EE 178/278A Probabilistic Systems Analysis. Spring 2014 Tse/Hussami Lecture 11. A Brief Introduction to Continuous Probability


 Hilary Stephens
 2 years ago
 Views:
Transcription
1 EE 178/278A Probabiistic Systems Anaysis Spring 2014 Tse/Hussami Lecture 11 A Brief Introduction to Continuous Probabiity Up to now we have focused excusivey on discrete probabiity spaces Ω, where the number of outcomes ω Ω is either finite or countaby infinite (such as the integers). As a consequence we have ony been abe to tak about discrete random variabes, which take on ony a finite (or countaby infinite) number of vaues. But in rea ife many quantities that we wish to mode probabiisticay are continuousvaued; exampes incude the position of a partice in a box, the time at which a certain incident happens, or the direction of trave of a meteorite. In this ecture, we discuss how to extend the concepts we ve seen in the discrete setting to this continuous setting. As we sha see, everything transates in a natura way once we have set up the right framework. The framework invoves some eementary cacuus. Continuous uniform probabiity spaces Suppose we spin a whee of fortune and record the position of the pointer on the outer circumference of the whee. Assuming that the circumference is of ength and that the whee is unbiased, the position is presumaby equay ikey to take on any vaue in the rea interva [0,]. How do we mode this experiment using a probabiity space? Consider for a moment the (amost) anaogous discrete setting, where the pointer can stop ony at a finite number m of positions distributed eveny around the whee. (If m is very arge, then presumaby this is in some sense simiar to the continuous setting.) Then we woud mode this situation using the discrete sampe space Ω = {0, m, 2 (m 1) m,..., m }, with uniform probabiities P(ω) = 1 m for each ω Ω. In the continuous word, however, we get into troube if we try the same approach. If we et ω range over a rea numbers in [0,], what vaue shoud we assign to each P(ω)? By uniformity this probabiity shoud be the same for a ω, but then if we assign to it any positive vaue, the sum of a probabiities P(ω) for ω Ω wi be! Thus P(ω) must be zero for a ω Ω. But if a of our outcomes have probabiity zero, then we are unabe to assign meaningfu probabiities to any events! To rescue this situation, consider instead any nonempty interva [a, b] [0, ]. Can we assign a nonzero probabiity vaue to this interva? Since the tota probabiity assigned to [0,] must be 1, and since we want our probabiity to be uniform, the natura assignment of probabiity to the interva [a,b] is ength of [a,b] P([a,b]) = ength of [0,] = b a. (1) In other words, the probabiity of an interva is proportiona to its ength. Note that intervas are subsets of the sampe space Ω and are therefore events. So in continuous probabiity, we are assigning probabiities to certain basic events, in contrast to discrete probabiity, where we assigned probabiity to points in the sampe space. But what about probabiities of other events? Actuay, by specifying the probabiity of intervas we have aso specified the probabiity of any event E which can be written EE 178/278A, Spring 2014, Lecture 11 1
2 Figure 1: The event E as a subset of the sampe space Ω. as the disjoint union of (a finite or countaby infinite number of) intervas, E = i E i. For then we can write P(E) = i P(E i ), in anaogous fashion to the discrete case. Thus for exampe the probabiity that the pointer ends up in the first or third quadrants of the whee is /4 + /4 = 1 2. For a practica purposes, such events are a we reay need. 1 An exampe: Buffon s neede Here is a simpe appication of continuous probabiity to the anaysis of a cassica procedure for estimating the vaue of π known as Buffon s neede, after its 18th century inventor GeorgesLouis Lecerc, Comte de Buffon. Here we are given a neede of ength, and a board rued with horizonta ines at distance apart. The experiment consists of throwing the neede randomy onto the board and observing whether or not it crosses one of the ines. We sha see beow that (assuming a perfecty random throw) the probabiity of this event is exacty 2/π. This means that, if we perform the experiment many times and record the proportion of throws on which the neede crosses a ine, then the Law of Large Numbers (Lecture Note 10) tes us that we wi get a good estimate of the quantity 2/π, and therefore aso of π; and we can use Chebyshev s inequaity as in the other estimation probems we considered in that same Lecture Note to determine how many throws we need in order to achieve specified accuracy and confidence. To anayze the experiment, we first need to specify the probabiity space. Note that the position where the neede ands is competey specified by two numbers: the vertica distance y between the midpoint of the neede and the cosest horizonta ine, and the ange θ between the neede and the vertica. The vertica distance y ranges between 0 and /2, whie θ ranges between π/2 and π/2. Thus, the sampe space is the rectange Ω = [ π/2,π/2] [0,/2]. Note that, compared to the wheeoffortune exampe, the sampe space is twodimensiona rather than onedimensiona. But ike the wheeoffortune exampe, the sampe space is aso continuous. Now et E denote the event that the neede crosses a ine. It is a subset of the sampe space Ω. We need to identify this subset expicity. By eementary geometry the vertica distance of the endpoint of the neede from its midpoint is 2 cosθ, so the neede wi cross the ine if and ony if y 2 cosθ. The event E is sketched in Figure 1. 1 A forma treatment of which events can be assigned a wedefined probabiity requires a discussion of measure theory, which is beyond the scope of this course. EE 178/278A, Spring 2014, Lecture 11 2
3 Since we are assuming a competey random throw, probabiity of the event E is: P(E) = area of E area of Ω. This is the twodimensiona generaization of equation (1) in the wheeoffortune exampe, where the probabiity of anding in an interva is proportiona to the ength of the interva. The area of the whoe sampe space is π/2. The area of E is: π/2 [ ] π/2 cosθ dθ = π/2 2 2 sinθ =. π/2 Hence, P(E) = π/2 = 2 π. This is exacty what we caimed at the beginning of the section! Continuous random variabes Reca that in the discrete setting we typicay work with random variabes and their distributions, rather than directy with probabiity spaces and events. This is even more so in continuous probabiity, since numerica quantities are amost aways invoved. In the wheeoffortune exampe, the position X of the pointer is a random variabe. In the Buffon neede exampe, the vertica distance Y and the ange Θ are random variabes. In fact, they are a continuous random variabes. These random variabes are a reativey simpe, in the sense that they are a uniformy distributed on the range of vaues they can take on. (Because of this simpicity, we didn t even need to worry about the random variabes expicity when cacuating probabiities in these exampes, and instead reason directy with the sampe space.) But more compicated random variabes do not have a uniform distribution. How, precisey, shoud we define the distribution of a genera continuous random variabe? In the discrete case the distribution of a r.v. X is described by specifying, for each possibe vaue a, the probabiity P(X = a). But for the r.v. X corresponding to the position of the pointer, we have P(X = a) = 0 for every a, so we run into the same probem as we encountered above in defining the probabiity space. The resoution is essentiay the same: instead of specifying P(X = a), we instead specify P(a < X b) for a intervas [a,b]. 2 To do this formay, we need to introduce the concept of a probabiity density function (sometimes referred to just as a density, or a pdf ). Definition 11.1 (Density): A probabiity density function for a random variabe X is a function f : R R satisfying b P(a < X b) = f (x)dx for a a b. (2) a Let s examine this definition. Note that the definite integra is just the area under the curve f between the vaues a and b (Figure 2(a)). Thus f pays a simiar roe to the histogram we sometimes draw to picture the distribution of a discrete random variabe. In order for the definition to make sense, f must obey certain properties. Some of these are technica in nature, which basicay just ensure that the integra is aways we defined; we sha not dwe on this issue 2 Note that it does not matter whether or not we incude the endpoints a,b; since P(X = a) = P(X = b) = 0, we have P(a < X < b) = P(a < X b) = P(a < X < b). EE 178/278A, Spring 2014, Lecture 11 3
4 Figure 2: (a) The area under the density curve between a and b is the probabiity that the random variabe ies in that range. (b) For sma δ, the area under the curve between x and x + δ can be we approximated by the area of the rectange of height f (x) and width δ. here since a the densities that we wi meet wi be we behaved. What about some more basic properties of f? First, it must be the case that f is a nonnegative function; for if f took on negative vaues we coud find an interva in which the integra is negative, so we woud have a negative probabiity for some event! Second, since the r.v. X must take on some vaue everywhere in the space, we must have f (x)dx = P( < X < ) = 1. (3) In other words, the tota area under the curve f must be 1. A caveat is in order here. Foowing the histogram anaogy above, it is tempting to think of f (x) as a probabiity. However, f (x) doesn t itsef correspond to the probabiity of anything! For one thing, there is no requirement that f (x) be bounded by 1 (and indeed, we sha see exampes of densities in which f (x) is greater than 1 for some x). To connect f (x) with probabiities, we need to ook at a very sma interva [x,x +δ] cose to x. Assuming that the interva [x,x +δ] is so sma that the function f doesn t change much over that interva. we have x+δ P(x < X x + δ) = f (z)dz δ f (x). (4) x This approximation is iustrated in Figure 2(b). Equivaenty, f (x) P(x < X x + δ). δ The approximation in (5) becomes more accurate as δ becomes sma. Hence, more formay, we can reate density and probabiity by taking imits: P(x < X x + δ) f (x) = im. (6) δ 0 δ Thus we can interpret f (x) as the probabiity per unit ength in the vicinity of x. Note that whie the equation (2) aows us to compute probabiities given the probabiity density function, the equation (6) aows us to compute the probabiity density function given probabiities. Both reationships are usefu in probems. (5) EE 178/278A, Spring 2014, Lecture 11 4
5 Figure 3: The density function of the wheeoffortune r.v. X. Now et s go back and put our wheeoffortune r.v. X into this framework. What shoud be the density of X? We, we want X to have nonzero probabiity ony on the interva [0,], so we shoud certainy have f (x) = 0 for x < 0 and for x >. Within the interva [0,] we want the distribution of X to be uniform, which means we shoud take f (x) = c for 0 x. What shoud be the vaue of c? This is determined by the requirement (3) that the tota area under f is 1. The area under the above curve is f (x)dx = 0 cdx = c, so we must take c = 1. Summarizing, then, the density of the uniform distribution on [0,] is given by 0 for x < 0; f (x) = 1/ for 0 x ; 0 for x >. This is potted in Figure 3. Note that f (x) can certainy be greater than 1, depending on the vaue of. Another Exampe: Suppose you throw a dart and it ands uniformy at random on a target which is a disk of unit radius. What is the probabiity density function of the distance of the dart from the center of the disk? Let X be the distance of the dart from the center of the disk. We first cacuate the probabiity that X is between x and x + δ. If x is negative or greater than or equa to 1, this probabiity is zero, so we focus on the case that x is between 0 and 1. The event in question is that the dart ands in the ring (annuus) shown in Figure 4. Since the dart ands uniformy at random on the disk, the probabiity of the event is just the ratio of the area of the ring and the area of the disk. Hence, P(x < X x + δ) = π[(x + δ)2 x 2 ] π(1) 2 (7) = x 2 + 2δx + δ 2 x 2 = 2δx δ 2. Using equation (6), we can now compute the probabiity density function of X: P(x < X x + δ) 2δx δ 2 f (x) = im = f (x) = im = 2x. δ 0 δ δ 0 δ Summarizing, we have 0 for x 0; f (x) = 2x for 0 x < 1; 0 for x 1. EE 178/278A, Spring 2014, Lecture 11 5
6 Figure 4: The sampe space is the disk of unit radius. The event of interest is the ring. Figure 5: (a) The probabiity density function and (b) the cumuative distribution function of the distance X from the target center. It is potted in Figure 5(a). Note that athough the dart ands uniformy inside the target, the distance X from the center is not uniformy distributed in the range from 0 to 1. This is because an ring farther away from the center has a arger area than an ring coser to the center with the same width δ. Hence the probabiity of anding in the ring farther away from the center is arger. Cumuative Distribution Function Let us reinterpret equation (7) in the dart throwing exampe above. In words, we are saying: P(x < X x + δ) = area of ring area of target = (area of disk of radius x + δ) (area of disk of radius x) area of target = area of disk of radius x + δ area of disk of radius x area of target area of target = P(X x + δ) P(X x). EE 178/278A, Spring 2014, Lecture 11 6
7 This ast equaity can be understood directy as foows. The event A that X x + δ (dart ands inside disk of radius x + δ) can be decomposed as a union of two events: 1) the event B that X x (dart ands inside disk of radius x), and 2) the event C that x < X x + δ (dart ands inside ring). The two events are disjoint. (See Figure 4.) Hence, P(A) = P(B) + P(C) or P(x < X x + δ) = P(X x + δ) P(X x), (8) which is exacty the same as above. Ceary, the reasoning eading to (8) has nothing much to do the particuars of this exampe but in fact (8) hods true for any random variabe X. A we needed are the facts that A = B C and B and C are disjoint events, and the facts are true in genera. Substituting (8) into (6), we obtain: P(X x + δ) P(X x) f (x) = im. δ 0 δ What does this equation remind you of? To make things even more expicit, et us define the cumuative distribution function F(x) = P(X x). (9) Then we have: F(x + δ) F(x) f (x) = im = d F(x). (10) δ 0 δ dx The function F has a name: it is caed the cumuative distribution function of the random variabe X (sometimes abbreviated as cdf). It is caed cumuative because at each vaue x, F(x) is the cumuative probabiity up to x. Note that the cumuative distribution function and the probabiity density function of a random variabe contains exacty the same information. Given the cumuative distribution function F, one can differentiate to get the probabiity density function f. Given the probabiity density function f, one can integrate to get the cumuative distribution function: F(x) = x f (a)da. So stricty speaking, one does not need to introduce the concept of cumuative distribution function at a. However, for many probems, the cumuative distribution function is easier to compute first and from that one can then compute the probabiity density function. Summarizing, we have: Definition 11.2 (Cumuative Distribution Function): The cumuative distribution function for a random variabe X is a function F : R R defined to be: F(x) = P(X x). (11) Its reationship with the probabiity density function f of X is given by f (x) = d dx F(x), F(x) = x f (a)da. EE 178/278A, Spring 2014, Lecture 11 7
8 The cumuative distribution function satisfies the foowing properties: 1. 0 F(x) 1 2. im x F(x) = 0 3. im x F(x) = 1 EE 178/278A, Spring 2014, Lecture 11 8
Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 18. A Brief Introduction to Continuous Probability
CS 7 Discrete Mathematics and Probability Theory Fall 29 Satish Rao, David Tse Note 8 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces
More information3.5 Pendulum period. 20090210 19:40:05 UTC / rev 4d4a39156f1e. g = 4π2 l T 2. g = 4π2 x1 m 4 s 2 = π 2 m s 2. 3.5 Pendulum period 68
68 68 3.5 Penduum period 68 3.5 Penduum period Is it coincidence that g, in units of meters per second squared, is 9.8, very cose to 2 9.87? Their proximity suggests a connection. Indeed, they are connected
More informationSorting, Merge Sort and the DivideandConquer Technique
Inf2B gorithms and Data Structures Note 7 Sorting, Merge Sort and the DivideandConquer Technique This and a subsequent next ecture wi mainy be concerned with sorting agorithms. Sorting is an extremey
More informationENERGY AND BOLTZMANN DISTRIBUTIONS
MISN159 NRGY AND BOLTZMANN DISTRIBUTIONS NRGY AND BOLTZMANN DISTRIBUTIONS by J. S. Kovacs and O. McHarris Michigan State University 1. Introduction.............................................. 1 2.
More informationEnergy Density / Energy Flux / Total Energy in 3D
Lecture 5 Phys 75 Energy Density / Energy Fux / Tota Energy in D Overview and Motivation: In this ecture we extend the discussion of the energy associated with wave otion to waves described by the D wave
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow Facts & Formuas Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More informationComputing the depth of an arrangement of axisaligned rectangles in parallel
Computing the depth of an arrangement of axisaigned rectanges in parae Hemut At Ludmia Scharf Abstract We consider the probem of computing the depth of the arrangement of n axisaigned rectanges in the
More informationSAT Math Facts & Formulas
Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:
More informationSecure Network Coding with a Cost Criterion
Secure Network Coding with a Cost Criterion Jianong Tan, Murie Médard Laboratory for Information and Decision Systems Massachusetts Institute of Technoogy Cambridge, MA 0239, USA Emai: {jianong, medard}@mit.edu
More informationRisk Margin for a NonLife Insurance RunOff
Risk Margin for a NonLife Insurance RunOff Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas February 2, 2011 Abstract For sovency purposes insurance companies need to cacuate socaed bestestimate
More informationFinance 360 Problem Set #6 Solutions
Finance 360 Probem Set #6 Soutions 1) Suppose that you are the manager of an opera house. You have a constant margina cost of production equa to $50 (i.e. each additiona person in the theatre raises your
More informationTERM INSURANCE CALCULATION ILLUSTRATED. This is the U.S. Social Security Life Table, based on year 2007.
This is the U.S. Socia Security Life Tabe, based on year 2007. This is avaiabe at http://www.ssa.gov/oact/stats/tabe4c6.htm. The ife eperiences of maes and femaes are different, and we usuay do separate
More informationRisk Margin for a NonLife Insurance RunOff
Risk Margin for a NonLife Insurance RunOff Mario V. Wüthrich, Pau Embrechts, Andreas Tsanakas August 15, 2011 Abstract For sovency purposes insurance companies need to cacuate socaed bestestimate reserves
More informationDiscounted Cash Flow Analysis (aka Engineering Economy)
Discounted Cash Fow Anaysis (aka Engineering Economy) Objective: To provide economic comparison of benefits and costs that occur over time Assumptions: Future benefits and costs can be predicted A Benefits,
More informationAn Idiot s guide to Support vector machines (SVMs)
An Idiot s guide to Support vector machines (SVMs) R. Berwick, Viage Idiot SVMs: A New Generation of Learning Agorithms Pre 1980: Amost a earning methods earned inear decision surfaces. Linear earning
More informationTeaching fractions in elementary school: A manual for teachers
Teaching fractions in eementary schoo: A manua for teachers H. Wu Apri 0, 998 [Added December, 200] I have decided to resurrect this fie of 998 because, as a reativey short summary of the basic eements
More information5. Introduction to Robot Geometry and Kinematics
V. Kumar 5. Introduction to Robot Geometry and Kinematics The goa of this chapter is to introduce the basic terminoogy and notation used in robot geometry and kinematics, and to discuss the methods used
More informationChapter 3: JavaScript in Action Page 1 of 10. How to practice reading and writing JavaScript on a Web page
Chapter 3: JavaScript in Action Page 1 of 10 Chapter 3: JavaScript in Action In this chapter, you get your first opportunity to write JavaScript! This chapter introduces you to JavaScript propery. In addition,
More informationPhysics 100A Homework 11 Chapter 11 (part 1) The force passes through the point A, so there is no arm and the torque is zero.
Physics A Homework  Chapter (part ) Finding Torque A orce F o magnitude F making an ange with the x axis is appied to a partice ocated aong axis o rotation A, at Cartesian coordinates (,) in the igure.
More informationMeanfield Dynamics of LoadBalancing Networks with General Service Distributions
Meanfied Dynamics of LoadBaancing Networks with Genera Service Distributions Reza Aghajani 1, Xingjie Li 2, and Kavita Ramanan 1 1 Division of Appied Mathematics, Brown University, Providence, RI, USA.
More informationInductance. Bởi: OpenStaxCollege
Inductance Bởi: OpenStaxCoege Inductors Induction is the process in which an emf is induced by changing magnetic fux. Many exampes have been discussed so far, some more effective than others. Transformers,
More informationFast Robust Hashing. ) [7] will be remapped (and therefore discarded), due to the loadbalancing property of hashing.
Fast Robust Hashing Manue Urueña, David Larrabeiti and Pabo Serrano Universidad Caros III de Madrid E89 Leganés (Madrid), Spain Emai: {muruenya,darra,pabo}@it.uc3m.es Abstract As statefu fowaware services
More informationSimultaneous Routing and Power Allocation in CDMA Wireless Data Networks
Simutaneous Routing and Power Aocation in CDMA Wireess Data Networks Mikae Johansson *,LinXiao and Stephen Boyd * Department of Signas, Sensors and Systems Roya Institute of Technoogy, SE 00 Stockhom,
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We show that if H is an effectivey competey decomposabe computabe torsionfree abeian group, then
More informationThe Simple Pendulum. by Dr. James E. Parks
by Dr. James E. Parks Department of Physics and Astronomy 401 Niesen Physics Buidin The University of Tennessee Knoxvie, Tennessee 37996100 Copyriht June, 000 by James Edar Parks* *A rihts are reserved.
More informationPricing Internet Services With Multiple Providers
Pricing Internet Services With Mutipe Providers Linhai He and Jean Warand Dept. of Eectrica Engineering and Computer Science University of Caifornia at Berkeey Berkeey, CA 94709 inhai, wr@eecs.berkeey.edu
More informationLESSON LEVERAGE ANALYSIS 21.0 AIMS AND OBJECTIVES 21.1 INTRODUCTION 21.2 OPERATING LEVERAGE CONTENTS
LESSON 21 LEVERAGE ANALYSIS CONTENTS 21.0 Aims and Objectives 21.1 Introduction 21.2 Operating Leverage 21.3 Financia Leverage 21.4 EBITEPS Anaysis 21.5 Combined Leverage 21.6 Let us Sum up 21.7 Lessonend
More informationElasticities. The range of values of price elasticity of demand. Price elasticity of demand [PED] Formula and definition
46 4 Easticities By the end of this chapter, you shoud be abe to: expain the concept of easticity define easticity of demand define and cacuate price easticity of demand iustrate different vaues of price
More informationBetting on the Real Line
Betting on the Rea Line Xi Gao 1, Yiing Chen 1,, and David M. Pennock 2 1 Harvard University, {xagao,yiing}@eecs.harvard.edu 2 Yahoo! Research, pennockd@yahooinc.com Abstract. We study the probem of designing
More informationOligopoly in Insurance Markets
Oigopoy in Insurance Markets June 3, 2008 Abstract We consider an oigopoistic insurance market with individuas who differ in their degrees of accident probabiities. Insurers compete in coverage and premium.
More informationAngles formed by 2 Lines being cut by a Transversal
Chapter 4 Anges fored by 2 Lines being cut by a Transversa Now we are going to nae anges that are fored by two ines being intersected by another ine caed a transversa. 1 2 3 4 t 5 6 7 8 If I asked you
More informationELEVATING YOUR GAME FROM TRADE SPEND TO TRADE INVESTMENT
Initiatives Strategic Mapping Success in The Food System: Discover. Anayze. Strategize. Impement. Measure. ELEVATING YOUR GAME FROM TRADE SPEND TO TRADE INVESTMENT Foodservice manufacturers aocate, in
More informationFRAME BASED TEXTURE CLASSIFICATION BY CONSIDERING VARIOUS SPATIAL NEIGHBORHOODS. Karl Skretting and John Håkon Husøy
FRAME BASED TEXTURE CLASSIFICATION BY CONSIDERING VARIOUS SPATIAL NEIGHBORHOODS Kar Skretting and John Håkon Husøy University of Stavanger, Department of Eectrica and Computer Engineering N4036 Stavanger,
More informationMagnetic circuits. Chapter 7. 7.1 Introduction to magnetism and magnetic circuits. At the end of this chapter you should be able to:
Chapter 7 Magnetic circuits At the end of this chapter you shoud be abe to: appreciate some appications of magnets describe the magnetic fied around a permanent magnet state the aws of magnetic attraction
More informationFigure 1. A Simple Centrifugal Speed Governor.
ENGINE SPEED CONTROL Peter Westead and Mark Readman, contro systems principes.co.uk ABSTRACT: This is one of a series of white papers on systems modeing, anaysis and contro, prepared by Contro Systems
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
1 DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS 2 ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We show that if H is an effectivey competey decomposabe computabe torsionfree abeian group,
More informationASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007
ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007 FRANÇOIS SIMENHAUS Université Paris 7, Mathématiques, case 7012, 2, pace Jussieu, 75251 Paris, France
More informationThe Use of CoolingFactor Curves for Coordinating Fuses and Reclosers
he Use of ooingfactor urves for oordinating Fuses and Recosers arey J. ook Senior Member, IEEE S& Eectric ompany hicago, Iinois bstract his paper describes how to precisey coordinate distribution feeder
More informationpage 11, Problem 1.14(a), line 1: Ψ(x, t) 2 Ψ(x, t) 2 ; line 2: h 2 2mk B page 13, Problem 1.18(b), line 2, first inequality: 3mk B
Corrections to the Instructor s Soution Manua Introduction to Quantum Mechanics, nd ed by David Griffiths Cumuative errata for the print version corrected in the current eectronic version I especiay thank
More informationBetting Strategies, Market Selection, and the Wisdom of Crowds
Betting Strategies, Market Seection, and the Wisdom of Crowds Wiemien Kets Northwestern University wkets@keogg.northwestern.edu David M. Pennock Microsoft Research New York City dpennock@microsoft.com
More informationAustralian Bureau of Statistics Management of Business Providers
Purpose Austraian Bureau of Statistics Management of Business Providers 1 The principa objective of the Austraian Bureau of Statistics (ABS) in respect of business providers is to impose the owest oad
More information1 Basic concepts in geometry
1 asic concepts in geometry 1.1 Introduction We start geometry with the simpest idea a point. It is shown using a dot, which is abeed with a capita etter. The exampe above is the point. straight ine is
More informationCapacity of Multiservice Cellular Networks with TransmissionRate Control: A Queueing Analysis
Capacity of Mutiservice Ceuar Networs with TransmissionRate Contro: A Queueing Anaysis Eitan Atman INRIA, BP93, 2004 Route des Lucioes, 06902 SophiaAntipois, France aso CESIMO, Facutad de Ingeniería,
More informationBreakeven analysis and shortterm decision making
Chapter 20 Breakeven anaysis and shortterm decision making REAL WORLD CASE This case study shows a typica situation in which management accounting can be hepfu. Read the case study now but ony attempt
More informationA quantum model for the stock market
A quantum mode for the stock market Authors: Chao Zhang a,, Lu Huang b Affiiations: a Schoo of Physics and Engineering, Sun Yatsen University, Guangzhou 5175, China b Schoo of Economics and Business Administration,
More informationChapter 1 Structural Mechanics
Chapter Structura echanics Introduction There are many different types of structures a around us. Each structure has a specific purpose or function. Some structures are simpe, whie others are compex; however
More informationMT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...
MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 20042012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................
More informationLife Contingencies Study Note for CAS Exam S. Tom Struppeck
Life Contingencies Study Note for CAS Eam S Tom Struppeck (Revised 9/19/2015) Introduction Life contingencies is a term used to describe surviva modes for human ives and resuting cash fows that start or
More informationA Description of the California Partnership for LongTerm Care Prepared by the California Department of Health Care Services
2012 Before You Buy A Description of the Caifornia Partnership for LongTerm Care Prepared by the Caifornia Department of Heath Care Services Page 1 of 13 Ony ongterm care insurance poicies bearing any
More informationLecture 7: Continuous Random Variables
Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider
More informationMAGNETS Opposites attract
MAGNETS Opposites attract This guide offers exercises and experiments at a variety of eves on magnetism. It s an opportunity to revisit the subject matter covered in the Magnetism chapter of the Expor
More informationCOMPARISON OF DIFFUSION MODELS IN ASTRONOMICAL OBJECT LOCALIZATION
COMPARISON OF DIFFUSION MODELS IN ASTRONOMICAL OBJECT LOCALIZATION Františe Mojžíš Department of Computing and Contro Engineering, ICT Prague, Technicá, 8 Prague frantise.mojzis@vscht.cz Abstract This
More informationBusiness schools are the academic setting where. The current crisis has highlighted the need to redefine the role of senior managers in organizations.
c r o s os r oi a d s REDISCOVERING THE ROLE OF BUSINESS SCHOOLS The current crisis has highighted the need to redefine the roe of senior managers in organizations. JORDI CANALS Professor and Dean, IESE
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationArtificial neural networks and deep learning
February 20, 2015 1 Introduction Artificia Neura Networks (ANNs) are a set of statistica modeing toos originay inspired by studies of bioogica neura networks in animas, for exampe the brain and the centra
More informationFace Hallucination and Recognition
Face Haucination and Recognition Xiaogang Wang and Xiaoou Tang Department of Information Engineering, The Chinese University of Hong Kong {xgwang1, xtang}@ie.cuhk.edu.hk http://mmab.ie.cuhk.edu.hk Abstract.
More informationAA Fixed Rate ISA Savings
AA Fixed Rate ISA Savings For the road ahead The Financia Services Authority is the independent financia services reguator. It requires us to give you this important information to hep you to decide whether
More informationOn Capacity Scaling in Arbitrary Wireless Networks
On Capacity Scaing in Arbitrary Wireess Networks Urs Niesen, Piyush Gupta, and Devavrat Shah 1 Abstract arxiv:07112745v3 [csit] 3 Aug 2009 In recent work, Özgür, Lévêque, and Tse 2007) obtained a compete
More informationCONDENSATION. Prabal Talukdar. Associate Professor Department of Mechanical Engineering IIT Delhi Email: prabal@mech.iitd.ac.in
CONDENSATION Praba Taukdar Associate Professor Department of Mechanica Engineering IIT Dehi Emai: praba@mech.iitd.ac.in Condensation When a vapor is exposed to a surface at a temperature beow T sat, condensation
More informationEconomic Shocks and Internal Migration
DISCUSSION PAPER SERIES IZA DP No. 8840 Economic Shocks and Interna Migration Joan Monras February 2015 Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor Economic Shocks and Interna
More informationPayondelivery investing
Payondeivery investing EVOLVE INVESTment range 1 EVOLVE INVESTMENT RANGE EVOLVE INVESTMENT RANGE 2 Picture a word where you ony pay a company once they have deivered Imagine striking oi first, before
More informationLet s get usable! Usability studies for indexes. Susan C. Olason. Study plan
Let s get usabe! Usabiity studies for indexes Susan C. Oason The artice discusses a series of usabiity studies on indexes from a systems engineering and human factors perspective. The purpose of these
More informationEarly access to FAS payments for members in poor health
Financia Assistance Scheme Eary access to FAS payments for members in poor heath Pension Protection Fund Protecting Peope s Futures The Financia Assistance Scheme is administered by the Pension Protection
More informationFX Hedging:10 Common Pitfalls. A Structured Approach to Financial Risk Management
FX Hedging:10 Common Pitfas Executive Summary The design and impementation of an effective FX risk management strategy can be a chaenge for many businesses. The extreme eve of voatiity experienced in the
More informationPricing and Revenue Sharing Strategies for Internet Service Providers
Pricing and Revenue Sharing Strategies for Internet Service Providers Linhai He and Jean Warand Department of Eectrica Engineering and Computer Sciences University of Caifornia at Berkeey {inhai,wr}@eecs.berkeey.edu
More informationWHITE PAPER UndERsTAndIng THE VAlUE of VIsUAl data discovery A guide To VIsUAlIzATIons
Understanding the Vaue of Visua Data Discovery A Guide to Visuaizations WHITE Tabe of Contents Executive Summary... 3 Chapter 1  Datawatch Visuaizations... 4 Chapter 2  Snapshot Visuaizations... 5 Bar
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special DistributionsVI Today, I am going to introduce
More informationOn the change in surpluses equivalence: measuring benefits from transport infrastructure investments
On the change in surpuses equivaence: measuring benefits from transport infrastructure investments José MeéndezHidago 1, Piet Rietved 1, Erik Verhoef 1,2 1 Department of Spatia Economics, Free University
More informationThe Radix4 and the Class of Radix2 s FFTs
Chapter 11 The Radix and the Cass of Radix s FFTs The divideandconuer paradigm introduced in Chapter 3 is not restricted to dividing a probem into two subprobems. In fact, as expained in Section. and
More informationMATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 BASIC INTEGRATION
MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 ASIC INTEGRATION This tutorial is essential prerequisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning
More informationComparison of Traditional and OpenAccess Appointment Scheduling for Exponentially Distributed Service Time
Journa of Heathcare Engineering Vo. 6 No. 3 Page 34 376 34 Comparison of Traditiona and OpenAccess Appointment Scheduing for Exponentiay Distributed Service Chongjun Yan, PhD; Jiafu Tang *, PhD; Bowen
More informationMarket Design & Analysis for a P2P Backup System
Market Design & Anaysis for a P2P Backup System Sven Seuken Schoo of Engineering & Appied Sciences Harvard University, Cambridge, MA seuken@eecs.harvard.edu Denis Chares, Max Chickering, Sidd Puri Microsoft
More informationJoint distributions Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014
Joint distributions Math 17 Probability and Statistics Prof. D. Joyce, Fall 14 Today we ll look at joint random variables and joint distributions in detail. Product distributions. If Ω 1 and Ω are sample
More informationChange of Continuous Random Variable
Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make
More informationarxiv:0712.3028v2 [astroph] 7 Jan 2008
A practica guide to Basic Statistica Techniques for Data Anaysis in Cosmoogy Licia Verde arxiv:0712.3028v2 [astroph] 7 Jan 2008 ICREA & Institute of Space Sciences (IEECCSIC), Fac. Ciencies, Campus UAB
More informationHybrid Process Algebra
Hybrid Process Agebra P.J.L. Cuijpers M.A. Reniers Eindhoven University of Technoogy (TU/e) Den Doech 2 5600 MB Eindhoven, The Netherands Abstract We deveop an agebraic theory, caed hybrid process agebra
More informationArt of Java Web Development By Neal Ford 624 pages US$44.95 Manning Publications, 2004 ISBN: 1932394060
IEEE DISTRIBUTED SYSTEMS ONLINE 15414922 2005 Pubished by the IEEE Computer Society Vo. 6, No. 5; May 2005 Editor: Marcin Paprzycki, http://www.cs.okstate.edu/%7emarcin/ Book Reviews: Java Toos and Frameworks
More informationAvailable online Journal of Scientific and Engineering Research, 2016, 3(2): Research Article
Avaiabe onine www.jsaer.com Journa of Scientific and Engineering Research, 016, ():1 Research Artice ISSN: 960 CODEN(USA): JSERBR A simpe design of a aboratory testing rig for the experimenta demonstration
More informationMath 447/547 Partial Differential Equations Prof. Carlson Homework 7 Text section 4.2 1. Solve the diffusion problem. u(t,0) = 0 = u x (t,l).
Math 447/547 Partia Differentia Equations Prof. Carson Homework 7 Text section 4.2 1. Sove the diffusion probem with the mixed boundary conditions u t = ku xx, < x
More informationChapter 4  Lecture 1 Probability Density Functions and Cumul. Distribution Functions
Chapter 4  Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the
More information11 TRANSFORMING DENSITY FUNCTIONS
11 TRANSFORMING DENSITY FUNCTIONS It can be expedient to use a transformation function to transform one probability density function into another. As an introduction to this topic, it is helpful to recapitulate
More informationCSCI 8260 Spring 2016. Computer Network Attacks and Defenses. Syllabus. Prof. Roberto Perdisci perdisci@cs.uga.edu
CSCI 8260 Spring 2016 Computer Network Attacks and Defenses Syabus Prof. Roberto Perdisci perdisci@cs.uga.edu Who is this course for? Open to graduate students ony Students who compete this course successfuy
More information7. Dry Lab III: Molecular Symmetry
0 7. Dry Lab III: Moecuar Symmetry Topics: 1. Motivation. Symmetry Eements and Operations. Symmetry Groups 4. Physica Impications of Symmetry 1. Motivation Finite symmetries are usefu in the study of moecues.
More informationDistribution of Income Sources of Recent Retirees: Findings From the New Beneficiary Survey
Distribution of Income Sources of Recent Retirees: Findings From the New Beneficiary Survey by Linda Drazga Maxfied and Virginia P. Rena* Using data from the New Beneficiary Survey, this artice examines
More informationMATH 2300 review problems for Exam 3 ANSWERS
MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test
More informationBooks. Planning. Learning. through. by Rachel Sparks Linfield. Illustrated by Cathy Hughes. Contents. Making plans 23
Match your theme to the 2012 EYFS Books Panning by Rache Sparks Linfied. Iustrated by Cathy Hughes Contents Making pans 23 Using the Eary Goas 46 EYFS Panning Chart 7 Theme 1: Favourite games 89 Theme
More informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 101634501 Probability and Statistics for Engineers Winter 20102011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete
More informationBudgeting Loans from the Social Fund
Budgeting Loans from the Socia Fund tes sheet Pease read these notes carefuy. They expain the circumstances when a budgeting oan can be paid. Budgeting Loans You may be abe to get a Budgeting Loan if:
More informationLearning framework for NNs. Introduction to Neural Networks. Learning goal: Inputs/outputs. x 1 x 2. y 1 y 2
Introduction to Neura Networks Learning framework for NNs What are neura networks? Noninear function approimators How do they reate to pattern recognition/cassification? Noninear discriminant functions
More informationInsertion and deletion correcting DNA barcodes based on watermarks
Kracht and Schober BMC Bioinformatics (2015) 16:50 DOI 10.1186/s1285901504827 METHODOLOGY ARTICLE Open Access Insertion and deetion correcting DNA barcodes based on watermarks David Kracht * and Steffen
More informationSimulationBased Booking Limits for Airline Revenue Management
OPERATIONS RESEARCH Vo. 53, No. 1, January February 2005, pp. 90 106 issn 0030364X eissn 15265463 05 5301 0090 informs doi 10.1287/opre.1040.0164 2005 INFORMS SimuationBased Booking Limits for Airine
More informationUnderstanding. nystagmus. RCOphth
Understanding nystagmus RCOphth RNIB s understanding series The understanding series is designed to hep you, your friends and famiy understand a itte bit more about your eye condition. Other tites in the
More informationWhat makes a good Chair? A good chair will also: l always aim to draw a balance between hearing everyone s views and getting through the business.
Chairing a meeting An important job of the Chairperson is chairing meetings. Prior House 6 Tibury Pace Brighton BN2 0GY Te. 01273 606160 Fax. 01273 673663 info@resourcecentre.org.uk www.resourcecentre.org.uk
More informationSales and Use Tax Implications of Loyalty Programs
EDITED BY WALTER HELLERSTEIN, J.D. STATE & LOCAL Saes and Use Tax Impications of Loyaty Programs MARY T. BENTON, MATTHEW P. HEDSTROM, AND GREGG D. BARTON Examining a oyaty program reward s form, structure,
More informationGWPD 4 Measuring water levels by use of an electric tape
GWPD 4 Measuring water eves by use of an eectric tape VERSION: 2010.1 PURPOSE: To measure the depth to the water surface beow andsurface datum using the eectric tape method. Materias and Instruments 1.
More information25 Indirect taxes, subsidies, and price controls
5 Indirect taxes, subsidies, and price contros By the end of this chapter, you shoud be abe to: define and give exampes of an indirect tax expain the difference between a specific tax and a percentage
More informationEstimation of Tail Development Factors in the PaidIncurred Chain Reserving Method
Estimation of Tai Deveopment Factors in the aidincurred Chain Reserving Method by Michae Merz and Mario V Wüthrich AbSTRACT In many appied caims reserving probems in &C insurance the caims settement process
More informationAPPENDIX 10.1: SUBSTANTIVE AUDIT PROGRAMME FOR PRODUCTION WAGES: TROSTON PLC
Appendix 10.1: substantive audit programme for production wages: Troston pc 389 APPENDIX 10.1: SUBSTANTIVE AUDIT PROGRAMME FOR PRODUCTION WAGES: TROSTON PLC The detaied audit programme production wages
More informationJournal of Economic Behavior & Organization
Journa of Economic Behavior & Organization 85 (23 79 96 Contents ists avaiabe at SciVerse ScienceDirect Journa of Economic Behavior & Organization j ourna ho me pag e: www.esevier.com/ocate/j ebo Heath
More informationTeamwork. Abstract. 2.1 Overview
2 Teamwork Abstract This chapter presents one of the basic eements of software projects teamwork. It addresses how to buid teams in a way that promotes team members accountabiity and responsibiity, and
More information