3.4 Numerical integration


 Lee Hunter
 9 months ago
 Views:
Transcription
1 3.4. Numericl integrtion Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [, b], i.e. I( f )= w(x) f (x) dx. The weight function my be the identity function w(x) =1, so tht the integrl represents the re under the function f in the intervl. In other pplictions the weight function could lso be the probbility density function of continuous rndom vrible x with support [, b], sothti( f ), represents the expected vlue of f ( x). In the following we discuss soclled numericl qudrture methods. In order to pproximte the bove integrl, qudrture methods choose discrete set of qudrture nodes x i nd pproprite weights w i in wy tht n I( f ) w i f (x i ). The integrl is therefore pproximted by the sum of function vlues t the nodes x 0 < < x n b nd the respective qudrture weights w i. Of course, the pproximtion improves with rising number n. Different qudrture methods only differ with respect to the chosen nodes x i nd weights w i. In the following we will concentrte on NewtonCotes method nd Gussin qudrture method. The former pproximtes the integrnd f between nodes using loworder polynomils nd sum the integrls of the polynomils to estimte the integrl of f. The ltter methods choose the nodes nd weights in order to mtch specific moments(such s expected vlue, vrince etc.) of the pproximted function Summed NewtonCotes methods In this section we set w(x) =1. A weighted integrl cn then simply be computed by setting g(x) =w(x) f (x) nd pproximting g(x) dx. Summed NewtonCotes formuls prtition the intervl [, b] into n subintervls of equl length h by computing the qudrture nodes x i := + ih, i = 0, 1,..., n, mit h := b n. A summed NewtonCotes formul of degree k then interpoltes the function f on every subintervl [x i, x i+1 ] with polynomil of degree k. Wefinlly integrte the interpolting polynomil on every subintervl nd sum up ll the resulting subres. Figure 3.10 shows this pproch for k = 0ndk = 1.
2 64 Chpter 3 Numericl Solution Methods f(x) f(x) f(x) f(x) = x 1 x x 3 x 4 x 5 b = x 6 x = x 1 x x 3 x 4 x 5 b = x 6 x Figure 3.10: Summed NewtonCotes formuls for k = 0 nd k = 1 Summed rectngle rule If k = 0, the interpolting functions re polynomils of degree 0, i.e. constnt functions. We cn write ny re below this constnt functions on the intervl [x i, x i+1 ] s I (0) [x i,x i+1 ] ( f )=(x i+1 x i ) f (x i )=hf(x i ), which is the surfce of rectngle. The degree 0 formul is therefore clled the summed rectngle rule, the explicit form of which is given by I (0) n 1 ( f )= hf(x i ). The weights of the summed rectngle rule consequently re w i = h for i = 0,...,n 1 nd w n = 0. Summed trpezoid rule If k = 1, the function f in the ith subintervl [x i, x i+1 ] is pproximted by the line segment pssing through the two points (x i, f (x i )) nd (x i+1, f (x i+1 )). The re under this line segment is the surfce of trpezoid, i.e. I (1) [x i,x i+1 ] ( f )=hf(x i)+ 1 h[ f (x i+1) f (x i )] = h [ f (x i)+ f (x i+1 )]. It is immeditely cler tht the summed trpezoid rule improves the pproximtion of I( f ) compred to the summed rectngle rule discussed bove. Summing up the res of the trpezoids cross subintervls yields the pproximtion { } I (1) = h n 1 f ()+ f (x i )+ f (b) i=1 of the whole integrl I( f ). The weights of the rule therefore re w 0 = w n = h nd w i = h for i = 1,...,n 1. (3.5)
3 3.4. Numericl integrtion 65 Summed rectngle nd trpezoid rule re simple nd robust. Hence, computtion does not need much effort. In ddition, the ccurcy of the pproximtion of I( f ) increses with rising n. It is lso cler tht the trpezoid rule will exctly compute the integrl of ny firstorder polynomil, i.e. line. Progrm 3.11 shows how to pply the summed trpezoid rule to the function cos(x). We Progrm 3.11: Summed trpezoid rule with cos(x) progrm NewtonCotes! vrible declrtion implicit none integer, prmeter :: n = 10 rel*8, prmeter :: = 0d0, b = d0 rel*8 :: h, x(0:n), w(0:n), f(0:n) integer :: i! clculte qudrture nodes h = (b)/dble(n) x = (/( + dble(i)*h,,n)/)! get weights w(0) = h/d0 w(n) = h/d0 w(1:n1) = h! clculte function vlues t nodes f = cos(x)! Output numericl nd nlyticl solution write(*, (,f10.6) ) Numericl:,sum(w*f, 1) write(*, (,f10.6) ) Anlyticl:,sin(d0)sin(0d0) end progrm thereby first declre ll the vribles needed. Specil ttention should be devoted to the declrtion of x, w nd f. These rrys will store the qudrture nodes x i, the weights w i nd the function vlues t the nodes f (x i ), respectively. We then first compute h nd the nodes x i = + ih nd the weights w i s in (3.5). With the respective function vlues, the pproximtion to the integrl 0 cos(x) dx = sin() sin(0) is the given by sum(w*f, 1). Note tht the pproximtion of the integrl is quite bd. If we increse the number of qudrture nodes n increses the ccurcy, however, we need bout 600 nodes in order to perfectly mtch numericl nd nlyticl result on 6 digits.
4 66 Chpter 3 Numericl Solution Methods Summed Simpson rule Finlly if k =, the integrnd f in the ith subintervl [x i, x i+1 ] is pproximted by secondorder polynomil function c 0 + c 1 x + c x.nowthreegrph points re required to specify the polynomil prmeters c i in the subintervls. Given these prmeters, one cn compute the integrl of the qudrtic function in the subintervl [x i, x i+1 ] s [ ( ) ] I () [x i,x i+1 ] ( f )=h xi + x f (x i )+4f i+1 + f (x i+1 ). 3 Summing up the different res on the subintervls yields { I () ( f )= h n 1 n 1 ( ) } xi + x f ()+ 6 f (x i )+4 f i+1 + f (b). i=1 The summed Simpson rule is lmost s simple to implement s the trpezoid rule. If the integrnd is smooth, Simpson s rule yields pproximtion error tht with rising n flls twice s fst s tht of the trpezoid rule. For this reson Simpson s rule is usully preferred to the trpezoid nd rectngle rule. Note, however, tht the trpezoid rule will often be more ccurte thn Simpson s rule if the integrnd exhibits discontinuities in its first derivtive, which cn occur in economic pplictions exhibiting corner solutions. Of course, summed NewtonCotes rules lso exist for higher order piecewise polynomil pproximtions, but they re more difficult to work with nd thus rrely used Gussin Qudrture In opposite to the NewtonCotes methods, Gussin qudrture tkes explicitly into ccount weight functions w(x). Here obviously, qudrture nodes x i nd weights w i re computed differently. GussLegendre qudrture Suppose gin, for the moment, tht w(x) =1. The GussLegendre qudrture nodes x i [, b] nd weights w i re computed in wy tht they stisfy the n + momentmtching conditions x k w(x) dx = x k n dx = w i xi k for k = 0, 1,..., n 1. (3.6) With the bove conditions holding, we cn write every integrl over polynomil with degree m n 1s p(x) dx = c 0 p(x) =c 0 + c 1 x c m x m 1 dx +c 1 xdx } {{} = n w ix i }{{} = n w i c m x m dx } {{} = n w ixi m
5 3.4. Numericl integrtion 67 n = w i [c 0 + c 1 x i c m x m n i ] = w i p(x i ). Consequently, qudrture nodes nd weights tht stisfy the momentmtching conditions in (3.6) re ble to integrte ny polynomil p of degree m n 1exctly. Clculting the nodes nd weights of the GussLegendre qudrture formul is not so esy. One method is to set up the nonliner eqution system defined in (3.6). For n = 3 e.g. we hve w 0 x 0 + w 1 x 1 + w x = 1 ) (b = 1 dx, w 0 x 0 + w 1 x 1 + w x = 1 ( b ) = xdx,. w 0 x0 5 + w 1x1 5 + w x 5 = 1 ( b 6 6) = x 5 dx. 6 This nonliner eqution system cn be solved by rootfinding lgorithm like fzero. However, there is more efficient, but lso less intuitive wy to clculte qudrture nodes nd weights of the GussLegendre qudrture implemented in the subroutine legendre in the module gussin_int. Progrm3.1 demonstrtes how to use it. The Progrm 3.1: GussLegendre qudrture with cos(x) progrm GussLegendre! module use use gussin_int! vrible declrtion implicit none integer, prmeter :: n = 10 rel*8, prmeter :: = 0d0, b = d0 rel*8 :: x(0:n), w(0:n), f(0:n)! clculte nodes nd weights cll legendre(0d0, d0, x, w)! clculte function vlues t nodes f = cos(x)! Output numericl nd nlyticl solution write(*, (,f10.6) ) Numericl:,sum(w*f, 1) write(*, (,f10.6) ) Anlyticl:,sin(d0)sin(0d0) end progrm subroutine tkes two rel*8 rguments defining the left nd right intervl borders nd
6 68 Chpter 3 Numericl Solution Methods b. In ddition, we hve to pss two rrys of equl length to the routine. The first of these will be filled with the nodes x i, wheres the second will be given the weights w i. After hving clculted the function vlues f (x i ), we cn gin clculte the numericl pproximtion of the integrl like in Progrm Here, with 10 qudrture nodes, we lredy mtch the nlyticl integrl vlue by 6 digits. Note tht we needed bout 600 nodes with the trpezoid rule. GussHermite qudrture If we set the weight function to w(x) = 1 π exp( x ),weresultinthegusshermite qudrture method. Note tht the weight function now is equl to the density function of the stndrd norml distribution. Now, with m n + 1 nd normlly distributed rndom vrible x, wehve E( x) = 1 exp( x )x m n dx = π w i xi m = I GH (x m ). Consequently, with GussHermite qudrture, we re ble to perfectly mtch the first n 1 moments of normlly distributed rndom vrible x. An pproximtion procedure for normlly distributed rndom vribles is included in the module normlprob. The procedure norml_discrete(x, w, mu, sig) is exctly bsed on the GussHermite qudrture method. The subroutine receives four input rguments, the lst of which re expected vlue nd vrince of the norml distribution tht should be pproximted. The routine then stores pproprite nodes x i nd weights w i in the rrys x nd w (tht should be of sme length) which cn be used to compute moments of normlly distributed rndom vrible x with expecttion mu nd vrince sig. For pplying this subroutine we consider the following exmple: Exmple Consider n griculturl commodity mrket, where plnting decisions re bsed on the price expected t hrvest A = E(p), (3.7) with A denoting crege supply nd E(p) defining expected price. After the crege is plnted, normlly distributed rndom yield y N(1, 0.1) is relized, giving rise to the quntity q s = Ay which is sold t the mrket clering price p = 3 q s. In order to solve this system we substitute nd therefore q s = [ E(p)] y p = 3 [ E(p)] y.
7 3.4. Numericl integrtion 69 Tking expecttions on both sides leds to E(p) =3 [ E(p)] E(y) nd therefore E(p) = 1. Consequently, equilibrium crege is A = 1. Finlly, the equilibrium price distribution hs vrince of Vr(p) =4 [ E(p)] Vr(y) =4Vr(y) =0.4. Suppose now tht the government introduces price support progrm which gurntees ech producer minimum price of 1. If the mrket price flls below this level, the government pys the producer the difference per unit produced. Consequently, the producer now receives n effective price of mx(p,1) nd the expected price in (3.7) thenis clculted vi E(p) =E [mx (3 Ay,1)]. (3.8) The equilibrium crege supply finlly is the supply A tht fulfills (3.7) with the bove price expecttion. Agin, this problem cnnot be solved nlyticlly. In order to compute the solution of the bove crege problem, one hs to use two modules. normlprob on the one hnd provides the method to discretize the normlly distributed rndom vrible y. The solution to (3.7) cn on the other hnd be clculted by using the method fzero from the rootfinding module. Progrm 3.13 shows how to do this. Due to spce restrictions, we do not show the whole progrm. For running the progrm, we lso need module globls in which we store the expecttion mu nd vrince sig of the normlly distributed rndom vrible y s well s the minimum price minp gurnteed by the government. In ddition, the module stores the qudrture nodes y nd weights w obtined by discretizing the norml distribution for y. In the progrm, we first discretize the distribution for y by mens of the subroutine norml_discrete. This routine receives the expected vlue nd vrince of the norml distribution nd stores the respective nodes y nd weights w in two rrys of sme size. Next we set strting guess for crege supply. We then let fzero find the root of the function mrket tht clcultes the mrket equilibrium condition in (3.7). This function only gets A s n input. From A we cn clculte the expected price E(p) by mens of (3.8) nd finlly the mrket clering condition. Hving found the root of mrket, i.e. the equilibrium crege supply, we cn clculte the expected vlue nd vrince of the price p. Inorder to showthe effects of minimum price, wefirst set the minimum price gurntee minp to lrge negtive vlue, sy 100. This lrge negtive minimum price will never be binding, hence, the outcome is exctly the sme s in the model without price gurntee, see the bove exmple description. If we now set the minimum price t 1, equilibrium crege supply increses by bout 9.7percent. This is becuse the expected effective price
8 70 Chpter 3 Numericl Solution Methods Progrm 3.13: Agriculturl problem progrm griculture...! discretize y cll norml_discrete(y, w, mu, sig)! initilize vribles A = 1d0! get optimum cll fzero(a, mrket, check)! get expecttion nd vrince of price Ep = sum(w*mx(3d0d0*a*y, minp)) Vrp = sum(w*(mx(3d0d0*a*y, minp)  Ep)**)... end progrm function mrket(a) use globls rel*8, intent(in) :: A rel*8 :: mrket rel*8 :: Ep! clculte expected price Ep = sum(w*mx(3d0d0*a*y, minp))! get equilibrium eqution mrket = A  (0.5d0+0.5d0*Ep) end function for the producer rises from the minimum price gurntee. In ddition, the uncertinty for the producers is reduced, since the vrince of the price distribution decreses from 0.4 to bout 0.115, which is gin due to the fct tht the possible price reliztion now hve lower limit of 1.
Numerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationIII. Lecture on Numerical Integration. File faclib/dattab/lecturenotes/numericalinter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecturenotes/numericalinter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationCOSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III)  Gauss Quadrature and Adaptive Quadrature
COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III  Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More information7  Continuous random variables
71 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7  Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin
More informationOrthogonal Polynomials and LeastSquares Approximations to Functions
Chpter Orthogonl Polynomils nd LestSqures Approximtions to Functions **4/5/3 ET. Discrete LestSqures Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationTHE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS
THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2Dimensionl Gussin Qudrture 20 4
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More information31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes
Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More informationNumerical Integration
Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationOPEN NEWTON  COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS
IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 96 pissn: 78 OPEN NEWTON  COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS T. Rmchndrn R.Priml
More informationS. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:
FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy
More information7. Numerical evaluation of definite integrals
7. Numericl evlution of definite integrls Tento učení text yl podpořen z Operčního progrmu Prh  Adptilit Hn Hldíková Numericl pproximtion of definite integrl is clled numericl qudrture, the formuls re
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationNew Integral Inequalities for ntime Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353362 New Integrl Inequlities for ntime Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationHarmonic Mean Derivative  Based Closed Newton Cotes Quadrature
IOSR Journl of Mthemtics (IOSRJM) eissn:  pissn: 9X. Volume Issue Ver. IV (My.  Jun. 0) PP  www.iosrjournls.org Hrmonic Men Derivtive  Bsed Closed Newton Cotes Qudrture T. Rmchndrn D.Udykumr nd
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. LmiAthens Lmi 3500 Greece Abstrct Using
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationCLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTONCOTES QUADRATURES
Filomt 27:4 (2013) 649 658 DOI 10.2298/FIL1304649M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED
More informationIntroduction to Numerical Analysis
Introduction to Numericl Anlysis Doron Levy Deprtment of Mthemtics nd Center for Scientific Computtion nd Mthemticl Modeling (CSCAMM) University of Mrylnd June 14, 2012 D. Levy CONTENTS Contents 1 Introduction
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationNormal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution
Norml Distribution Lecture 6: More Binomil Distribution If X is rndom vrible with norml distribution with men µ nd vrince σ 2, X N (µ, σ 2, then P(X = x = f (x = 1 e 1 (x µ 2 2 σ 2 σ Sttistics 104 Colin
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More information38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes
The Uniform Distribution 8. Introduction This Section introduces the simplest type of continuous probbility distribution which fetures continuous rndom vrible X with probbility density function f(x) which
More informationProblem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume
Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom r vrible. Define new rndom vrible Y = g. Find the pdf of Y. Method: Step : Step : Step 3: Plot Y = g( ). Find F ( y) by mpping
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More informationInterpolation. Gaussian Quadrature. September 25, 2011
Gussin Qudrture September 25, 2011 Approximtion of integrls Approximtion of integrls by qudrture Mny definite integrls cnnot be computed in closed form, nd must be pproximted numericlly. Bsic building
More informationMapping the delta function and other Radon measures
Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
More informationMath 115 ( ) YumTong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx
Mth 5 20062007) YumTong Siu Lgrnge Multipliers nd Vritionl Problems with Constrints Integrl Constrints. Consider the vritionl problem of finding the extremls for the functionl J[y] = F x,y,y )dx with
More informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationLECTURE NOTE #12 PROF. ALAN YUILLE
LECTURE NOTE #12 PROF. ALAN YUILLE 1. Clustering, Kmens, nd EM Tsk: set of unlbeled dt D = {x 1,..., x n } Decompose into clsses w 1,..., w M where M is unknown. Lern clss models p(x w)) Discovery of
More informationD01BBF NAG Fortran Library Routine Document
D01 Qudrture D01BBF NAG Fortrn Librry Routine Document Note. Before using this routine, plese red the Users Note for your implementtion to check the interprettion of bold itlicised terms nd other implementtiondependent
More informationMETHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS
Journl of Young Scientist Volume III 5 ISSN 448; ISSN CDROM 449; ISSN Online 445; ISSNL 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationWeek 10: Riemann integral and its properties
Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the
More informationNumerical Methods 2007
Physics Mster Course: Numericl Methods 2007 Hns Mssen Rdboud Universiteit Nijmegen Onderwijsinstituut Wiskunde, Ntuur en Sterrenkunde Toernooiveld 1 6525 ED Nijmegen September 2007 1 Introduction In these
More information(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35
7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know
More informationChapter 4. Additional Variational Concepts
Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.
More information4.1. Probability Density Functions
STT 1 4.14. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile  vers  discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationWeek 7 Riemann Stieltjes Integration: Lectures 1921
Week 7 Riemnn Stieltjes Integrtion: Lectures 1921 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0
More informationEfficient Computation of a Class of Singular Oscillatory Integrals by Steepest Descent Method
Applied Mthemticl Sciences, Vol. 8, 214, no. 31, 15351542 HIKARI Ltd, www.mhikri.com http://dx.doi.org/1.12988/ms.214.43166 Efficient Computtion of Clss of Singulr Oscilltory Integrls by Steepest Descent
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationThe problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.
ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion
More informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI Question No: 1 { Mrks: 1 )  Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...
More informationLecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations
18330 Lecture Notes: Orthogonl Polynomils, Gussin Qudrture, nd Integrl Equtions Homer Reid My 1, 2014 In the previous set of notes we rrived t the definition of Chebyshev polynomils T n (x) vi the following
More informationNumerical Methods I. Olof Widlund Transcribed by Ian Tobasco
Numericl Methods I Olof Widlund Trnscribed by In Tobsco Abstrct. This is prt one of two semester course on numericl methods. The course ws offered in Fll 011 t the Cournt Institute for Mthemticl Sciences,
More information1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.
Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the
More informationThe Definite Integral
CHAPTER 3 The Definite Integrl Key Words nd Concepts: Definite Integrl Questions to Consider: How do we use slicing to turn problem sttement into definite integrl? How re definite nd indefinite integrls
More informationOrdinary Differential Equations Boundary Value Problem
Ordinry Differentil Equtions Boundry Vlue Problem Shooting method Runge Kutt method Computerbsed solutions o BVPFD subroutine (Fortrn IMSL subroutine tht Solves (prmeterized) system of differentil equtions
More informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More informationECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance
Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between
More informationSolutions to Problems in Merzbacher, Quantum Mechanics, Third Edition. Chapter 7
Solutions to Problems in Merzbcher, Quntum Mechnics, Third Edition Homer Reid April 5, 200 Chpter 7 Before strting on these problems I found it useful to review how the WKB pproimtion works in the first
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationThe Dirac distribution
A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, BbeşBolyi University Str. Koglnicenu
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More informationChapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses
Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n nsided regulr polygon of perimeter p n with vertices on C. Form cone
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The twodimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationPDE Notes. Paul Carnig. January ODE s vs PDE s 1
PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................
More information1 Online Learning and Regret Minimization
2.997 DecisionMking in LrgeScle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.
More informationMath 324 Course Notes: Brief description
Brief description These re notes for Mth 324, n introductory course in Mesure nd Integrtion. Students re dvised to go through ll sections in detil nd ttempt ll problems. These notes will be modified nd
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCKKURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When relvlued
More informationMATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.
MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded
More informationC1M14. Integrals as Area Accumulators
CM Integrls s Are Accumultors Most tetbooks do good job of developing the integrl nd this is not the plce to provide tht development. We will show how Mple presents Riemnn Sums nd the ccompnying digrms
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More informationBIFURCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS
BIFRCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationMACsolutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationSummary of Elementary Calculus
Summry of Elementry Clculus Notes by Wlter Noll (1971) 1 The rel numbers The set of rel numbers is denoted by R. The set R is often visulized geometriclly s numberline nd its elements re often referred
More informationProblem Set 3
14.102 Problem Set 3 Due Tuesdy, October 18, in clss 1. Lecture Notes Exercise 208: Find R b log(t)dt,where0
More informationTest , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes
Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10
University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin
More informationChapter 4 Models for Stationary Time Series
Chpter 4 Models for Sttionry Time Series This chpter discusses the bsic concepts of brod clss of prmetric time series models the utoregressivemoving verge models (ARMA. These models hve ssumed gret importnce
More informationNonLinear & Logistic Regression
NonLiner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationAcceptance Sampling by Attributes
Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire
More information