) through (d)
Discrete Probability Distributions Columns (a) through (d) Probability Mass Distribution Description Notes Notation or Density Function --------------------(PMF or PDF)-------------------- (a) (b) (c)
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions
More 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 informationDUBLIN CITY UNIVERSITY
DUBLIN CITY UNIVERSITY SAMPLE EXAMINATIONS 2017/2018 MODULE: QUALIFICATIONS: Simulation for Finance MS455 B.Sc. Actuarial Mathematics ACM B.Sc. Financial Mathematics FIM YEAR OF STUDY: 4 EXAMINERS: Mr
More informationChapter 4: Monte Carlo Methods. Paisan Nakmahachalasint
Chapter 4: Monte Carlo Methods Paisan Nakmahachalasint Introduction Monte Carlo Methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. 60 minutes.
Closed book and notes. 60 minutes. A summary table of some univariate continuous distributions is provided. Four Pages. In this version of the Key, I try to be more complete than necessary to receive full
More informationISyE 6644 Fall 2014 Test 3 Solutions
1 NAME ISyE 6644 Fall 14 Test 3 Solutions revised 8/4/18 You have 1 minutes for this test. You are allowed three cheat sheets. Circle all final answers. Good luck! 1. [4 points] Suppose that the joint
More information3 Continuous Random Variables
Jinguo Lian Math437 Notes January 15, 016 3 Continuous Random Variables Remember that discrete random variables can take only a countable number of possible values. On the other hand, a continuous random
More 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 informationMoments. Raw moment: February 25, 2014 Normalized / Standardized moment:
Moments Lecture 10: Central Limit Theorem and CDFs Sta230 / Mth 230 Colin Rundel Raw moment: Central moment: µ n = EX n ) µ n = E[X µ) 2 ] February 25, 2014 Normalized / Standardized moment: µ n σ n Sta230
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2017 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Exercise 4.1 Let X be a random variable with p(x)
More informationChapter 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 informationStatistical Methods for Astronomy
Statistical Methods for Astronomy If your experiment needs statistics, you ought to have done a better experiment. -Ernest Rutherford Lecture 1 Lecture 2 Why do we need statistics? Definitions Statistical
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 informationPseudo-Random Numbers Generators. Anne GILLE-GENEST. March 1, Premia Introduction Definitions Good generators...
14 pages 1 Pseudo-Random Numbers Generators Anne GILLE-GENEST March 1, 2012 Contents Premia 14 1 Introduction 2 1.1 Definitions............................. 2 1.2 Good generators..........................
More informationMAS1302 Computational Probability and Statistics
MAS1302 Computational Probability and Statistics April 23, 2008 3. Simulating continuous random behaviour 3.1 The Continuous Uniform U(0,1) Distribution We have already used this random variable a great
More informationChapter 3.3 Continuous distributions
Chapter 3.3 Continuous distributions In this section we study several continuous distributions and their properties. Here are a few, classified by their support S X. There are of course many, many more.
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 informationStochastic Simulation of
Stochastic Simulation of Communication Networks -WS 2014/2015 Part 2 Random Number Generation Prof. Dr. C. Görg www.comnets.uni-bremen.de VSIM 2-1 Table of Contents 1 General Introduction 2 Random Number
More informationApplied Statistics and Probability for Engineers. Sixth Edition. Chapter 4 Continuous Random Variables and Probability Distributions.
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 4 Continuous Random Variables and Probability Distributions 4 Continuous CHAPTER OUTLINE Random
More informationChapter 4 Continuous Random Variables and Probability Distributions
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 4 Continuous Random Variables and Probability Distributions 4 Continuous CHAPTER OUTLINE 4-1
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 informationB. Maddah ENMG 622 Simulation 11/11/08
B. Maddah ENMG 622 Simulation 11/11/08 Random-Number Generators (Chapter 7, Law) Overview All stochastic simulations need to generate IID uniformly distributed on (0,1), U(0,1), random numbers. 1 f X (
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 14: Continuous random variables Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin www.cs.cmu.edu/
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional
More informationDepartment of Electrical- and Information Technology. ETS061 Lecture 3, Verification, Validation and Input
ETS061 Lecture 3, Verification, Validation and Input Verification and Validation Real system Validation Model Verification Measurements Verification Break model down into smaller bits and test each bit
More informationContinuous Probability Distributions. Uniform Distribution
Continuous Probability Distributions Uniform Distribution Important Terms & Concepts Learned Probability Mass Function (PMF) Cumulative Distribution Function (CDF) Complementary Cumulative Distribution
More informationp. 6-1 Continuous Random Variables p. 6-2
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability (>). Often, there is interest in random variables
More informationLecture 4. Continuous Random Variables and Transformations of Random Variables
Math 408 - Mathematical Statistics Lecture 4. Continuous Random Variables and Transformations of Random Variables January 25, 2013 Konstantin Zuev (USC) Math 408, Lecture 4 January 25, 2013 1 / 13 Agenda
More 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 informationNational Sun Yat-Sen University CSE Course: Information Theory. Maximum Entropy and Spectral Estimation
Maximum Entropy and Spectral Estimation 1 Introduction What is the distribution of velocities in the gas at a given temperature? It is the Maxwell-Boltzmann distribution. The maximum entropy distribution
More informationNotation: X = random variable; x = particular value; P(X = x) denotes probability that X equals the value x.
Ch. 16 Random Variables Def n: A random variable is a numerical measurement of the outcome of a random phenomenon. A discrete random variable is a random variable that assumes separate values. # of people
More informationECE 302, Final 3:20-5:20pm Mon. May 1, WTHR 160 or WTHR 172.
ECE 302, Final 3:20-5:20pm Mon. May 1, WTHR 160 or WTHR 172. 1. Enter your name, student ID number, e-mail address, and signature in the space provided on this page, NOW! 2. This is a closed book exam.
More informationBrief Review of Probability
Maura Department of Economics and Finance Università Tor Vergata Outline 1 Distribution Functions Quantiles and Modes of a Distribution 2 Example 3 Example 4 Distributions Outline Distribution Functions
More informationChapter 4: CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
Chapter 4: CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Gamma Distribution Weibull Distribution Lognormal Distribution Sections 4-9 through 4-11 Another exponential distribution example
More informationGeneral Principles in Random Variates Generation
General Principles in Random Variates Generation E. Moulines and G. Fort Telecom ParisTech June 2015 Bibliography : Luc Devroye, Non-Uniform Random Variate Generator, Springer-Verlag (1986) available on
More informationProbability Review. Yutian Li. January 18, Stanford University. Yutian Li (Stanford University) Probability Review January 18, / 27
Probability Review Yutian Li Stanford University January 18, 2018 Yutian Li (Stanford University) Probability Review January 18, 2018 1 / 27 Outline 1 Elements of probability 2 Random variables 3 Multiple
More informationProperties of Continuous Probability Distributions The graph of a continuous probability distribution is a curve. Probability is represented by area
Properties of Continuous Probability Distributions The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve. The curve is called the probability
More informationSimulation. Where real stuff starts
Simulation Where real stuff starts March 2019 1 ToC 1. What is a simulation? 2. Accuracy of output 3. Random Number Generators 4. How to sample 5. Monte Carlo 6. Bootstrap 2 1. What is a simulation? 3
More informationECO220Y Continuous Probability Distributions: Uniform and Triangle Readings: Chapter 9, sections
ECO220Y Continuous Probability Distributions: Uniform and Triangle Readings: Chapter 9, sections 9.8-9.9 Fall 2011 Lecture 8 Part 1 (Fall 2011) Probability Distributions Lecture 8 Part 1 1 / 19 Probability
More informationSimulation. Where real stuff starts
1 Simulation Where real stuff starts ToC 1. What is a simulation? 2. Accuracy of output 3. Random Number Generators 4. How to sample 5. Monte Carlo 6. Bootstrap 2 1. What is a simulation? 3 What is a simulation?
More informationRandom variables, Expectation, Mean and Variance. Slides are adapted from STAT414 course at PennState
Random variables, Expectation, Mean and Variance Slides are adapted from STAT414 course at PennState https://onlinecourses.science.psu.edu/stat414/ Random variable Definition. Given a random experiment
More informationContinuous Random Variables
MATH 38 Continuous Random Variables Dr. Neal, WKU Throughout, let Ω be a sample space with a defined probability measure P. Definition. A continuous random variable is a real-valued function X defined
More informationLecture 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 informationChapter Learning Objectives. Probability Distributions and Probability Density Functions. Continuous Random Variables
Chapter 4: Continuous Random Variables and Probability s 4-1 Continuous Random Variables 4-2 Probability s and Probability Density Functions 4-3 Cumulative Functions 4-4 Mean and Variance of a Continuous
More informationModeling Uncertainty in the Earth Sciences Jef Caers Stanford University
Probability theory and statistical analysis: a review Modeling Uncertainty in the Earth Sciences Jef Caers Stanford University Concepts assumed known Histograms, mean, median, spread, quantiles Probability,
More informationGenerating pseudo- random numbers
Generating pseudo- random numbers What are pseudo-random numbers? Numbers exhibiting statistical randomness while being generated by a deterministic process. Easier to generate than true random numbers?
More information1 Probability and Random Variables
1 Probability and Random Variables The models that you have seen thus far are deterministic models. For any time t, there is a unique solution X(t). On the other hand, stochastic models will result in
More informationSemester , Example Exam 1
Semester 1 2017, Example Exam 1 1 of 10 Instructions The exam consists of 4 questions, 1-4. Each question has four items, a-d. Within each question: Item (a) carries a weight of 8 marks. Item (b) carries
More informationECE302 Exam 2 Version A April 21, You must show ALL of your work for full credit. Please leave fractions as fractions, but simplify them, etc.
ECE32 Exam 2 Version A April 21, 214 1 Name: Solution Score: /1 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully. Please check your answers
More information15-388/688 - Practical Data Science: Basic probability. J. Zico Kolter Carnegie Mellon University Spring 2018
15-388/688 - Practical Data Science: Basic probability J. Zico Kolter Carnegie Mellon University Spring 2018 1 Announcements Logistics of next few lectures Final project released, proposals/groups due
More informationCommon ontinuous random variables
Common ontinuous random variables CE 311S Earlier, we saw a number of distribution families Binomial Negative binomial Hypergeometric Poisson These were useful because they represented common situations:
More information1.1 Review of Probability Theory
1.1 Review of Probability Theory Angela Peace Biomathemtics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology. CRC Press,
More informationSimulation. Version 1.1 c 2010, 2009, 2001 José Fernando Oliveira Maria Antónia Carravilla FEUP
Simulation Version 1.1 c 2010, 2009, 2001 José Fernando Oliveira Maria Antónia Carravilla FEUP Systems - why simulate? System any object or entity on which a particular study is run. Model representation
More informationChapter 5. Chapter 5 sections
1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationAPPM/MATH 4/5520 Solutions to Exam I Review Problems. f X 1,X 2. 2e x 1 x 2. = x 2
APPM/MATH 4/5520 Solutions to Exam I Review Problems. (a) f X (x ) f X,X 2 (x,x 2 )dx 2 x 2e x x 2 dx 2 2e 2x x was below x 2, but when marginalizing out x 2, we ran it over all values from 0 to and so
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 informationMidterm Exam 1 Solution
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2015 Kannan Ramchandran September 22, 2015 Midterm Exam 1 Solution Last name First name SID Name of student on your left:
More informationDepartment of Statistical Science FIRST YEAR EXAM - SPRING 2017
Department of Statistical Science Duke University FIRST YEAR EXAM - SPRING 017 Monday May 8th 017, 9:00 AM 1:00 PM NOTES: PLEASE READ CAREFULLY BEFORE BEGINNING EXAM! 1. Do not write solutions on the exam;
More informationContinuous Random Variables and Continuous Distributions
Continuous Random Variables and Continuous Distributions Continuous Random Variables and Continuous Distributions Expectation & Variance of Continuous Random Variables ( 5.2) The Uniform Random Variable
More informationRecall the Basics of Hypothesis Testing
Recall the Basics of Hypothesis Testing The level of significance α, (size of test) is defined as the probability of X falling in w (rejecting H 0 ) when H 0 is true: P(X w H 0 ) = α. H 0 TRUE H 1 TRUE
More information( x) ( ) F ( ) ( ) ( ) Prob( ) ( ) ( ) X x F x f s ds
Applied Numerical Analysis Pseudo Random Number Generator Lecturer: Emad Fatemizadeh What is random number: A sequence in which each term is unpredictable 29, 95, 11, 60, 22 Application: Monte Carlo Simulations
More informationGamma and Normal Distribuions
Gamma and Normal Distribuions Sections 5.4 & 5.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 15-3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationLecture 3 Continuous Random Variable
Lecture 3 Continuous Random Variable 1 Cumulative Distribution Function Definition Theorem 3.1 For any random variable X, 2 Continuous Random Variable Definition 3 Example Suppose we have a wheel of circumference
More information