Geometry of Transformations of Random Variables
|
|
- Rafe Tyler
- 6 years ago
- Views:
Transcription
1 Geometry of Transformations of Random Variables Univariate distributions We are interested in the problem of finding the distribution of Y = h(x) when the transformation h is one-to-one so that there is a unique x = h (y) for eah x and y with positive probability or density. In the ase of disrete random variables, the transformation is simple. P (Y = y) = P (h(x) = y) = P ( X = h (y) ) In ontrast, for absolutely ontinuous random variables, the density f Y (y) is in general not equal to f X (h (y)). The reason is that the geometry of the transformation beomes more omplex as the dimension inreases. For disrete distributions, probability is loated at zero-dimensional points, and the transformations do not affet the size of points. For univariate absolutely ontinuous distributions, however, probability is assoiated with the integral of a density over a one-dimensional line segment. Transformations an hange the lengths of intervals, as shown here where an interval of length dx is transformed to smaller interval of length dy. h y+dy y x x+dx Figure : Transformation Y = h(x). The figure shows Y = h(x) over a very small interval so that h appears to be essentially linear. For small dx, the probability in the interval (x, x + dx) is approximately f X (x)dx. The density at y = h(x) will be the limit of the ratio of this probability over the length of the interval between h(x) and h(x + dx) whih is h(x + dx) h(x). (If h (x) < 0, then h(x + dx) < h(x) so the absolute value is needed.) As h is differentiable, the approximation h(x + dx) h(x) + h (x)dx is aurate for very small dx and it follows that the transformed interval has approximate length h (x) dx. The density at y is then f Y (y) = f X(x)dx h (x) dx = f X(h (y)) h (h (y)) after applying x = h (y). Multivariate Distributions We would like to extend this idea to joint densities. If random variables X = (X,..., X n ) have joint density f X, we aim find the joint density f Y of the random variables Y = (Y,..., Y n ) where
2 we write Y = h(x) to mean Y i = h i (X,..., X n ) for i =,..., n. We will assume that h is a differentiable bijetion whih means that all partial derivatives h i / x j exist and that the vetor equation (y,..., y n ) = h(x,..., x n ) has a unique solution (suh that f X > 0) with (x,..., x n ) = h (y,..., y n ). Bivariate Distributions. We motivate the general answer by examining the bivariate ase (Y, Y 2 ) = h(x, X 2 ). The density at h(x, x 2 ) is the limiting ratio of the probability in a retangle with a orner at (x, x 2 ) with sides of length dx and over the area of the retangle dx. The density at (y, y 2 ) = h(x, x 2 ) will depend on the geometry of the transformation of the orners of this retangle. (x,x 2 + ) (x + dx,x 2 + ) (x,x 2 ) (x + dx,x 2 ) Figure 2: Retangle before transformation. By the partial differentiability of h in eah dimension, the following approximation is true. h (x + dx, x 2 + ) h (x, x 2 ) + h dx + h h 2 (x + dx, x 2 + ) h 2 (x, x 2 ) + h 2 dx + h 2 To simplify the expressions, let y = h (x, x 2 ), y 2 = h 2 (x, x 2 ), a = h dx, b = h, = h 2 dx, and d = h 2. With this notation, the four orners of the retangle are mapped approximately as follows: (x, x 2 ) (y, y 2 ) (x + dx, x 2 ) (y + a, y 2 + b) (x + dx, x 2 + ) (y + a +, y 2 + b + d) (x, x 2 + ) (y +, y 2 + d) These points will not be arranged as a retangle in general, but will be a parallelogram. The parallelogram an be understood geometrially as being formed by the two vetors (a, b) and (, d) 2
3 extending from (y, y 2 ) to form two adjaent sides with the other sides then being parallel and equal length to these. The proper saling of the density f Y (y) will then depend on the relative area of this parallelogram to the original retangle. The following figure shows a parallelogram where lower left orner orresponds to the point (y, y 2 ) and the two adjaent sides are desribed by the vetors (a, b) and (, d). In this figure, a, b,, d > 0, whih orresponds to all of the partial derivatives h i / x j being positive. In addition, a > and d > b so that ad > b. The following geometri argument relies on these hoies, but the result will be true in general. d b a Figure 3: Parallelogram after transformation. There are two retangles with dashed lines added to the figure. The larger of these retangles has width a and height d and the smaller one has width and height b. In addition, there are two small dotted lines added to the figure whih reate six triangles and two larger polygons. Notie that the six triangles ome in pairs whih are the same size and orientation. Eah pair inludes one shaded triangle within the parallelogram and one outside. When the shading of the triangles are reversed, we get the following figure. The total shaded area is the same and is equal to ad b as it is the differene in the areas of the retangles. Thus, the area of the parallelogram depends only on the lengths and orientations of the vetors (a, b) and (, d). When these vetors are ombined to form a matrix, we see that the area is equal to the absolute value of the determinant of this matrix. ( ) a b ad b = det d In fat, the absolute value of this determinant measures the area of the orresponding parallelogram for any real a, b,, d whih an be shown by working through all ases. If we substitute bak in our original expressions, we see that the area of the parallelogram is ad b = ( h dx ) ( ) h2 3 ( h ) ( ) h2 dx = J dx
4 d b a Figure 4: Equal area. The area of the parallelogram is equal to the differene in the areas of the retangles. where J = det ( h h h 2 h 2 is alled the Jaobian or Jaobian derivative of the transformation. The ratio of the area of the parallelogram to the area of the original retangle is J and it follows then that the joint density of the random variables Y and Y 2 is f Y (y, y 2 ) = ) J(h (y, y 2 )) f X(h (y, y 2 )). More than two dimensions. It is natural to then ask how this extends to joint distributions of n random variables. The answer is that the density requires a resaling whih is found by alulating the reiproal of the absolute value of the Jaobian derivative for this larger transformation whih is simply a determinant of a larger matrix of partial derivatives. The derivation above found the Jaobian deriavative by omputing y j / x i for eah i, j, but it is also possible to take derivatives of the inverse relationships x j / y j and find the orresponding Jaobian deriavative. The value of this seond derivative is the reiproal of the first. In order to better distinguish these ases, it is useful to introdue a different notation that make the diretion of differentiation lear. We an define the Jaobian derivative as follows. y (y,..., y n ) (x,..., x n ) = det. y x n y n.... y n x n 4
5 The density of Y = (Y,..., Y n ) an then be omputed by finding one of two Jaobian derivatives. f Y (y,..., y n ) = = (y,...,y n) f X (h (x,..., x n )) (x,...,x n) (x,..., x n ) (y,..., y n ) f X(h (x,..., x n )) If you simply memorize the expression f Y (y,..., y n ) (y,..., y n ) = f X (x,..., x n ) (x,..., x n ) you an rerrange this algebraially to find either Jaobian and then properly use it or its reiproal to find the desired density after the transformation. 5
Maximum Entropy and Exponential Families
Maximum Entropy and Exponential Families April 9, 209 Abstrat The goal of this note is to derive the exponential form of probability distribution from more basi onsiderations, in partiular Entropy. It
More informationAn Integrated Architecture of Adaptive Neural Network Control for Dynamic Systems
An Integrated Arhiteture of Adaptive Neural Network Control for Dynami Systems Robert L. Tokar 2 Brian D.MVey2 'Center for Nonlinear Studies, 2Applied Theoretial Physis Division Los Alamos National Laboratory,
More information23.1 Tuning controllers, in the large view Quoting from Section 16.7:
Lesson 23. Tuning a real ontroller - modeling, proess identifiation, fine tuning 23.0 Context We have learned to view proesses as dynami systems, taking are to identify their input, intermediate, and output
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 4
Advaned Computational Fluid Dynamis AA5A Leture 4 Antony Jameson Winter Quarter,, Stanford, CA Abstrat Leture 4 overs analysis of the equations of gas dynamis Contents Analysis of the equations of gas
More informationWhere as discussed previously we interpret solutions to this partial differential equation in the weak sense: b
Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential
More informationThe Hanging Chain. John McCuan. January 19, 2006
The Hanging Chain John MCuan January 19, 2006 1 Introdution We onsider a hain of length L attahed to two points (a, u a and (b, u b in the plane. It is assumed that the hain hangs in the plane under a
More informationRelativistic Dynamics
Chapter 7 Relativisti Dynamis 7.1 General Priniples of Dynamis 7.2 Relativisti Ation As stated in Setion A.2, all of dynamis is derived from the priniple of least ation. Thus it is our hore to find a suitable
More informationHankel Optimal Model Order Reduction 1
Massahusetts Institute of Tehnology Department of Eletrial Engineering and Computer Siene 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Hankel Optimal Model Order Redution 1 This leture overs both
More informationA model for measurement of the states in a coupled-dot qubit
A model for measurement of the states in a oupled-dot qubit H B Sun and H M Wiseman Centre for Quantum Computer Tehnology Centre for Quantum Dynamis Griffith University Brisbane 4 QLD Australia E-mail:
More informationz k sin(φ)(x ı + y j + z k)da = R 1 3 cos3 (φ) π 2π dθ = div(z k)dv = E curl(e x ı + e x j + e z k) d S = S
Mathematis 2443-6H Name (please print) Final xamination May 7, 28 Instrutions: Give brief, lear answers. Use theorems whenever possible. I. Verify the Divergene Theorem for the vetor field F(x,y,z) z k
More informationUPPER-TRUNCATED POWER LAW DISTRIBUTIONS
Fratals, Vol. 9, No. (00) 09 World Sientifi Publishing Company UPPER-TRUNCATED POWER LAW DISTRIBUTIONS STEPHEN M. BURROUGHS and SARAH F. TEBBENS College of Marine Siene, University of South Florida, St.
More informationProperties of Quarks
PHY04 Partile Physis 9 Dr C N Booth Properties of Quarks In the earlier part of this ourse, we have disussed three families of leptons but prinipally onentrated on one doublet of quarks, the u and d. We
More informationQ2. [40 points] Bishop-Hill Model: Calculation of Taylor Factors for Multiple Slip
27-750, A.D. Rollett Due: 20 th Ot., 2011. Homework 5, Volume Frations, Single and Multiple Slip Crystal Plastiity Note the 2 extra redit questions (at the end). Q1. [40 points] Single Slip: Calulating
More informationPlanning with Uncertainty in Position: an Optimal Planner
Planning with Unertainty in Position: an Optimal Planner Juan Pablo Gonzalez Anthony (Tony) Stentz CMU-RI -TR-04-63 The Robotis Institute Carnegie Mellon University Pittsburgh, Pennsylvania 15213 Otober
More informationQuantum Mechanics: Wheeler: Physics 6210
Quantum Mehanis: Wheeler: Physis 60 Problems some modified from Sakurai, hapter. W. S..: The Pauli matries, σ i, are a triple of matries, σ, σ i = σ, σ, σ 3 given by σ = σ = σ 3 = i i Let stand for the
More informationMath 32B Review Sheet
Review heet Tau Beta Pi - Boelter 6266 Contents ouble Integrals 2. Changing order of integration.................................... 4.2 Integrating over more general domains...............................
More informationF = F x x + F y. y + F z
ECTION 6: etor Calulus MATH20411 You met vetors in the first year. etor alulus is essentially alulus on vetors. We will need to differentiate vetors and perform integrals involving vetors. In partiular,
More informationDeveloping Excel Macros for Solving Heat Diffusion Problems
Session 50 Developing Exel Maros for Solving Heat Diffusion Problems N. N. Sarker and M. A. Ketkar Department of Engineering Tehnology Prairie View A&M University Prairie View, TX 77446 Abstrat This paper
More informationEvaluation of effect of blade internal modes on sensitivity of Advanced LIGO
Evaluation of effet of blade internal modes on sensitivity of Advaned LIGO T0074-00-R Norna A Robertson 5 th Otober 00. Introdution The urrent model used to estimate the isolation ahieved by the quadruple
More informationMAC Calculus II Summer All you need to know on partial fractions and more
MC -75-Calulus II Summer 00 ll you need to know on partial frations and more What are partial frations? following forms:.... where, α are onstants. Partial frations are frations of one of the + α, ( +
More informationMath 220A - Fall 2002 Homework 8 Solutions
Math A - Fall Homework 8 Solutions 1. Consider u tt u = x R 3, t > u(x, ) = φ(x) u t (x, ) = ψ(x). Suppose φ, ψ are supported in the annular region a < x < b. (a) Find the time T 1 > suh that u(x, t) is
More informationAcoustic Waves in a Duct
Aousti Waves in a Dut 1 One-Dimensional Waves The one-dimensional wave approximation is valid when the wavelength λ is muh larger than the diameter of the dut D, λ D. The aousti pressure disturbane p is
More informationDiscrete Bessel functions and partial difference equations
Disrete Bessel funtions and partial differene equations Antonín Slavík Charles University, Faulty of Mathematis and Physis, Sokolovská 83, 186 75 Praha 8, Czeh Republi E-mail: slavik@karlin.mff.uni.z Abstrat
More informationWe will show that: that sends the element in π 1 (P {z 1, z 2, z 3, z 4 }) represented by l j to g j G.
1. Introdution Square-tiled translation surfaes are lattie surfaes beause they are branhed overs of the flat torus with a single branhed point. Many non-square-tiled examples of lattie surfaes arise from
More informationSingular Event Detection
Singular Event Detetion Rafael S. Garía Eletrial Engineering University of Puerto Rio at Mayagüez Rafael.Garia@ee.uprm.edu Faulty Mentor: S. Shankar Sastry Researh Supervisor: Jonathan Sprinkle Graduate
More informationMillennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six
More information3 Tidal systems modelling: ASMITA model
3 Tidal systems modelling: ASMITA model 3.1 Introdution For many pratial appliations, simulation and predition of oastal behaviour (morphologial development of shorefae, beahes and dunes) at a ertain level
More informationMOLECULAR ORBITAL THEORY- PART I
5.6 Physial Chemistry Leture #24-25 MOLECULAR ORBITAL THEORY- PART I At this point, we have nearly ompleted our rash-ourse introdution to quantum mehanis and we re finally ready to deal with moleules.
More informationA Spatiotemporal Approach to Passive Sound Source Localization
A Spatiotemporal Approah Passive Sound Soure Loalization Pasi Pertilä, Mikko Parviainen, Teemu Korhonen and Ari Visa Institute of Signal Proessing Tampere University of Tehnology, P.O.Box 553, FIN-330,
More informationAdvances in Radio Science
Advanes in adio Siene 2003) 1: 99 104 Copernius GmbH 2003 Advanes in adio Siene A hybrid method ombining the FDTD and a time domain boundary-integral equation marhing-on-in-time algorithm A Beker and V
More informationDIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS
CHAPTER 4 DIGITAL DISTANCE RELAYING SCHEME FOR PARALLEL TRANSMISSION LINES DURING INTER-CIRCUIT FAULTS 4.1 INTRODUCTION Around the world, environmental and ost onsiousness are foring utilities to install
More informationExperiment 03: Work and Energy
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.01 Purpose of the Experiment: Experiment 03: Work and Energy In this experiment you allow a art to roll down an inlined ramp and run into
More informationAharonov-Bohm effect. Dan Solomon.
Aharonov-Bohm effet. Dan Solomon. In the figure the magneti field is onfined to a solenoid of radius r 0 and is direted in the z- diretion, out of the paper. The solenoid is surrounded by a barrier that
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E')
22.54 Neutron Interations and Appliations (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') Referenes -- J. R. Lamarsh, Introdution to Nulear Reator Theory (Addison-Wesley, Reading, 1966),
More informationAnalysis of discretization in the direct simulation Monte Carlo
PHYSICS OF FLUIDS VOLUME 1, UMBER 1 OCTOBER Analysis of disretization in the diret simulation Monte Carlo iolas G. Hadjionstantinou a) Department of Mehanial Engineering, Massahusetts Institute of Tehnology,
More informationFINITE WORD LENGTH EFFECTS IN DSP
FINITE WORD LENGTH EFFECTS IN DSP PREPARED BY GUIDED BY Snehal Gor Dr. Srianth T. ABSTRACT We now that omputers store numbers not with infinite preision but rather in some approximation that an be paed
More informationMeasuring & Inducing Neural Activity Using Extracellular Fields I: Inverse systems approach
Measuring & Induing Neural Ativity Using Extraellular Fields I: Inverse systems approah Keith Dillon Department of Eletrial and Computer Engineering University of California San Diego 9500 Gilman Dr. La
More informationRelative Maxima and Minima sections 4.3
Relative Maxima and Minima setions 4.3 Definition. By a ritial point of a funtion f we mean a point x 0 in the domain at whih either the derivative is zero or it does not exists. So, geometrially, one
More informationIn this case it might be instructive to present all three components of the current density:
Momentum, on the other hand, presents us with a me ompliated ase sine we have to deal with a vetial quantity. The problem is simplified if we treat eah of the omponents of the vet independently. s you
More informationParticle-wave symmetry in Quantum Mechanics And Special Relativity Theory
Partile-wave symmetry in Quantum Mehanis And Speial Relativity Theory Author one: XiaoLin Li,Chongqing,China,hidebrain@hotmail.om Corresponding author: XiaoLin Li, Chongqing,China,hidebrain@hotmail.om
More informationA NETWORK SIMPLEX ALGORITHM FOR THE MINIMUM COST-BENEFIT NETWORK FLOW PROBLEM
NETWORK SIMPLEX LGORITHM FOR THE MINIMUM COST-BENEFIT NETWORK FLOW PROBLEM Cen Çalışan, Utah Valley University, 800 W. University Parway, Orem, UT 84058, 801-863-6487, en.alisan@uvu.edu BSTRCT The minimum
More informationLikelihood-confidence intervals for quantiles in Extreme Value Distributions
Likelihood-onfidene intervals for quantiles in Extreme Value Distributions A. Bolívar, E. Díaz-Franés, J. Ortega, and E. Vilhis. Centro de Investigaión en Matemátias; A.P. 42, Guanajuato, Gto. 36; Méxio
More informationNormative and descriptive approaches to multiattribute decision making
De. 009, Volume 8, No. (Serial No.78) China-USA Business Review, ISSN 57-54, USA Normative and desriptive approahes to multiattribute deision making Milan Terek (Department of Statistis, University of
More information9 Geophysics and Radio-Astronomy: VLBI VeryLongBaseInterferometry
9 Geophysis and Radio-Astronomy: VLBI VeryLongBaseInterferometry VLBI is an interferometry tehnique used in radio astronomy, in whih two or more signals, oming from the same astronomial objet, are reeived
More informationmax min z i i=1 x j k s.t. j=1 x j j:i T j
AM 221: Advaned Optimization Spring 2016 Prof. Yaron Singer Leture 22 April 18th 1 Overview In this leture, we will study the pipage rounding tehnique whih is a deterministi rounding proedure that an be
More informationEinstein s Three Mistakes in Special Relativity Revealed. Copyright Joseph A. Rybczyk
Einstein s Three Mistakes in Speial Relativity Revealed Copyright Joseph A. Rybzyk Abstrat When the evidene supported priniples of eletromagneti propagation are properly applied, the derived theory is
More informationControl Theory association of mathematics and engineering
Control Theory assoiation of mathematis and engineering Wojieh Mitkowski Krzysztof Oprzedkiewiz Department of Automatis AGH Univ. of Siene & Tehnology, Craow, Poland, Abstrat In this paper a methodology
More information5.1 Composite Functions
SECTION. Composite Funtions 7. Composite Funtions PREPARING FOR THIS SECTION Before getting started, review the following: Find the Value of a Funtion (Setion., pp. 9 ) Domain of a Funtion (Setion., pp.
More informationThe Lorenz Transform
The Lorenz Transform Flameno Chuk Keyser Part I The Einstein/Bergmann deriation of the Lorentz Transform I follow the deriation of the Lorentz Transform, following Peter S Bergmann in Introdution to the
More informationChapter 2: Solution of First order ODE
0 Chapter : Solution of irst order ODE Se. Separable Equations The differential equation of the form that is is alled separable if f = h g; In order to solve it perform the following steps: Rewrite the
More informationLECTURE NOTES FOR , FALL 2004
LECTURE NOTES FOR 18.155, FALL 2004 83 12. Cone support and wavefront set In disussing the singular support of a tempered distibution above, notie that singsupp(u) = only implies that u C (R n ), not as
More informationNonreversibility of Multiple Unicast Networks
Nonreversibility of Multiple Uniast Networks Randall Dougherty and Kenneth Zeger September 27, 2005 Abstrat We prove that for any finite direted ayli network, there exists a orresponding multiple uniast
More informationBerry s phase for coherent states of Landau levels
Berry s phase for oherent states of Landau levels Wen-Long Yang 1 and Jing-Ling Chen 1, 1 Theoretial Physis Division, Chern Institute of Mathematis, Nankai University, Tianjin 300071, P.R.China Adiabati
More informationPanos Kouvelis Olin School of Business Washington University
Quality-Based Cometition, Profitability, and Variable Costs Chester Chambers Co Shool of Business Dallas, TX 7575 hamber@mailosmuedu -768-35 Panos Kouvelis Olin Shool of Business Washington University
More informationMost results in this section are stated without proof.
Leture 8 Level 4 v2 he Expliit formula. Most results in this setion are stated without proof. Reall that we have shown that ζ (s has only one pole, a simple one at s =. It has trivial zeros at the negative
More informationA Differential Equation for Specific Catchment Area
Proeedings of Geomorphometry 2009. Zurih, Sitzerland, 3 ugust - 2 September, 2009 Differential Equation for Speifi Cathment rea J. C. Gallant, M. F. Huthinson 2 CSIRO Land and Water, GPO Box 666, Canberra
More informationBottom Shear Stress Formulations to Compute Sediment Fluxes in Accelerated Skewed Waves
Journal of Coastal Researh SI 5 453-457 ICS2009 (Proeedings) Portugal ISSN 0749-0258 Bottom Shear Stress Formulations to Compute Sediment Fluxes in Aelerated Skewed Waves T. Abreu, F. Sanho and P. Silva
More informationBeams on Elastic Foundation
Professor Terje Haukaas University of British Columbia, Vanouver www.inrisk.ub.a Beams on Elasti Foundation Beams on elasti foundation, suh as that in Figure 1, appear in building foundations, floating
More informationA variant of Coppersmith s Algorithm with Improved Complexity and Efficient Exhaustive Search
A variant of Coppersmith s Algorithm with Improved Complexity and Effiient Exhaustive Searh Jean-Sébastien Coron 1, Jean-Charles Faugère 2, Guénaël Renault 2, and Rina Zeitoun 2,3 1 University of Luxembourg
More informationGeneral Equilibrium. What happens to cause a reaction to come to equilibrium?
General Equilibrium Chemial Equilibrium Most hemial reations that are enountered are reversible. In other words, they go fairly easily in either the forward or reverse diretions. The thing to remember
More informationThe shape of a hanging chain. a project in the calculus of variations
The shape of a hanging hain a projet in the alulus of variations April 15, 218 2 Contents 1 Introdution 5 2 Analysis 7 2.1 Model............................... 7 2.2 Extremal graphs.........................
More information6.4 Dividing Polynomials: Long Division and Synthetic Division
6 CHAPTER 6 Rational Epressions 6. Whih of the following are equivalent to? y a., b. # y. y, y 6. Whih of the following are equivalent to 5? a a. 5, b. a 5, 5. # a a 6. In your own words, eplain one method
More informationIntegration of the Finite Toda Lattice with Complex-Valued Initial Data
Integration of the Finite Toda Lattie with Complex-Valued Initial Data Aydin Huseynov* and Gusein Sh Guseinov** *Institute of Mathematis and Mehanis, Azerbaijan National Aademy of Sienes, AZ4 Baku, Azerbaijan
More informationFinal Review. A Puzzle... Special Relativity. Direction of the Force. Moving at the Speed of Light
Final Review A Puzzle... Diretion of the Fore A point harge q is loated a fixed height h above an infinite horizontal onduting plane. Another point harge q is loated a height z (with z > h) above the plane.
More informationA Queueing Model for Call Blending in Call Centers
A Queueing Model for Call Blending in Call Centers Sandjai Bhulai and Ger Koole Vrije Universiteit Amsterdam Faulty of Sienes De Boelelaan 1081a 1081 HV Amsterdam The Netherlands E-mail: {sbhulai, koole}@s.vu.nl
More informationStudy of EM waves in Periodic Structures (mathematical details)
Study of EM waves in Periodi Strutures (mathematial details) Massahusetts Institute of Tehnology 6.635 partial leture notes 1 Introdution: periodi media nomenlature 1. The spae domain is defined by a basis,(a
More informationINTRO VIDEOS. LESSON 9.5: The Doppler Effect
DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS INTRO VIDEOS Big Bang Theory of the Doppler Effet Doppler Effet LESSON 9.5: The Doppler Effet 1. Essential Idea: The Doppler Effet desribes the phenomenon
More informationCavity flow with surface tension past a flat plate
Proeedings of the 7 th International Symposium on Cavitation CAV9 Paper No. ## August 7-, 9, Ann Arbor, Mihigan, USA Cavity flow with surfae tension past a flat plate Yuriy Savhenko Institute of Hydromehanis
More informationMathematics II. Tutorial 5 Basic mathematical modelling. Groups: B03 & B08. Ngo Quoc Anh Department of Mathematics National University of Singapore
Mathematis II Tutorial 5 Basi mathematial modelling Groups: B03 & B08 February 29, 2012 Mathematis II Ngo Quo Anh Ngo Quo Anh Department of Mathematis National University of Singapore 1/13 : The ost of
More informationChapter 8 Hypothesis Testing
Leture 5 for BST 63: Statistial Theory II Kui Zhang, Spring Chapter 8 Hypothesis Testing Setion 8 Introdution Definition 8 A hypothesis is a statement about a population parameter Definition 8 The two
More informationNon-Markovian study of the relativistic magnetic-dipole spontaneous emission process of hydrogen-like atoms
NSTTUTE OF PHYSCS PUBLSHNG JOURNAL OF PHYSCS B: ATOMC, MOLECULAR AND OPTCAL PHYSCS J. Phys. B: At. Mol. Opt. Phys. 39 ) 7 85 doi:.88/953-75/39/8/ Non-Markovian study of the relativisti magneti-dipole spontaneous
More informationDirectional Coupler. 4-port Network
Diretional Coupler 4-port Network 3 4 A diretional oupler is a 4-port network exhibiting: All ports mathed on the referene load (i.e. S =S =S 33 =S 44 =0) Two pair of ports unoupled (i.e. the orresponding
More informationCDS 101/110: Lecture 2.1 System Modeling. Model-Based Analysis of Feedback Systems
CDS 11/11: Leture 21 System Modeling Rihard M Murray 5 Otober 215 Goals:! Define a model and its use in answering questions about a system! Introdue the onepts of state, dynamis, inputs and outputs! Review
More informationA population of 50 flies is expected to double every week, leading to a function of the x
4 Setion 4.3 Logarithmi Funtions population of 50 flies is epeted to doule every week, leading to a funtion of the form f ( ) 50(), where represents the numer of weeks that have passed. When will this
More information7 Max-Flow Problems. Business Computing and Operations Research 608
7 Max-Flow Problems Business Computing and Operations Researh 68 7. Max-Flow Problems In what follows, we onsider a somewhat modified problem onstellation Instead of osts of transmission, vetor now indiates
More informationSampler-A. Secondary Mathematics Assessment. Sampler 521-A
Sampler-A Seondary Mathematis Assessment Sampler 521-A Instrutions for Students Desription This sample test inludes 14 Seleted Response and 4 Construted Response questions. Eah Seleted Response has a
More informationFrequency Domain Analysis of Concrete Gravity Dam-Reservoir Systems by Wavenumber Approach
Frequeny Domain Analysis of Conrete Gravity Dam-Reservoir Systems by Wavenumber Approah V. Lotfi & A. Samii Department of Civil and Environmental Engineering, Amirkabir University of Tehnology, Tehran,
More informationAppendix A Market-Power Model of Business Groups. Robert C. Feenstra Deng-Shing Huang Gary G. Hamilton Revised, November 2001
Appendix A Market-Power Model of Business Groups Roert C. Feenstra Deng-Shing Huang Gary G. Hamilton Revised, Novemer 200 Journal of Eonomi Behavior and Organization, 5, 2003, 459-485. To solve for the
More information4.4 Solving Systems of Equations by Matrices
Setion 4.4 Solving Systems of Equations by Matries 1. A first number is 8 less than a seond number. Twie the first number is 11 more than the seond number. Find the numbers.. The sum of the measures of
More informationU S A Mathematical Talent Search. PROBLEMS / SOLUTIONS / COMMENTS Round 4 - Year 11 - Academic Year
U S A Mathematial Talent Searh PROBLEMS / SOLUTIONS / COMMENTS Round 4 - Year 11 - Aademi Year 1999-000 Gene A. Berg, Editor 1/4/11. Determine the unique 9-digit integer M that has the following properties:
More informationComputer Engineering 4TL4: Digital Signal Processing (Fall 2003) Solutions to Final Exam
Computer Engineering TL: Digital Signal Proessing (Fall 3) Solutions to Final Exam The step response ynof a ausal, stable LTI system is: n [ ] = [ yn ] un, [ ] where un [ ] is the unit step funtion a Find
More informationModeling Probabilistic Measurement Correlations for Problem Determination in Large-Scale Distributed Systems
009 9th IEEE International Conferene on Distributed Computing Systems Modeling Probabilisti Measurement Correlations for Problem Determination in Large-Sale Distributed Systems Jing Gao Guofei Jiang Haifeng
More information(a) We desribe physics as a sequence of events labelled by their space time coordinates: x µ = (x 0, x 1, x 2 x 3 ) = (c t, x) (12.
2 Relativity Postulates (a) All inertial observers have the same equations of motion and the same physial laws. Relativity explains how to translate the measurements and events aording to one inertial
More informationFour-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field
Four-dimensional equation of motion for visous ompressible substane with regard to the aeleration field, pressure field and dissipation field Sergey G. Fedosin PO box 6488, Sviazeva str. -79, Perm, Russia
More informationDifferential Equations 8/24/2010
Differential Equations A Differential i Equation (DE) is an equation ontaining one or more derivatives of an unknown dependant d variable with respet to (wrt) one or more independent variables. Solution
More informationEECS 120 Signals & Systems University of California, Berkeley: Fall 2005 Gastpar November 16, Solutions to Exam 2
EECS 0 Signals & Systems University of California, Berkeley: Fall 005 Gastpar November 6, 005 Solutions to Exam Last name First name SID You have hour and 45 minutes to omplete this exam. he exam is losed-book
More informationGrasp Planning: How to Choose a Suitable Task Wrench Space
Grasp Planning: How to Choose a Suitable Task Wrenh Spae Ch. Borst, M. Fisher and G. Hirzinger German Aerospae Center - DLR Institute for Robotis and Mehatronis 8223 Wessling, Germany Email: [Christoph.Borst,
More informationarxiv: v2 [cs.dm] 4 May 2018
Disrete Morse theory for the ollapsibility of supremum setions Balthazar Bauer INRIA, DIENS, PSL researh, CNRS, Paris, Frane Luas Isenmann LIRMM, Université de Montpellier, CNRS, Montpellier, Frane arxiv:1803.09577v2
More informationNEW MEANS OF CYBERNETICS, INFORMATICS, COMPUTER ENGINEERING, AND SYSTEMS ANALYSIS
Cybernetis and Systems Analysis, Vol. 43, No. 5, 007 NEW MEANS OF CYBERNETICS, INFORMATICS, COMPUTER ENGINEERING, AND SYSTEMS ANALYSIS ARCHITECTURAL OPTIMIZATION OF A DIGITAL OPTICAL MULTIPLIER A. V. Anisimov
More informationInternational Journal of Advanced Engineering Research and Studies E-ISSN
Researh Paper FINIE ELEMEN ANALYSIS OF A CRACKED CANILEVER BEAM Mihir Kumar Sutar Address for Correspondene Researh Sholar, Department of Mehanial & Industrial Engineering Indian Institute of ehnology
More informationLATTICE BOLTZMANN METHOD FOR MICRO CHANNEL AND MICRO ORIFICE FLOWS TAIHO YEOM. Bachelor of Science in Mechanical Engineering.
LATTICE BOLTZMANN METHOD FOR MICRO CHANNEL AND MICRO ORIFICE FLOWS By TAIHO YEOM Bahelor of Siene in Mehanial Engineering Ajou University Suwon, South Korea 2005 Submitted to the Faulty of the Graduate
More information1 sin 2 r = 1 n 2 sin 2 i
Physis 505 Fall 005 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.5, 7.8, 7.16 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with
More informationChapter 9. The excitation process
Chapter 9 The exitation proess qualitative explanation of the formation of negative ion states Ne and He in He-Ne ollisions an be given by using a state orrelation diagram. state orrelation diagram is
More informationSURFACE WAVES OF NON-RAYLEIGH TYPE
SURFACE WAVES OF NON-RAYLEIGH TYPE by SERGEY V. KUZNETSOV Institute for Problems in Mehanis Prosp. Vernadskogo, 0, Mosow, 75 Russia e-mail: sv@kuznetsov.msk.ru Abstrat. Existene of surfae waves of non-rayleigh
More informationRobust Recovery of Signals From a Structured Union of Subspaces
Robust Reovery of Signals From a Strutured Union of Subspaes 1 Yonina C. Eldar, Senior Member, IEEE and Moshe Mishali, Student Member, IEEE arxiv:87.4581v2 [nlin.cg] 3 Mar 29 Abstrat Traditional sampling
More informationMATHEMATICAL AND NUMERICAL BASIS OF BINARY ALLOY SOLIDIFICATION MODELS WITH SUBSTITUTE THERMAL CAPACITY. PART II
Journal of Applied Mathematis and Computational Mehanis 2014, 13(2), 141-147 MATHEMATICA AND NUMERICA BAI OF BINARY AOY OIDIFICATION MODE WITH UBTITUTE THERMA CAPACITY. PART II Ewa Węgrzyn-krzypzak 1,
More informationChapter 2 Lecture 8 Longitudinal stick fixed static stability and control 5 Topics
Flight dynamis II Stability and ontrol hapter 2 Leture 8 Longitudinal stik fied stati stability and ontrol 5 Topis 2.6 ontributions of power plant to mg and mα 2.6.1 Diret ontributions of powerplant to
More informationReview of Force, Stress, and Strain Tensors
Review of Fore, Stress, and Strain Tensors.1 The Fore Vetor Fores an be grouped into two broad ategories: surfae fores and body fores. Surfae fores are those that at over a surfae (as the name implies),
More informationDevelopment of a user element in ABAQUS for modelling of cohesive laws in composite structures
Downloaded from orbit.dtu.dk on: Jan 19, 2019 Development of a user element in ABAQUS for modelling of ohesive laws in omposite strutures Feih, Stefanie Publiation date: 2006 Doument Version Publisher's
More informationCounting Idempotent Relations
Counting Idempotent Relations Beriht-Nr. 2008-15 Florian Kammüller ISSN 1436-9915 2 Abstrat This artile introdues and motivates idempotent relations. It summarizes haraterizations of idempotents and their
More information