APPLICATION TO THE BINOMIAL SUMMATION OF A LAPLACIAN METHOD FOR THE EVALUATION OF DEFINITE INTEGRALS. Introduction.
|
|
- Theodore Boone
- 6 years ago
- Views:
Transcription
1 E. C. MOLINA (New York - U. S. A) APPLICATION TO THE BINOMIAL SUMMATION OF A LAPLACIAN METHOD FOR THE EVALUATION OF DEFINITE INTEGRALS Introduction. The numerical evaluation of the incomplete Binomial Summation, a problem of major importance for many statistical and engineering applications of the Theory of ProbabiKty, is a question for which a satisfactory solution has not as yet been obtained. Several approximation formulas have been presented (*), each of which gives good results for some Kmited range of values of the variables involved; but a formula of wide applicabikty is still a desideratum. The purpose of this paper is to submit for consideration an approximation formula which seems to meet the situation to a measurable extent. The writer derived it by applying to the equation p _ / a;, '~ (l x)' l ~''dx (i) s t) px{ - p)n ~~ x=? -i ' X==G /V - (l x)"- ( 'dx d a method which is pecukarly efficacious for approximately evaluating definite integrals when the integrands contain factors raised to high powers. The method used constitutes the subject matter of Chapter I, Part II, Book I of Laplace's, Théorie Analytique des Probabilités. POISSON applied the method to the integrals in the equation fx'^'/il + xy'+^dx () s ÇW -^œ= ^ x ~ c fx n^/(l + x) n + i dx ó and published a first approximation, together with its derivation, in his Recherches sur la Probabilité des Jugements. Poisson's approximation seems (*) For an excellent resumé of some well-known formulas, together with a discussion of their limitations, reference may be had to C. JORDAN : Statistique Mathématique, articles 7 and.
2 COMUNICAZIONI never to have been used and was less fortunate than his famous limit to the binomial expansion which also was lost sight of until it reappeared under the caption «law of smak numbers v. While the integrals in equations () and () are wek known equivalent forms for the complete and incomplete Beta functions, the equations themselves are not so famikar although one or the other wik be found in LAPLACE, POISSON, BOOLE (Differential Equations) and at least two other places. Approximate Formula. The approximate formula derived from equation () and submitted herewith for consideration la T () Sß^-^-sKlK' where Si is the i th approximation to the infinite series B,T*-*[ + (s- DV-* + (a-)(»- )7\-...] () i=i, + S [l 5... (s l)]- s s l T, = Ti, (5) r» = ( B _i)iog_^_ +(c_i)iog c J + ( w _ e )iog?-j», and a=np\ T to be taken negative when a<(c l)n/(n l). The first, second and third approximations to the infinite series S are where Q T? Q i + B T Q B L -\-B T-\-B {l-j- T ) i =[(w-c)-(c-l)]/f / (w-l)(to-c)(c-l), R = (l/)[l/(n-e) + l/(e-l)-/(n-l)], R =- (/5) J R [ J R + /(n -)]. It will be noted that R», \R ± \ and i are symmetric functions of (n o) and (c ). For the limiting case (Poisson's Exponential Binomial Limit) where ra=oo, jp= but np = a, we have T* = l + (c-l)log(c-l)/a+(a-c). Ä t =/^=), JB,=/(C-), ^=-(/5)^^.
3 E. C. MOLINA : Evaluation of definite integrals 7 Numerical Results. Since it is easy to compute the binomial summation directly when either c or n c is small, the practical value of an approximate formula depends on its efficiency for large values of these quantities. The analysis given below under the beading «Derivation of the Approximate Formula» indicates that the successive R s 's in the series for S decrease when ic and in c increase. Therefore, when these two quantities are large, a few terms of the approximate formula () may be expected to give satisfactory results. As a matter of fact, the formula gives good results when ^c and \n c are not large. To confirm this statement the Tables ( L ) given at the end of this paper are submitted. In the th column of each table are given times the true values of x=n In P^QPHI-PY -*- In the columns headed A i9 A and A are given times the differences between the true values and those obtained by applying formula () with the first, second and third approximations to S respectively. Table I in Czuber's, Wahrscheinlichkeitsrechnung, was used for evaluating the probabkity integral in equation (). The range of values of P covered by the tables is such that at the lower end of each section P>.5 while at the upper end P<.9995, except where this latter condition would call for a value of c<. Of course, a larger or smaker range might have been given. The decision as to this question was based on the fact that several writers on the theory of statistics, when dealing with the normal law of errors, speak of an error exceeding or times the standard deviation as being a very improbable event. In order to keep the number of pages required for the tables within reasonable bounds computations were made only for even values of c. The values of a=np used are such that each of the values p = l/, p = l/ and p = l/ occurs twice; likewise each of the values n=, n=5 and ^= occurs twice. A greater degree of accuracy than that indicated by the tables can, of course, be obtained by working out and using i, JR 5,...; for this purpose, recourse should be had to equation () below and the details immediately fokowing it. The only practical limitation to the use of formula () would appear to be the number of places given by the existing tables for the probability integral. (*) I am greatly indebted to Miss NELLIEMAE Z. PEARSON of the Department of Development and Research both for supervising the work of my computers and contributing personally several sections of the tables.
4 COMUNICAZIONI However, this difficulty is encountered only when P, or ( P), is smak, in which case T is large and the integral T [er t: dt may be readily evaluated by computing the first few terms of the series [e- T '/T^][l-Tr + (l )Tr -(l. ö)^-*...], where, as above, Ti = Ttf. When Pis very small, the difference c a=c np is relatively large compared to a, and for this latter case recourse may be had to the approximate formula published by the writer in the American Mathematical Monthly for June, 9. Derivation of the Approximate Formula. FoKowing LAPLACE closely, let us set () y(x) = Ye-t% where Y=y(x Q ) is the maximum value of y(x). Then (7) fydx=y)e-^)dt, oo the upper limit T being given by the equation () y&)=y(z*)e- T \ Assuming dx/dt expanded in powers of t so that (9) dx/dt==^d, q+i t s and setting R s =D s+i /D i, equation (7) reduces to p [ydx= YD, R s fpér*'dt. Ó s== Our fundamental equation () may now be written T x=~n ^B s j t s e~ r 'dt < > ïtw-^- -^ - s= oo
5 E. C. MOLINA: Evaluation of definite integrals 9 Integrating by parts and separating the terms involving (e-t'dt from the terms containing e~ v, we obtain equations () and (). To determine R s =D s+i /D l, note that equation () gives t=(log Y logy) i and set v(x) = (x x )/(log Y logy)! so that x may be written in the form x=x + v(x)t. This form for x gives the expansion (Lagrange's Theorem for the simple case where f(x)=x; see Modern Analysis by WHITTAKER and WATSON) _ V ** ( d *~ iv '\ x ~^osi\^=ï) x^q- Comparing this expansion for x with the previous expansion (9) for dx/dt, we obtain D i =v(x ) aild A + / l d^h\ s \\v(x) daf ) x = Xo ' D i Up to this point no particular form has been attributed to the function y(x). From now on we deal with the function which constitutes the integrand of the integrals in equation (). The function.. t/a y(x)=x c - i (l-x) n - c gives the expansion (log Y logy) = (x Xo) [Ao + A l (x x ) + A (x x Q )...], where. x = (c-l)/(n-l) is the value of x for which y(x) is a maximum and. l \d s + {\o g Y-logij) 9 (*+)! dx'+ Jx=x Q We are now prepared to evaluate R s. g=a + Ai(x x ) + A (x x Q )... and 7o, *» g s =d*g/dx s. Then Set g=(/)^/ [(5/)^-^],
6 COMUNICAZIONI Therefore, since g s =sla s when x=x, P = (/)^-[(5/)^f-^^], R = -A^I [A A -A A i A + Afl Substituting for A, Ai, A and A the expressions derived by giving s the values,, and respectively in equation (), we obtain for R i9 R and R the functions of n and c given on page. For values of s greater than the direct evaluation of d v s+i ldx? by successive differentiation becomes very tedious. It will be found much more practical to use the fokowing procedure (*), where D is a symbol of operation, A=A, and b=a ± /d*a~( s+i) ' \ da-is+i)li UdA D s ~ b + or () + \da* D* ô ds-, A -(S+i)ß (s l)\da'-\ d>n A -(s+d/ (Ds-mfyn). m! da» w=l Db '^ rfm-(s+d/ slda* The fokowing equations give the details requisite for the formation of R s to R inclusive; A s can be computed from equation (). Db=A, D b=a, D b=a, D*b=A 5, D 5 b=a, D«b=A, D b=a s, Db =A i A, D b =A i A + A\, D*b =A i A i + A A, D i b =A i A, + A A é + A, I) 5 b =A i AB + A A 5 + A A D*b =A i Ai + A A + A As + A, Db* = A A, D b = SA A + SA L A, D b =SA A i + A i A A + AI, D^ = SA A 5 + A i A A + SA i Al + SA A, D 5 b = SA A + A L A A b + A i AA i + SA A i + SA A, Db* = ±A\A, ( l ) See De Morgan's: Differential and Integral Calculus, () page, art..
7 E. C. MOLINA: Evaluation of definite integrals ZW=AlA + AtAî, D i b i =AAfA 5 + AlA A i + QAlAl+A l AtA,. + At, Db*=òAtA, D*b*=5A\A + \.A\A\, DW=5AiA i + AlA î A + AlAl, Db s =A{A, D b 5 =AlA s +5 A\A\, DW=A\A,_. or To illustrate the use of the procedure given above, let us evaluate R. We have *-*- m) *>+$ )»»+{% ) +^> = - (5/) Afl*(A t ) + (I)(5/)(J/)A^I*(A A, + AÏ) -(l/)(5/)(7/)(9/)^-"/wm t ) + (l/)(5/)(7/)(9/)(ll/)il -»/»^î R i -*(5/)A;*[-AlA t + (7/)Al(A i A, + Al/) - (l/)(7/)(9/) A AtA + (/) (7/) (9/) (/) At]. Tables Indicating Degree of Accuracy Obtainable by Use of Formula () for Evaluating. X=C Pi = lst approximation, A l = (P P l ) 5, P =d approximation, A = (P P ), P =d approximation, A = (P P ) G, a=np, Z"=(n-) log^ +(d-l) log ^ +(n-c) log ^, 7=-^ rut! r ferodi.
8 COMUNICAZIONI TABLE I. /(IO ) P( c ) ^ a =.5; n = oc; p-= A a =.5; ^=; # = TABLE II. 7( tì ) P( ) A a = = 5 ; n = c _ 7 5 a 5; n=; p=m ; 9 i 99, * a = = 5; w = 5; p =.l \p-p \>\p-p,\
9 E. C. MOLINA: Evaluation of definite integrals TABLE III. T ( G ) P( ) A*, «= ; n = DO; p = a- = ; w = ; p = A * 9* \P-P \ > \P-P i \ TABLE IV e T J( ) P( ) ^ ^ a = 5; w = oo; jt? = is g 5
10 COMUNICAZIONI TABLE IV. (Continued) a = 5; w=; p= _ TABLE V. ( ) P( ) = 5; = oo: p = : a = 5; w==5; p =.h
Math RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5
Math 201-203-RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 What is the Antiderivative? In a derivative problem, a function f(x) is given and you find the derivative f (x) using
More information16.4. Power Series. Introduction. Prerequisites. Learning Outcomes
Power Series 6.4 Introduction In this Section we consider power series. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. We use the ratio
More informationMultiple Random Variables
Multiple Random Variables Joint Probability Density Let X and Y be two random variables. Their joint distribution function is F ( XY x, y) P X x Y y. F XY ( ) 1, < x
More information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More informationReview of Vector Analysis in Cartesian Coordinates
Review of Vector Analysis in Cartesian Coordinates 1 Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers. Scalars are usually
More informationLipschitz Robustness of Finite-state Transducers
Lipschitz Robustness of Finite-state Transducers Roopsha Samanta IST Austria Joint work with Tom Henzinger and Jan Otop December 16, 2014 Roopsha Samanta Lipschitz Robustness of Finite-state Transducers
More information1 Review of di erential calculus
Review of di erential calculus This chapter presents the main elements of di erential calculus needed in probability theory. Often, students taking a course on probability theory have problems with concepts
More informationChapter 6 - Random Processes
EE385 Class Notes //04 John Stensby Chapter 6 - Random Processes Recall that a random variable X is a mapping between the sample space S and the extended real line R +. That is, X : S R +. A random process
More informationSTAT 430/510 Probability
STAT 430/510 Probability Hui Nie Lecture 16 June 24th, 2009 Review Sum of Independent Normal Random Variables Sum of Independent Poisson Random Variables Sum of Independent Binomial Random Variables Conditional
More informationProblem 1 HW3. Question 2. i) We first show that for a random variable X bounded in [0, 1], since x 2 apple x, wehave
Next, we show that, for any x Ø, (x) Æ 3 x x HW3 Problem i) We first show that for a random variable X bounded in [, ], since x apple x, wehave Var[X] EX [EX] apple EX [EX] EX( EX) Since EX [, ], therefore,
More informationVector Calculus. A primer
Vector Calculus A primer Functions of Several Variables A single function of several variables: f: R $ R, f x (, x ),, x $ = y. Partial derivative vector, or gradient, is a vector: f = y,, y x ( x $ Multi-Valued
More informationSection Taylor and Maclaurin Series
Section.0 Taylor and Maclaurin Series Ruipeng Shen Feb 5 Taylor and Maclaurin Series Main Goal: How to find a power series representation for a smooth function us assume that a smooth function has a power
More information16.4. Power Series. Introduction. Prerequisites. Learning Outcomes
Power Series 6.4 Introduction In this section we consider power series. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. We use the ratio
More informationy mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent
Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()
More informationSTAT 430/510: Lecture 15
STAT 430/510: Lecture 15 James Piette June 23, 2010 Updates HW4 is up on my website. It is due next Mon. (June 28th). Starting today back at section 6.4... Conditional Distribution: Discrete Def: The conditional
More informationChapter 4: Partial differentiation
Chapter 4: Partial differentiation It is generally the case that derivatives are introduced in terms of functions of a single variable. For example, y = f (x), then dy dx = df dx = f. However, most of
More informationStatistics, Data Analysis, and Simulation SS 2013
Statistics, Data Analysis, and Simulation SS 213 8.128.73 Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, 23. April 213 What we ve learned so far Fundamental
More informationBefore you begin read these instructions carefully:
NATURAL SCIENCES TRIPOS Part IB & II (General) Tuesday, 30 May, 2017 9:00 am to 12:00 pm MATHEMATICS (1) Before you begin read these instructions carefully: You may submit answers to no more than six questions.
More informationDEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 208 Version A refers to the regular exam and Version B to the make-up. Version A. A particle
More informationprobability of k samples out of J fall in R.
Nonparametric Techniques for Density Estimation (DHS Ch. 4) n Introduction n Estimation Procedure n Parzen Window Estimation n Parzen Window Example n K n -Nearest Neighbor Estimation Introduction Suppose
More informationCoordinate systems and vectors in three spatial dimensions
PHYS2796 Introduction to Modern Physics (Spring 2015) Notes on Mathematics Prerequisites Jim Napolitano, Department of Physics, Temple University January 7, 2015 This is a brief summary of material on
More informationON THE AVERAGE NUMBER OF REAL ROOTS OF A RANDOM ALGEBRAIC EQUATION
ON THE AVERAGE NUMBER OF REAL ROOTS OF A RANDOM ALGEBRAIC EQUATION M. KAC 1. Introduction. Consider the algebraic equation (1) Xo + X x x + X 2 x 2 + + In-i^" 1 = 0, where the X's are independent random
More informationStatistics, Data Analysis, and Simulation SS 2017
Statistics, Data Analysis, and Simulation SS 2017 08.128.730 Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, 27. April 2017 Dr. Michael O. Distler
More informationAN ARCSINE LAW FOR MARKOV CHAINS
AN ARCSINE LAW FOR MARKOV CHAINS DAVID A. FREEDMAN1 1. Introduction. Suppose {xn} is a sequence of independent, identically distributed random variables with mean 0. Under certain mild assumptions [4],
More informationLecture 2. Distributions and Random Variables
Lecture 2. Distributions and Random Variables Igor Rychlik Chalmers Department of Mathematical Sciences Probability, Statistics and Risk, MVE300 Chalmers March 2013. Click on red text for extra material.
More informationContinuous Random Variables
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability. Often, there is interest in random variables
More informationDiscrete abstractions of hybrid systems for verification
Discrete abstractions of hybrid systems for verification George J. Pappas Departments of ESE and CIS University of Pennsylvania pappasg@ee.upenn.edu http://www.seas.upenn.edu/~pappasg DISC Summer School
More informationT i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a. A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r )
v e r. E N G O u t l i n e T i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r ) C o n t e n t s : T h i s w o
More informationlim Prob. < «_ - arc sin «1 / 2, 0 <_ «< 1.
ON THE NUMBER OF POSITIVE SUMS OF INDEPENDENT RANDOM VARIABLES P. ERDÖS AND M. KAC 1 1. Introduction. In a recent paper' the authors have introduced a method for proving certain limit theorems of the theory
More informationMathematics Self-Assessment for Physics Majors
Mathematics Self-Assessment for Physics Majors Steven Detweiler and Yoonseok Lee November 29, 202 The lack of mathematical sophistication is a leading cause of difficulty for students in Classical Mechanics
More informationStatistics, Data Analysis, and Simulation SS 2015
Statistics, Data Analysis, and Simulation SS 2015 08.128.730 Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, 27. April 2015 Dr. Michael O. Distler
More informationNumerical Quadrature over a Rectangular Domain in Two or More Dimensions
Numerical Quadrature over a Rectangular Domain in Two or More Dimensions Part 2. Quadrature in several dimensions, using special points By J. C. P. Miller 1. Introduction. In Part 1 [1], several approximate
More informationECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo Economics, HKU September 17, 2018 Luo, Y. (Economics, HKU) ME September 17, 2018 1 / 46 Static Optimization and Extreme Values In this topic, we will study goal
More informationReview of Integration Techniques
A P P E N D I X D Brief Review of Integration Techniques u-substitution The basic idea underlying u-substitution is to perform a simple substitution that converts the intergral into a recognizable form
More informationf(w) f(a) = 1 2πi w a Proof. There exists a number r such that the disc D(a,r) is contained in I(γ). For any ǫ < r, w a dw
Proof[section] 5. Cauchy integral formula Theorem 5.. Suppose f is holomorphic inside and on a positively oriented curve. Then if a is a point inside, f(a) = w a dw. Proof. There exists a number r such
More informationENGI Partial Differentiation Page y f x
ENGI 3424 4 Partial Differentiation Page 4-01 4. Partial Differentiation For functions of one variable, be found unambiguously by differentiation: y f x, the rate of change of the dependent variable can
More informationLecture 5: Moment generating functions
Lecture 5: Moment generating functions Definition 2.3.6. The moment generating function (mgf) of a random variable X is { x e tx f M X (t) = E(e tx X (x) if X has a pmf ) = etx f X (x)dx if X has a pdf
More informationCHAPTER 2. Techniques for Solving. Second Order Linear. Homogeneous ODE s
A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES A COLLECTION OF HANDOUTS ON SCALAR LINEAR ORDINARY DIFFERENTIAL
More informationBrief introduction to groups and group theory
Brief introduction to groups and group theory In physics, we often can learn a lot about a system based on its symmetries, even when we do not know how to make a quantitative calculation Relevant in particle
More informationNumerical Methods School of Mechanical Engineering Chung-Ang University
Part 5 Chapter 19 Numerical Differentiation Prof. Hae-Jin Choi hjchoi@cau.ac.kr 1 Chapter Objectives l Understanding the application of high-accuracy numerical differentiation formulas for equispaced data.
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 3. Calculaus in Deterministic and Stochastic Environments Steve Yang Stevens Institute of Technology 01/31/2012 Outline 1 Modeling Random Behavior
More informationThe Particle-Field Hamiltonian
The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and
More informationCHARACTERISTIC PARAMETER VALUES FOR AN INTEGRAL EQUATION*
1925.] INTEGRAL EQUATIONS 503 CHARACTERISTIC PARAMETER VALUES FOR AN INTEGRAL EQUATION* BY W. A. HURWITZ 1. Introduction ; Hermitian Kernel. It is well known, though seldom explicitly mentioned, that the
More informationChapter 2. Linear Algebra. rather simple and learning them will eventually allow us to explain the strange results of
Chapter 2 Linear Algebra In this chapter, we study the formal structure that provides the background for quantum mechanics. The basic ideas of the mathematical machinery, linear algebra, are rather simple
More informationA-LEVEL Mathematics. MPC4 Pure Core 4 Mark scheme June Version: 1.0 Final
A-LEVEL Mathematics MPC4 Pure Core 4 Mark scheme 660 June 06 Version:.0 Final Mark schemes are prepared by the Lead Assessment Writer and considered, together with the relevant questions, by a panel of
More informationSolving x 2 Dy 2 = N in integers, where D > 0 is not a perfect square. Keith Matthews
Solving x 2 Dy 2 = N in integers, where D > 0 is not a perfect square. Keith Matthews Abstract We describe a neglected algorithm, based on simple continued fractions, due to Lagrange, for deciding the
More informationParametric Curves. Calculus 2 Lia Vas
Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider
More informationCalculus of Variations Summer Term 2014
Calculus of Variations Summer Term 2014 Lecture 2 25. April 2014 c Daria Apushkinskaya 2014 () Calculus of variations lecture 2 25. April 2014 1 / 20 Purpose of Lesson Purpose of Lesson: To discuss the
More informationWe introduce methods that are useful in:
Instructor: Shengyu Zhang Content Derived Distributions Covariance and Correlation Conditional Expectation and Variance Revisited Transforms Sum of a Random Number of Independent Random Variables more
More informationMARK SCHEME for the November 2004 question paper 9709 MATHEMATICS 8719 HIGHER MATHEMATICS
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary and Advanced Level MARK SCHEME for the November 004 question paper 9709 MATHEMATICS 879 HIGHER MATHEMATICS 9709/03, 879/03 Paper
More informationONE - DIMENSIONAL INTEGRATION
Chapter 4 ONE - DIMENSIONA INTEGRATION 4. Introduction Since the finite element method is based on integral relations it is logical to expect that one should strive to carry out the integrations as efficiently
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES We have: Seen how to interpret derivatives as slopes and rates of change Seen how to estimate derivatives of functions given by tables of values Learned how
More informationI M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o
I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o u l d a l w a y s b e t a k e n, i n c l u d f o l
More informationA Note on the Differential Equations with Distributional Coefficients
MATEMATIKA, 24, Jilid 2, Bil. 2, hlm. 115 124 c Jabatan Matematik, UTM. A Note on the Differential Equations with Distributional Coefficients Adem Kilicman Department of Mathematics, Institute for Mathematical
More informationDetermination of f(00) from the asymptotic series for f(x) about x=0
Determination of f(00) from the asymptotic series for f(x) about x=0 Carl M. Bender and Stefan Boettcher Department of Physics, Washington University, St. Louis, Missouri 63130 (Received 20 January 1993;
More information5.9 Representations of Functions as a Power Series
5.9 Representations of Functions as a Power Series Example 5.58. The following geometric series x n + x + x 2 + x 3 + x 4 +... will converge when < x
More informationx 2 y = 1 2. Problem 2. Compute the Taylor series (at the base point 0) for the function 1 (1 x) 3.
MATH 8.0 - FINAL EXAM - SOME REVIEW PROBLEMS WITH SOLUTIONS 8.0 Calculus, Fall 207 Professor: Jared Speck Problem. Consider the following curve in the plane: x 2 y = 2. Let a be a number. The portion of
More informationChapter 3 Relational Model
Chapter 3 Relational Model Table of Contents 1. Structure of Relational Databases 2. Relational Algebra 3. Tuple Relational Calculus 4. Domain Relational Calculus Chapter 3-1 1 1. Structure of Relational
More informationChapter 8 Indeterminate Forms and Improper Integrals Math Class Notes
Chapter 8 Indeterminate Forms and Improper Integrals Math 1220-004 Class Notes Section 8.1: Indeterminate Forms of Type 0 0 Fact: The it of quotient is equal to the quotient of the its. (book page 68)
More informationt t t ér t rs r t ét q s
rés té t rs té s é té r t q r r ss r t t t ér t rs r t ét q s s t t t r2 sé t Pr ss r rs té P r s 2 t Pr ss r rs té r t r r ss s Pr ss r rs té P r q r Pr ss r t r t r r t r r Prés t r2 r t 2s Pr ss r rs
More informationAlgebra 2 Honors: Final Exam Review
Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt
More informationAn Introduction to Partial Differential Equations
An Introduction to Partial Differential Equations Ryan C. Trinity University Partial Differential Equations Lecture 1 Ordinary differential equations (ODEs) These are equations of the form where: F(x,y,y,y,y,...)
More information2. Write your full name and section on the space provided at the top of each odd numbered page.
I NAME: E - SECTION: Page 1 MATH 152 - COMMON FINAL Spring 2005 General Instructions: 1. The exam consists of 10 pages, including this cover; the test is printed on both sides of the page, and contains
More informationECE 302 Division 1 MWF 10:30-11:20 (Prof. Pollak) Final Exam Solutions, 5/3/2004. Please read the instructions carefully before proceeding.
NAME: ECE 302 Division MWF 0:30-:20 (Prof. Pollak) Final Exam Solutions, 5/3/2004. Please read the instructions carefully before proceeding. If you are not in Prof. Pollak s section, you may not take this
More informationMATH 317 Fall 2016 Assignment 5
MATH 37 Fall 26 Assignment 5 6.3, 6.4. ( 6.3) etermine whether F(x, y) e x sin y îı + e x cos y ĵj is a conservative vector field. If it is, find a function f such that F f. enote F (P, Q). We have Q x
More informationJoint Probability Distributions, Correlations
Joint Probability Distributions, Correlations What we learned so far Events: Working with events as sets: union, intersection, etc. Some events are simple: Head vs Tails, Cancer vs Healthy Some are more
More informationFLAP M6.1 Introducing differential equations COPYRIGHT 1998 THE OPEN UNIVERSITY S570 V1.1
F1 (a) This equation is of first order and of first degree. It is a linear equation, since the dependent variable y and its derivative only appear raised to the first power, and there are no products of
More informationSTAT/MA 416 Midterm Exam 2 Thursday, October 18, Circle the section you are enrolled in:
STAT/MA 46 Midterm Exam 2 Thursday, October 8, 27 Name Purdue student ID ( digits) Circle the section you are enrolled in: STAT/MA 46-- STAT/MA 46-2- 9: AM :5 AM 3: PM 4:5 PM REC 4 UNIV 23. The testing
More informationThe Random Variable for Probabilities Chris Piech CS109, Stanford University
The Random Variable for Probabilities Chris Piech CS109, Stanford University Assignment Grades 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 Frequency Frequency 10 20 30 40 50 60 70 80
More informationChapter 5 Joint Probability Distributions
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 5 Joint Probability Distributions 5 Joint Probability Distributions CHAPTER OUTLINE 5-1 Two
More informationSOLUTION FOR HOMEWORK 11, ACTS 4306
SOLUTION FOR HOMEWORK, ACTS 36 Welcome to your th homework. This is a collection of transformation, Central Limit Theorem (CLT), and other topics.. Solution: By definition of Z, Var(Z) = Var(3X Y.5). We
More informationI. Impulse Response and Convolution
I. Impulse Response and Convolution 1. Impulse response. Imagine a mass m at rest on a frictionless track, then given a sharp kick at time t =. We model the kick as a constant force F applied to the mass
More informationAsymptotic representation of Laplace transforms with an application to inverse factorial series
Asymptotic representation of Laplace transforms with an application to inverse factorial series By A. ERDBLYI. (Received 25th February, 1946. Read 1st March, 1946.) 1. In a recent paper 1 Professor A.
More informationStatistics and data analyses
Statistics and data analyses Designing experiments Measuring time Instrumental quality Precision Standard deviation depends on Number of measurements Detection quality Systematics and methology σ tot =
More informationSec. 1.1: Basics of Vectors
Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x
More informationNorthwestern University Department of Electrical Engineering and Computer Science
Northwestern University Department of Electrical Engineering and Computer Science EECS 454: Modeling and Analysis of Communication Networks Spring 2008 Probability Review As discussed in Lecture 1, probability
More informationCALCULUS I. Practice Problems Integrals. Paul Dawkins
CALCULUS I Practice Problems Integrals Paul Dawkins Table of Contents Preface... Integrals... Introduction... Indefinite Integrals... Comuting Indefinite Integrals... Substitution Rule for Indefinite Integrals...
More information2 tel
Us. Timeless, sophisticated wall decor that is classic yet modern. Our style has no limitations; from traditional to contemporar y, with global design inspiration. The attention to detail and hand- craf
More informationRESEARCH ARTICLE. Spectral Method for Solving The General Form Linear Fredholm Volterra Integro Differential Equations Based on Chebyshev Polynomials
Journal of Modern Methods in Numerical Mathematics ISSN: 29-477 online Vol, No, 2, c 2 Modern Science Publishers wwwm-sciencescom RESEARCH ARTICLE Spectral Method for Solving The General Form Linear Fredholm
More informationON THE NUMBER OF POSITIVE SUMS OF INDEPENDENT RANDOM VARIABLES
ON THE NUMBER OF POSITIVE SUMS OF INDEPENDENT RANDOM VARIABLES P. ERDÖS AND M. KAC 1 1. Introduction. In a recent paper 2 the authors have introduced a method for proving certain limit theorems of the
More information1 Boas, problem p.564,
Physics 6C Solutions to Homewor Set # Fall 0 Boas, problem p.564,.- Solve the following differential equations by series and by another elementary method and chec that the results agree: xy = xy +y ()
More information3. Numerical integration
3. Numerical integration... 3. One-dimensional quadratures... 3. Two- and three-dimensional quadratures... 3.3 Exact Integrals for Straight Sided Triangles... 5 3.4 Reduced and Selected Integration...
More informationConstructing Taylor Series
Constructing Taylor Series 8-8-200 The Taylor series for fx at x = c is fc + f cx c + f c 2! x c 2 + f c x c 3 + = 3! f n c x c n. By convention, f 0 = f. When c = 0, the series is called a Maclaurin series.
More informationUNIVERSITY OF CAMBRIDGE
UNIVERSITY OF CAMBRIDGE DOWNING COLLEGE MATHEMATICS FOR ECONOMISTS WORKBOOK This workbook is intended for students coming to Downing College Cambridge to study Economics 2018/ 19 1 Introduction Mathematics
More informationMATH Solutions to Probability Exercises
MATH 5 9 MATH 5 9 Problem. Suppose we flip a fair coin once and observe either T for tails or H for heads. Let X denote the random variable that equals when we observe tails and equals when we observe
More information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationO1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis. Monday 31st October 2016 (Week 4)
O1 History of Mathematics Lecture VIII Establishing rigorous thinking in analysis Monday 31st October 2016 (Week 4) Summary French institutions Fourier series Early-19th-century rigour Limits, continuity,
More information24. x 2 y xy y sec(ln x); 1 e x y 1 cos(ln x), y 2 sin(ln x) 25. y y tan x 26. y 4y sec 2x 28.
16 CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS 11. y 3y y 1 4. x yxyy sec(ln x); 1 e x y 1 cos(ln x), y sin(ln x) ex 1. y y y 1 x 13. y3yy sin e x 14. yyy e t arctan t 15. yyy e t ln t 16. y y y 41x
More informationThe Poisson Correlation Function
The Poisson Correlation Function By J. T. CAMPBELL, Edinburgh University. {Received 8th July, 9. Read Uh November, 9.). Introduction. In this study, the use of factorial moments and factorial moment generating
More informationAP Calculus BC Chapter 4 AP Exam Problems. Answers
AP Calculus BC Chapter 4 AP Exam Problems Answers. A 988 AB # 48%. D 998 AB #4 5%. E 998 BC # % 5. C 99 AB # % 6. B 998 AB #80 48% 7. C 99 AB #7 65% 8. C 998 AB # 69% 9. B 99 BC # 75% 0. C 998 BC # 80%.
More informationWeek 9 The Central Limit Theorem and Estimation Concepts
Week 9 and Estimation Concepts Week 9 and Estimation Concepts Week 9 Objectives 1 The Law of Large Numbers and the concept of consistency of averages are introduced. The condition of existence of the population
More information8/27/14. Kinematics and One-Dimensional Motion: Non-Constant Acceleration. Average Velocity. Announcements 8.01 W01D3. Δ r. v ave. = Δx Δt.
Kinematics and One-Dimensional Motion: Non-Constant Acceleration 8.01 W01D3 Announcements Familiarize Yourself with Website https://lms.mitx.mit.edu/courses/mitx/8.01/2014_fall/about Buy or Download Textbook
More informationMATH 151, FINAL EXAM Winter Quarter, 21 March, 2014
Time: 3 hours, 8:3-11:3 Instructions: MATH 151, FINAL EXAM Winter Quarter, 21 March, 214 (1) Write your name in blue-book provided and sign that you agree to abide by the honor code. (2) The exam consists
More informationBefore you begin read these instructions carefully.
MATHEMATICAL TRIPOS Part IA Wednesday, 4 June, 2014 9:00 am to 12:00 pm PAPER 2 Before you begin read these instructions carefully. The examination paper is divided into two sections. Each question in
More informationP1 Calculus II. Partial Differentiation & Multiple Integration. Prof David Murray. dwm/courses/1pd
P1 2017 1 / 39 P1 Calculus II Partial Differentiation & Multiple Integration Prof David Murray david.murray@eng.ox.ac.uk www.robots.ox.ac.uk/ dwm/courses/1pd 4 lectures, MT 2017 P1 2017 2 / 39 Motivation
More informationOn reaching head-to-tail ratios for balanced and unbalanced coins
Journal of Statistical Planning and Inference 0 (00) 0 0 www.elsevier.com/locate/jspi On reaching head-to-tail ratios for balanced and unbalanced coins Tamas Lengyel Department of Mathematics, Occidental
More information0.1 Rational Canonical Forms
We have already seen that it is useful and simpler to study linear systems using matrices. But matrices are themselves cumbersome, as they are stuffed with many entries, and it turns out that it s best
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More information) sm wl t. _.!!... e -pt sinh y t. Vo + mx" + cx' + kx = 0 (26) has a unique tions x(o) solution for t ;?; 0 satisfying given initial condi
1 48 Chapter 2 Linear Equations of Higher Order 28. (Overdamped) If Xo = 0, deduce from Problem 27 that x(t) Vo = e -pt sinh y t. Y 29. (Overdamped) Prove that in this case the mass can pass through its
More information