Stochastic Simulation Variance reduction methods Bo Friis Nielsen
|
|
- Dominick Snow
- 5 years ago
- Views:
Transcription
1 Stochastic Simulation Variance reduction methods Bo Friis Nielsen Applied Mathematics and Computer Science Technical University of Denmark 2800 Kgs. Lyngby Denmark
2 Variance reduction methods To obtain better estimates with the same ressources Exploit analytical knowledge and/or correlation Methods: Antithetic variables Control variates Stratified sampling Importance sampling Bo Friis Nielsen 6/ lecture 7 2
3 Case: Monte Carlo evaluation of integral Consider the integral 1 0 We can interpret this interval as e x dx E ( e U) = 1 0 e x dx = θ U U(0,1) To estimate the integral: sample of the random variable e U and take the average. X i = e U i X = n i=1 X i n This is the crude Monte Carlo estimator, crude because we use no refinements whatsoever. Bo Friis Nielsen 6/ lecture 7 3
4 Analytical considerations It is straightforward to calculate the integral in this case The estimator X 1 0 e x dx = e E(X) = e 1 V(X) = E(X 2 ) E(X) 2 E(X 2 ) = Based on one observation 1 0 (e x ) 2 dx = 1 2 ( e 2 1 ) V(X) = 1 2 ( e 2 1 ) (e 1) 2 = Bo Friis Nielsen 6/ lecture 7 4
5 Antithetic variables General idea: to exploit correlation If the estimator is positively correlated with U i (monotone function): Use 1 U also Y i = eu i +e 1 U i 2 = eu i + e e U i 2 Ȳ = n i=0 Y i n The computational effort of calculating the effort needed to compute X. Ȳ should be similar to By the latter expression of Y i we can generate the same number of Y s as X s Bo Friis Nielsen 6/ lecture 7 5
6 Antithetic variables - analytical We can analyse the example analytically due to its simplicity E(Ȳ) = E( X) = θ To calculate V(Ȳ) we start with V(Y i). V(Y i ) = 1 4 V( e U i ) V( e 1 U i ) Cov( e U i,e 1 U i ) = 1 2 V( e U i ) Cov( e U i (e 1 U i ) Cov ( e U i,e 1 U i ) = E ( e U i e 1 U i ) E ( e U i ) E ( e 1 U i ) = e (e 1) 2 = 3e e 2 1 = V(Y i ) = 1 ( ) = Bo Friis Nielsen 6/ lecture 7 6
7 Comparison: Crude method vs. antithetic Crude method: V(X i ) = 1 2 ( e 2 1 ) (e 1) 2 = Antithetic method: V(Y i ) = 1 ( ) = I.e, a reduction by 98 %, almost for free. The variance on X - and samples. Ȳ - will scale with 1/n, the number of Going from crude to antithetic method, reduces the variance as much as increasing number of samples with a factor 50. Bo Friis Nielsen 6/ lecture 7 7
8 Antethetic variables in more complex models If X = h(u 1,...,U n ) where h is monotone in each of its coordinates, then we can use antithetic variables to reduce the variance, because Y = h(1 U 1,...,1 U n ) Cov(X,Y) 0 and therefore V( 1 2 (X +Y)) 1 2 V(X). Bo Friis Nielsen 6/ lecture 7 8
9 Antithetic variables in the queue simulation Can you device the queueing model of yesterday, so that the number of rejections is a monotone function of the underlying U i s? Yes: Make sure that we always use either U i or 1 U i, so that a large U i implies customers arriving quickly and remaining long. Bo Friis Nielsen 6/ lecture 7 9
10 Control variates Use of covariates Z = X +c(y µ y ) E(Y) = µ y ( known) V(Z) = V(X)+c 2 V(Y)+2cCov(Y,X) We can minimize V(Z) by choosing to get c = Cov(X,Y) V(Y) V(Z) = V(X) Cov(X,Y)2 V(Y) Bo Friis Nielsen 6/ lecture 7 10
11 Example Use U as control variate Z i = X i +c ( U i 1 ) 2 X i = e U i The optimal value can be found by Cov(X,Y) = Cov ( U,e U) = E ( Ue U) E(U)E ( e U) In practice we would not know this covariance, but estimate it empirically. V(Z c= /12 ) = V ( e U) Cov( e U,U ) 2 V(U) = Bo Friis Nielsen 6/ lecture 7 11
12 Stratified sampling This is a general survey technique: We sample in predetermined areas, using knowledge of structure of the sampling space eui,1 10 +e U i, e U i,10 10 W i = 10 What is an appropriate number of strata? (In this case there is a simple answer; for complex problems not so) Bo Friis Nielsen 6/ lecture 7 12
13 Importance sampling Suppose we want to evaluate θ = E(h(X)) = h(x)f(x)dx For g(x) > 0 whenever f(x) > 0 this is equivalent to ( ) h(x)f(x) h(y)f(y) θ = g(x)dx = E g(x) g(y) where Y is distributed according to g(y) This is an efficient estimator ) of θ, if we have chosen g such that the variance of is small. ( h(y)f(y) g(y) Such a g will lead to more Y s where h(y) is large. More important regions will be sampled more often. Bo Friis Nielsen 6/ lecture 7 13
14 Re-using the random numbers We want to compare two different queueing systems. We can estimate the rejection rate of system i = 1,2 by θ i = E(g i (U 1,...,U n )) and then rate the two systems according to ˆθ 2 ˆθ 1 But typically g 1 ( ) and g 2 ( ) are positively correlated: Long service times imply many rejections. Bo Friis Nielsen 6/ lecture 7 14
15 Then are more efficient estimator is based on θ 2 θ 1 = E(g 2 (U 1,...,U n ) g 1 (U 1,...,U n )) This amounts to letting the two systems run with the same input sequence of random numbers, i.e. same arrival and service time for each customer. With some program flows, this is easily obtained by re-setting the seed of the RNG. When this is not sufficient, you must store the sequence of arrival and service times, so they can be re-used. Bo Friis Nielsen 6/ lecture 7 15
16 Exercise 5: Variance reduction methods Estimate the integral 1 0 ex dx by simulation (the crude Monte Carlo estimator). Use eg. an estimator based on 100 samples and present the result as the point estimator and a confidence interval. Estimate the integral 1 0 ex dx using antithetic variables, with comparable computer ressources. Estimate the integral 1 0 ex dx using a control variable, with comparable computer ressources. Estimate the integral 1 0 ex dx using stratified sampling, with comparable computer ressources. Use control variates to reduce the variance of the estimator in exercise 4 (Poisson arrivals). Bo Friis Nielsen 6/ lecture 7 16
Variance reduction. Michel Bierlaire. Transport and Mobility Laboratory. Variance reduction p. 1/18
Variance reduction p. 1/18 Variance reduction Michel Bierlaire michel.bierlaire@epfl.ch Transport and Mobility Laboratory Variance reduction p. 2/18 Example Use simulation to compute I = 1 0 e x dx We
More informationStochastic Simulation Discrete simulation/event-by-event
Stochastic Simulation Discrete simulation/event-by-event Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: bfni@dtu.dk Discrete event
More information7 Variance Reduction Techniques
7 Variance Reduction Techniques In a simulation study, we are interested in one or more performance measures for some stochastic model. For example, we want to determine the long-run average waiting time,
More informationMonte-Carlo MMD-MA, Université Paris-Dauphine. Xiaolu Tan
Monte-Carlo MMD-MA, Université Paris-Dauphine Xiaolu Tan tan@ceremade.dauphine.fr Septembre 2015 Contents 1 Introduction 1 1.1 The principle.................................. 1 1.2 The error analysis
More informationStochastic Simulation Introduction Bo Friis Nielsen
Stochastic Simulation Introduction Bo Friis Nielsen Applied Mathematics and Computer Science Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: bfn@imm.dtu.dk Practicalities Notes will handed
More informationOptimization and Simulation
Optimization and Simulation Variance reduction Michel Bierlaire Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering Ecole Polytechnique Fédérale de Lausanne M.
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 informationCSCI-6971 Lecture Notes: Monte Carlo integration
CSCI-6971 Lecture otes: Monte Carlo integration Kristopher R. Beevers Department of Computer Science Rensselaer Polytechnic Institute beevek@cs.rpi.edu February 21, 2006 1 Overview Consider the following
More informationF denotes cumulative density. denotes probability density function; (.)
BAYESIAN ANALYSIS: FOREWORDS Notation. System means the real thing and a model is an assumed mathematical form for the system.. he probability model class M contains the set of the all admissible models
More informationf X, Y (x, y)dx (x), where f(x,y) is the joint pdf of X and Y. (x) dx
INDEPENDENCE, COVARIANCE AND CORRELATION Independence: Intuitive idea of "Y is independent of X": The distribution of Y doesn't depend on the value of X. In terms of the conditional pdf's: "f(y x doesn't
More informationVariance reduction techniques
Variance reduction techniques Lecturer: Dmitri A. Moltchanov E-mail: moltchan@cs.tut.fi http://www.cs.tut.fi/kurssit/elt-53606/ OUTLINE: Simulation with a given accuracy; Variance reduction techniques;
More informationS6880 #13. Variance Reduction Methods
S6880 #13 Variance Reduction Methods 1 Variance Reduction Methods Variance Reduction Methods 2 Importance Sampling Importance Sampling 3 Control Variates Control Variates Cauchy Example Revisited 4 Antithetic
More informationNonlife Actuarial Models. Chapter 14 Basic Monte Carlo Methods
Nonlife Actuarial Models Chapter 14 Basic Monte Carlo Methods Learning Objectives 1. Generation of uniform random numbers, mixed congruential method 2. Low discrepancy sequence 3. Inversion transformation
More informationScientific Computing: Monte Carlo
Scientific Computing: Monte Carlo Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Spring 2012 April 5th and 12th, 2012 A. Donev (Courant Institute)
More informationNumerical Methods I Monte Carlo Methods
Numerical Methods I Monte Carlo Methods Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 Dec. 9th, 2010 A. Donev (Courant Institute) Lecture
More informationSimulation. Alberto Ceselli MSc in Computer Science Univ. of Milan. Part 4 - Statistical Analysis of Simulated Data
Simulation Alberto Ceselli MSc in Computer Science Univ. of Milan Part 4 - Statistical Analysis of Simulated Data A. Ceselli Simulation P.4 Analysis of Sim. data 1 / 15 Statistical analysis of simulated
More information[Chapter 6. Functions of Random Variables]
[Chapter 6. Functions of Random Variables] 6.1 Introduction 6.2 Finding the probability distribution of a function of random variables 6.3 The method of distribution functions 6.5 The method of Moment-generating
More informationVariance reduction. Timo Tiihonen
Variance reduction Timo Tiihonen 2014 Variance reduction techniques The most efficient way to improve the accuracy and confidence of simulation is to try to reduce the variance of simulation results. We
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 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 informationMonte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 4 Variance reduction for MC methods January
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 informationExercise 5 Release: Due:
Stochastic Modeling and Simulation Winter 28 Prof. Dr. I. F. Sbalzarini, Dr. Christoph Zechner (MPI-CBG/CSBD TU Dresden, 87 Dresden, Germany Exercise 5 Release: 8..28 Due: 5..28 Question : Variance of
More informationBindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 14: Monday, May 2
Logistics Week 14: Monday, May 2 Welcome to the last week of classes! Project 3 is due Friday, May 6. Final is Friday, May 13. Lecture Wednesday will be a review, and section this week will be devoted
More informationMonte Carlo-based statistical methods (MASM11/FMS091)
Monte Carlo-based statistical methods (MASM11/FMS091) Jimmy Olsson Centre for Mathematical Sciences Lund University, Sweden Lecture 4 Variance reduction for MC methods February 1, 2013 J. Olsson Monte
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Output Analysis for Monte-Carlo Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com Output Analysis
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Regression based methods, 1st part: Introduction (Sec.
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca October 22nd, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
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 informationACM 116: Lectures 3 4
1 ACM 116: Lectures 3 4 Joint distributions The multivariate normal distribution Conditional distributions Independent random variables Conditional distributions and Monte Carlo: Rejection sampling Variance
More informationMath 3215 Intro. Probability & Statistics Summer 14. Homework 5: Due 7/3/14
Math 325 Intro. Probability & Statistics Summer Homework 5: Due 7/3/. Let X and Y be continuous random variables with joint/marginal p.d.f. s f(x, y) 2, x y, f (x) 2( x), x, f 2 (y) 2y, y. Find the conditional
More informationComputer Systems Modelling
Computer Systems Modelling Computer Laboratory Computer Science Tripos, Part II Lent Term 2010/11 R. J. Gibbens Problem sheet William Gates Building 15 JJ Thomson Avenue Cambridge CB3 0FD http://www.cl.cam.ac.uk/
More informationS-Y0 U-G0 CANDIDATES
«< «X((«( (V«" j -- Y ? K «: :» V X K j 44 E GVE E E EY Y VE E 2 934 VE EEK E-EE E E 4 E -Y0 U-G0 E - Y - V Y ^ K - G --Y-G G - E K - : - ( > x 200 G < G E - : U x K K - " - z E V E E " E " " j j x V
More informationStochastic Processes - lesson 2
Stochastic Processes - lesson 2 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: bfn@imm.dtu.dk Outline Basic probability theory (from
More informationVariance reduction techniques
Variance reduction techniques Lecturer: Dmitri A. Moltchanov E-mail: moltchan@cs.tut.fi http://www.cs.tut.fi/ moltchan/modsim/ http://www.cs.tut.fi/kurssit/tlt-2706/ OUTLINE: Simulation with a given confidence;
More informationSimulation of stationary processes. Timo Tiihonen
Simulation of stationary processes Timo Tiihonen 2014 Tactical aspects of simulation Simulation has always a goal. How to organize simulation to reach the goal sufficiently well and without unneccessary
More informationChapter 4. Continuous Random Variables
Chapter 4. Continuous Random Variables Review Continuous random variable: A random variable that can take any value on an interval of R. Distribution: A density function f : R R + such that 1. non-negative,
More informationRobustness to Parametric Assumptions in Missing Data Models
Robustness to Parametric Assumptions in Missing Data Models Bryan Graham NYU Keisuke Hirano University of Arizona April 2011 Motivation Motivation We consider the classic missing data problem. In practice
More informationMarkov Chain Monte Carlo Lecture 1
What are Monte Carlo Methods? The subject of Monte Carlo methods can be viewed as a branch of experimental mathematics in which one uses random numbers to conduct experiments. Typically the experiments
More informationSTA 2101/442 Assignment 3 1
STA 2101/442 Assignment 3 1 These questions are practice for the midterm and final exam, and are not to be handed in. 1. Suppose X 1,..., X n are a random sample from a distribution with mean µ and variance
More informationL3 Monte-Carlo integration
RNG MC IS Examination RNG MC IS Examination Monte-Carlo and Empirical Methods for Statistical Inference, FMS091/MASM11 Monte-Carlo integration Generating pseudo-random numbers: Rejection sampling Conditional
More informationSTA 2201/442 Assignment 2
STA 2201/442 Assignment 2 1. This is about how to simulate from a continuous univariate distribution. Let the random variable X have a continuous distribution with density f X (x) and cumulative distribution
More informationChapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010
Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression
More informationCPSC 531: System Modeling and Simulation. Carey Williamson Department of Computer Science University of Calgary Fall 2017
CPSC 531: System Modeling and Simulation Carey Williamson Department of Computer Science University of Calgary Fall 2017 Quote of the Day A person with one watch knows what time it is. A person with two
More informationIEOR 6711: Stochastic Models I SOLUTIONS to the First Midterm Exam, October 7, 2008
IEOR 6711: Stochastic Models I SOLUTIONS to the First Midterm Exam, October 7, 2008 Justify your answers; show your work. 1. A sequence of Events. (10 points) Let {B n : n 1} be a sequence of events in
More informationLecture 22: Variance and Covariance
EE5110 : Probability Foundations for Electrical Engineers July-November 2015 Lecture 22: Variance and Covariance Lecturer: Dr. Krishna Jagannathan Scribes: R.Ravi Kiran In this lecture we will introduce
More informationIntroduction to Rare Event Simulation
Introduction to Rare Event Simulation Brown University: Summer School on Rare Event Simulation Jose Blanchet Columbia University. Department of Statistics, Department of IEOR. Blanchet (Columbia) 1 / 31
More information375 PU M Sc Statistics
375 PU M Sc Statistics 1 of 100 193 PU_2016_375_E For the following 2x2 contingency table for two attributes the value of chi-square is:- 20/36 10/38 100/21 10/18 2 of 100 120 PU_2016_375_E If the values
More information2905 Queueing Theory and Simulation PART IV: SIMULATION
2905 Queueing Theory and Simulation PART IV: SIMULATION 22 Random Numbers A fundamental step in a simulation study is the generation of random numbers, where a random number represents the value of a random
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 informationVariance Reduction. in Statistics, we deal with estimators or statistics all the time
Variance Reduction in Statistics, we deal with estiators or statistics all the tie perforance of an estiator is easured by MSE for biased estiators and by variance for unbiased ones hence it s useful to
More informationHochdimensionale Integration
Oliver Ernst Institut für Numerische Mathematik und Optimierung Hochdimensionale Integration 14-tägige Vorlesung im Wintersemester 2010/11 im Rahmen des Moduls Ausgewählte Kapitel der Numerik Contents
More informationStatistical Computing: Exam 1 Chapters 1, 3, 5
Statistical Computing: Exam Chapters, 3, 5 Instructions: Read each question carefully before determining the best answer. Show all work; supporting computer code and output must be attached to the question
More informationStat 451 Lecture Notes Monte Carlo Integration
Stat 451 Lecture Notes 06 12 Monte Carlo Integration Ryan Martin UIC www.math.uic.edu/~rgmartin 1 Based on Chapter 6 in Givens & Hoeting, Chapter 23 in Lange, and Chapters 3 4 in Robert & Casella 2 Updated:
More informationMaster s Written Examination - Solution
Master s Written Examination - Solution Spring 204 Problem Stat 40 Suppose X and X 2 have the joint pdf f X,X 2 (x, x 2 ) = 2e (x +x 2 ), 0 < x < x 2
More informationStatistics Ph.D. Qualifying Exam: Part I October 18, 2003
Statistics Ph.D. Qualifying Exam: Part I October 18, 2003 Student Name: 1. Answer 8 out of 12 problems. Mark the problems you selected in the following table. 1 2 3 4 5 6 7 8 9 10 11 12 2. Write your answer
More informationSTAT 515 MIDTERM 2 EXAM November 14, 2018
STAT 55 MIDTERM 2 EXAM November 4, 28 NAME: Section Number: Instructor: In problems that require reasoning, algebraic calculation, or the use of your graphing calculator, it is not sufficient just to write
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 informationECE531 Lecture 8: Non-Random Parameter Estimation
ECE531 Lecture 8: Non-Random Parameter Estimation D. Richard Brown III Worcester Polytechnic Institute 19-March-2009 Worcester Polytechnic Institute D. Richard Brown III 19-March-2009 1 / 25 Introduction
More informationOutline for today. Computation of the likelihood function for GLMMs. Likelihood for generalized linear mixed model
Outline for today Computation of the likelihood function for GLMMs asmus Waagepetersen Department of Mathematics Aalborg University Denmark Computation of likelihood function. Gaussian quadrature 2. Monte
More informationExercises and Answers to Chapter 1
Exercises and Answers to Chapter The continuous type of random variable X has the following density function: a x, if < x < a, f (x), otherwise. Answer the following questions. () Find a. () Obtain mean
More informationControl Variates for Markov Chain Monte Carlo
Control Variates for Markov Chain Monte Carlo Dellaportas, P., Kontoyiannis, I., and Tsourti, Z. Dept of Statistics, AUEB Dept of Informatics, AUEB 1st Greek Stochastics Meeting Monte Carlo: Probability
More informationEcon 2120: Section 2
Econ 2120: Section 2 Part I - Linear Predictor Loose Ends Ashesh Rambachan Fall 2018 Outline Big Picture Matrix Version of the Linear Predictor and Least Squares Fit Linear Predictor Least Squares Omitted
More informationStatistical Methods in Particle Physics
Statistical Methods in Particle Physics Lecture 3 October 29, 2012 Silvia Masciocchi, GSI Darmstadt s.masciocchi@gsi.de Winter Semester 2012 / 13 Outline Reminder: Probability density function Cumulative
More information1 Stat 605. Homework I. Due Feb. 1, 2011
The first part is homework which you need to turn in. The second part is exercises that will not be graded, but you need to turn it in together with the take-home final exam. 1 Stat 605. Homework I. Due
More informationf(x θ)dx with respect to θ. Assuming certain smoothness conditions concern differentiating under the integral the integral sign, we first obtain
0.1. INTRODUCTION 1 0.1 Introduction R. A. Fisher, a pioneer in the development of mathematical statistics, introduced a measure of the amount of information contained in an observaton from f(x θ). Fisher
More informationQ = (c) Assuming that Ricoh has been working continuously for 7 days, what is the probability that it will remain working at least 8 more days?
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 5-6 in Ross, Thursday, March 31, 11:00am-1:00pm Open Book: but only the Ross
More informationMath 362, Problem set 1
Math 6, roblem set Due //. (4..8) Determine the mean variance of the mean X of a rom sample of size 9 from a distribution having pdf f(x) = 4x, < x
More information180B Lecture Notes, W2011
Bruce K. Driver 180B Lecture Notes, W2011 January 11, 2011 File:180Lec.tex Contents Part 180B Notes 0 Course Notation List......................................................................................................................
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 informationMonte Carlo Methods for Computation and Optimization (048715)
Technion Department of Electrical Engineering Monte Carlo Methods for Computation and Optimization (048715) Lecture Notes Prof. Nahum Shimkin Spring 2015 i PREFACE These lecture notes are intended for
More informationClases 7-8: Métodos de reducción de varianza en Monte Carlo *
Clases 7-8: Métodos de reducció de variaza e Mote Carlo * 9 de septiembre de 27 Ídice. Variace reductio 2. Atithetic variates 2 2.. Example: Uiform radom variables................ 3 2.2. Example: Tail
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationChapter 5 Class Notes
Chapter 5 Class Notes Sections 5.1 and 5.2 It is quite common to measure several variables (some of which may be correlated) and to examine the corresponding joint probability distribution One example
More informationMultivariate Distributions CIVL 7012/8012
Multivariate Distributions CIVL 7012/8012 Multivariate Distributions Engineers often are interested in more than one measurement from a single item. Multivariate distributions describe the probability
More informationStatistics. Statistics
The main aims of statistics 1 1 Choosing a model 2 Estimating its parameter(s) 1 point estimates 2 interval estimates 3 Testing hypotheses Distributions used in statistics: χ 2 n-distribution 2 Let X 1,
More informationBayesian Inference. Chapter 2: Conjugate models
Bayesian Inference Chapter 2: Conjugate models Conchi Ausín and Mike Wiper Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master in
More informationNon-parametric Inference and Resampling
Non-parametric Inference and Resampling Exercises by David Wozabal (Last update. Juni 010) 1 Basic Facts about Rank and Order Statistics 1.1 10 students were asked about the amount of time they spend surfing
More informationThe Nonparametric Bootstrap
The Nonparametric Bootstrap The nonparametric bootstrap may involve inferences about a parameter, but we use a nonparametric procedure in approximating the parametric distribution using the ECDF. We use
More informationGEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs
STATISTICS 4 Summary Notes. Geometric and Exponential Distributions GEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs P(X = x) = ( p) x p x =,, 3,...
More informationMA 320 Introductory Probability, Section 1 Spring 2017 Final Exam Practice May. 1, Exam Scores. Question Score Total. Name:
MA 320 Introductory Probability, Section 1 Spring 2017 Final Exam Practice May. 1, 2017 Exam Scores Question Score Total 1 10 Name: Last 4 digits of student ID #: No books or notes may be used. Turn off
More informationLecture 10. Failure Probabilities and Safety Indexes
Lecture 10. Failure Probabilities and Safety Indexes Igor Rychlik Chalmers Department of Mathematical Sciences Probability, Statistics and Risk, MVE300 Chalmers May 2013 Safety analysis - General setup:
More informationJoint Distribution of Two or More Random Variables
Joint Distribution of Two or More Random Variables Sometimes more than one measurement in the form of random variable is taken on each member of the sample space. In cases like this there will be a few
More informationComputer Intensive Methods in Mathematical Statistics
Computer Intensive Methods in Mathematical Statistics Department of mathematics johawes@kth.se Lecture 4 Variance reduction for MC methods 30 March 2017 Computer Intensive Methods (1) What do we need to
More informationLecture 4: Proofs for Expectation, Variance, and Covariance Formula
Lecture 4: Proofs for Expectation, Variance, and Covariance Formula by Hiro Kasahara Vancouver School of Economics University of British Columbia Hiro Kasahara (UBC) Econ 325 1 / 28 Discrete Random Variables:
More informationISE/OR 762 Stochastic Simulation Techniques
ISE/OR 762 Stochastic Simulation Techniques Topic 0: Introduction to Discrete Event Simulation Yunan Liu Department of Industrial and Systems Engineering NC State University January 9, 2018 Yunan Liu (NC
More informationStatistics & Data Sciences: First Year Prelim Exam May 2018
Statistics & Data Sciences: First Year Prelim Exam May 2018 Instructions: 1. Do not turn this page until instructed to do so. 2. Start each new question on a new sheet of paper. 3. This is a closed book
More informationK-ANTITHETIC VARIATES IN MONTE CARLO SIMULATION ISSN k-antithetic Variates in Monte Carlo Simulation Abdelaziz Nasroallah, pp.
K-ANTITHETIC VARIATES IN MONTE CARLO SIMULATION ABDELAZIZ NASROALLAH Abstract. Standard Monte Carlo simulation needs prohibitive time to achieve reasonable estimations. for untractable integrals (i.e.
More informationChapter 5: Monte Carlo Integration and Variance Reduction
Chapter 5: Monte Carlo Integration and Variance Reduction Lecturer: Zhao Jianhua Department of Statistics Yunnan University of Finance and Economics Outline 5.2 Monte Carlo Integration 5.2.1 Simple MC
More informationExtra Topic: DISTRIBUTIONS OF FUNCTIONS OF RANDOM VARIABLES
Extra Topic: DISTRIBUTIONS OF FUNCTIONS OF RANDOM VARIABLES A little in Montgomery and Runger text in Section 5-5. Previously in Section 5-4 Linear Functions of Random Variables, we saw that we could find
More informationX i. X(n) = 1 n. (X i X(n)) 2. S(n) n
Confidence intervals Let X 1, X 2,..., X n be independent realizations of a random variable X with unknown mean µ and unknown variance σ 2. Sample mean Sample variance X(n) = 1 n S 2 (n) = 1 n 1 n i=1
More informationCHAPTER 4 MATHEMATICAL EXPECTATION. 4.1 Mean of a Random Variable
CHAPTER 4 MATHEMATICAL EXPECTATION 4.1 Mean of a Random Variable The expected value, or mathematical expectation E(X) of a random variable X is the long-run average value of X that would emerge after a
More informationStability of optimization problems with stochastic dominance constraints
Stability of optimization problems with stochastic dominance constraints D. Dentcheva and W. Römisch Stevens Institute of Technology, Hoboken Humboldt-University Berlin www.math.hu-berlin.de/~romisch SIAM
More informationStochastic Simulation
Stochastic Simulation Jan-Pieter Dorsman 1 & Michel Mandjes 1,2,3 1 Korteweg-de Vries Institute for Mathematics, University of Amsterdam 2 CWI, Amsterdam 3 Eurandom, Eindhoven University of Amsterdam,
More informationGaussian random variables inr n
Gaussian vectors Lecture 5 Gaussian random variables inr n One-dimensional case One-dimensional Gaussian density with mean and standard deviation (called N, ): fx x exp. Proposition If X N,, then ax b
More informationAsymptotic Statistics-III. Changliang Zou
Asymptotic Statistics-III Changliang Zou The multivariate central limit theorem Theorem (Multivariate CLT for iid case) Let X i be iid random p-vectors with mean µ and and covariance matrix Σ. Then n (
More informationParameter Estimation and Fitting to Data
Parameter Estimation and Fitting to Data Parameter estimation Maximum likelihood Least squares Goodness-of-fit Examples Elton S. Smith, Jefferson Lab 1 Parameter estimation Properties of estimators 3 An
More informationExercises sheet n 1 : Introduction to MC and IS
Exercises sheet n 1 : Introduction to MC and IS Exercise 1 - An example in Finance In financial applications, we have to calculate quantities of the type ( (e C = E βg K, (1 + G being a standard Gaussian
More information