Prior distributions. July 29, 2002
|
|
- Patience Owens
- 5 years ago
- Views:
Transcription
1 Prior distributios Aledre Tchourbov PKI 357, UNOmh, South 67th St. Omh, NE , USA Phoe: E-mil: July 9, Abstrct This documet itroduces prior distributios for the purposes of Byesi sttistics. Usig prior beliefs we c sigifictly improve our sttisticl ifereces bsed o observtios. Most of the books o sttistics do ot cover the mteril preseted. Here we try to collect the iformtio vilble o cojugte priors to certi distributios. Itroductio Accordig to the Byesi rule [], we c epress posterior probbility of certi evet H give some dt with the formul P dt HP H P H dt = P dt The probbility of H give the dt is clled the posterior probbility of H. The posterior equls to the likelihood time the prior divided by mrgil probbility of dt. The pper shows wht priors we c hve d how they ffect posterior distributios give likelihood. Prior d posterior distributios Sometimes prior distributio c be pproimted by oe tht is i coveiet fmily of distributios, which combies with the likelihood to produce posterior tht is mgeble. We see tht objective wy of buildig priors for the biomil prmeter ws to use the cojugte fmily distributio tht hs the property tht the updted distributio is i the sme fmily. I geerl, if the prior distributio belogs to fmily G, the dt hve distributio belogig to fmily H, d the posterior distributio lso belogs to G, the we sy tht G is fmily of cojugte priors to H. Thus, the bet distributio is cojugte prior to the biomil, d the orml is self cojugte. Cojugte priors my ot eist; whe they do, selectig member of the cojugte fmily s prior is doe mostly for mthemticl coveiece, sice the posterior c be evluted very simply. More geerlly, umericl methods of itegrtio would hve to be used to evlute the posterior. I would like to thk professors Heshm Ali d Jiteder Deogu for the opportuity to work o this project
2 Observtios Prior Posterior Beroulli Bet Bet Poisso Gmm Gmm Biomil Bet Bet Norml Norml Norml Norml Gmm Gmm Tble : Cojugte priors 3 Bet priors From Byes 763: A white billird bll W is rolled log lie d we look t where it stops, scle the tble from to. We suppose tht it hs uiform probbility of fllig ywhere o the lie. It stops t poit p. A red billird bll R is the rolled times uder the sme uiform ssumptio. X the deotes the umber of times R goes o further th W wet. Give X, wht iferece c we mke bout p? Here we re lookig for the posterior distributio of p give X. The prior distributio of p is uiform gp = Uiform, = Bet, =. Give p, X hs biomil distributio P X = p = p p The overll distributio of the umber of successes is the sum of probbilities for ll possible p s P < p < b, X = = p p dp P X = = p p dp Suppose we throw ll + blls o the tble, d choose the red oe. The the probbility tht the red oe hs whites to the left of it is. So we hve + P X = = p p dp = p p dp = + p p dp =!! +! ccordig to defiitio formul for bet fuctio Br, s = we hve X B, p p r p s dp = r!s! r + s! = ΓrΓs Γr + s
3 P X = = p p dp = B +, + P < p < b X = = p p dp B +, + P < p < b X = = p p dp B +, + which is bet distributio of p with prmeters + d +. The desity fuctio f of the bet distributio is fp = Γ + b ΓΓb p p b, p Emple Suppose tht the prior distributio of p is Bet, b, i.e. gp = p p b B, b Likelihood hs biomil distributio f p = p p The posterior distributio of p give is hp = f pgp f = B, b p p p p b p + p +b dp B, b = p+ p +b B +, + b = Bet +, + b This distributio is thus bet s well with prmeters = + d b = b +. 4 Norml prior Here we follow emple o pge 589 [], which proves the Norml cojugte prior for Norml distributio. The cojugte for Norml likelihood is the Norml distributio. Emple We cosider iferece cocerig ukow me with kow vrice. 3
4 First, suppose tht the prior distributio of µ is Nµ, σ. Nµ, σ is tke. The posterior distributio of µ is hµ = f µgµ = f µgµ f µgµ f µgµdµ A sigle observtio X µ µ σ π ep µ ep σ σ π σ µ ep µ µ σ σ ep µ σ + µ σ σ + µ + σ σ + µ σ Let, b d c be the coefficiets i the qudrtic polyomil i µ tht is the lst epressio. The my the be writte hµ = ep µ b µ + c To simplify this further, we use the techique of completig the squre d rewrite the epressio s hµ = ep ep µ b ep µ b c We see tht posterior distributio of µ is orml with me µ = + µ σ σ + σ σ b For prcticl resos, we defie the precisio s the iverse of the vrice: we deote by ξ = d d ξ σ = σ Theorem Suppose tht µ Nµ, σ. The the posterior distributio of µ is orml with me µ = σ µ + ξ ξ + ξ d precisio ξ = ξ + ξ The posterior me is weighted verge of the prior me d the dt, weights beig proportiol to the respective precisios. With very getle prior we would hve very low precisio ξ, very t prior d mostly the posterior is Norml with s its me. Of course wht we re usully iterested i is the posterior give iid smple of size, wht you could epect hppes it is equivlet to ddig oe observtio from distributio tht hs vrice σ. 4
5 5 Multiomil Dirichlet priors Dirichlet prior Dirichlet prior is cojugte to multiomil distributio. This is probbility distributio o o the simple. = { p = p, p,..., p, p p =, p i } The Dirichlet distributio c be writte s DΘ α = Γ K i= α i K i= Γα i K i= Θ α i i where α = α,..., α K, with α i > re costts specifyig the Dirichlet distributio Θ i stisfy Θ i d K i= Θ i = The multiomil distributio correspodig to k blls dropped ito boes with fied probbility p,..., p, with ith bo cotiig k i blls is k k,..., k p k p k For two vribles K = the Dirichlet distributio reduces to Bet distributio, d ormlizig costt becomes Bet fuctio. The Dirichlet is coveiet prior becuse the posterior p hvig observed k,..., k is Dirichlet with probbility α +k,..., α +k. A importt chrcteriztio of the Dirichlet: it is the oly prior tht predicts outcomes lierly i the pst. Oe frequetly used specil cse is the symmetric Dirichlet whe ll α i = c >. We deote this prior s D c. Dirichlet priors re importt becuse They re turl cojugte priors for multiomil distributios, i.e. posterior prmeter distributio, fter hvig observed some dt from multiomil distributio with Dirichlet prior, lso hve form of Dirichlet distributio The Dirichlet distributio c be see s multivrite geerliztio of the bet distributio, over the spce of distributios P, with costt o the verge distce reltive etropy to referece distributio determied by Θ d α. Refereces [] Rev. Thoms Byes, A essy towrds solvig problem i the doctrie of chces, Philosophicl Trsctios of the Royl Society of Lodo 763. [] Joh. A. Rice, Mthemticl sttistics d dt lysis, Dubury Press,
Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationSurds, Indices, and Logarithms Radical
MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio
More informationPROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
More information n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!
mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationStudents must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...
Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationChapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
More informationDouble Sums of Binomial Coefficients
Itertiol Mthemticl Forum, 3, 008, o. 3, 50-5 Double Sums of Biomil Coefficiets Athoy Sofo School of Computer Sciece d Mthemtics Victori Uiversity, PO Box 448 Melboure, VIC 800, Austrli thoy.sofo@vu.edu.u
More informationInference on One Population Mean Hypothesis Testing
Iferece o Oe Popultio Me ypothesis Testig Scerio 1. Whe the popultio is orml, d the popultio vrice is kow i. i. d. Dt : X 1, X,, X ~ N(, ypothesis test, for istce: Exmple: : : : : : 5'7" (ull hypothesis:
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More informationMath 3B Midterm Review
Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.1-7.8 i your ebook. The best idictio of wht
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationChapter System of Equations
hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More information1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
More informationIntroduction to mathematical Statistics
More ttistics tutoril t wwwdumblittledoctorcom Itroductio to mthemticl ttistics Chpter 7 ypothesis Testig Exmple (ull hypothesis) : the verge height is 5 8 or more ( : μ 5'8" ) (ltertive hypothesis) :
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Lecture 17
CS 70 Discrete Mthemtics d Proility Theory Sprig 206 Ro d Wlrd Lecture 7 Vrice We hve see i the previous ote tht if we toss coi times with is p, the the expected umer of heds is p. Wht this mes is tht
More informationLinear Programming. Preliminaries
Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationReview of Sections
Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,
More informationLesson 4 Linear Algebra
Lesso Lier Algebr A fmily of vectors is lierly idepedet if oe of them c be writte s lier combitio of fiitely my other vectors i the collectio. Cosider m lierly idepedet equtios i ukows:, +, +... +, +,
More information( ) k ( ) 1 T n 1 x = xk. Geometric series obtained directly from the definition. = 1 1 x. See also Scalars 9.1 ADV-1: lim n.
Sclrs-9.0-ADV- Algebric Tricks d Where Tylor Polyomils Come From 207.04.07 A.docx Pge of Algebric tricks ivolvig x. You c use lgebric tricks to simplify workig with the Tylor polyomils of certi fuctios..
More informationINFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
More informationThe Exponential Function
The Epoetil Fuctio Defiitio: A epoetil fuctio with bse is defied s P for some costt P where 0 d. The most frequetly used bse for epoetil fuctio is the fmous umber e.788... E.: It hs bee foud tht oyge cosumptio
More informationConvergence rates of approximate sums of Riemann integrals
Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationFrequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
More informationWhy study large deviations? The problem of estimating buer overow frequency The performance of many systems is limited by events which have a small pr
Why study lrge devitios? The problem of estimtig buer overow frequecy The performce of my systems is ited by evets which hve smll probbility of occurrig, but which hve severe cosequeces whe they occur.
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More information: : 8.2. Test About a Population Mean. STT 351 Hypotheses Testing Case I: A Normal Population with Known. - null hypothesis states 0
8.2. Test About Popultio Me. Cse I: A Norml Popultio with Kow. H - ull hypothesis sttes. X1, X 2,..., X - rdom smple of size from the orml popultio. The the smple me X N, / X X Whe H is true. X 8.2.1.
More informationAlgebra II, Chapter 7. Homework 12/5/2016. Harding Charter Prep Dr. Michael T. Lewchuk. Section 7.1 nth roots and Rational Exponents
Algebr II, Chpter 7 Hrdig Chrter Prep 06-07 Dr. Michel T. Lewchuk Test scores re vilble olie. I will ot discuss the test. st retke opportuit Sturd Dec. If ou hve ot tke the test, it is our resposibilit
More informationLecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields
Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive
More informationBackground 1. Cramer-Rao inequality
Bro-06, Lecture, 60506 D/Stt/Bro-06/tex wwwmstqueesuc/ blevit/ Observtio vector, dt: Bckgroud Crmer-Ro iequlity X (X,, X X R Here dt c be rel- or vector vlued, compoets of the dt c be idepedet or depedet,
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationChapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
More informationSchrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
More information334 MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION
MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION TEST SAMPLE TEST III - P APER Questio Distributio INSTRUCTIONS:. Attempt ALL questios.. Uless otherwise specified, ll worig must be
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More informationKing Fahd University of Petroleum & Minerals
Kig Fhd Uiversity of Petroleum & Mierls DEPARTMENT OF MATHEMATICAL CIENCE Techicl Report eries TR 434 April 04 A Direct Proof of the Joit Momet Geertig Fuctio of mple Me d Vrice Awr H. Jorder d A. Lrdji
More informationVectors. Vectors in Plane ( 2
Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector
More informationThe total number of permutations of S is n!. We denote the set of all permutations of S by
DETERMINNTS. DEFINITIONS Def: Let S {,,, } e the set of itegers from to, rrged i scedig order. rerrgemet jjj j of the elemets of S is clled permuttio of S. S. The totl umer of permuttios of S is!. We deote
More informationIn an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case
Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationAvd. Matematisk statistik
Avd. Mtemtisk sttistik TENTAMEN I SF94 SANNOLIKHETSTEORI/EAM IN SF94 PROBABILITY THE- ORY WEDNESDAY 8th OCTOBER 5, 8-3 hrs Exmitor : Timo Koski, tel. 7 3747, emil: tjtkoski@kth.se Tillåt hjälpmedel Mes
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probbility d Stochstic Processes: A Friedly Itroductio for Electricl d Computer Egieers Roy D. Ytes d Dvid J. Goodm Problem Solutios : Ytes d Goodm,4..4 4..4 4..7 4.4. 4.4. 4..6 4.6.8 4.6.9 4.7.4 4.7.
More information2017/2018 SEMESTER 1 COMMON TEST
07/08 SEMESTER COMMON TEST Course : Diplom i Electroics, Computer & Commuictios Egieerig Diplom i Electroic Systems Diplom i Telemtics & Medi Techology Diplom i Electricl Egieerig with Eco-Desig Diplom
More informationy udv uv y v du 7.1 INTEGRATION BY PARTS
7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product
More informationStatistics for Financial Engineering Session 1: Linear Algebra Review March 18 th, 2006
Sttistics for Ficil Egieerig Sessio : Lier Algebr Review rch 8 th, 6 Topics Itroductio to trices trix opertios Determits d Crmer s rule Eigevlues d Eigevectors Quiz The cotet of Sessio my be fmilir to
More informationA general theory of minimal increments for Hirsch-type indices and applications to the mathematical characterization of Kosmulski-indices
Mlysi Jourl of Librry & Iformtio Sciece, Vol. 9, o. 3, 04: 4-49 A geerl theory of miiml icremets for Hirsch-type idices d pplictios to the mthemticl chrcteriztio of Kosmulski-idices L. Egghe Uiversiteit
More informationSolutions to Problem Set 7
8.0 Clculus Jso Strr Due by :00pm shrp Fll 005 Fridy, Dec., 005 Solutios to Problem Set 7 Lte homework policy. Lte work will be ccepted oly with medicl ote or for other Istitute pproved reso. Coopertio
More informationlecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
More informationImportant Facts You Need To Know/Review:
Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t
More informationCertain sufficient conditions on N, p n, q n k summability of orthogonal series
Avilble olie t www.tjs.com J. Nolier Sci. Appl. 7 014, 7 77 Reserch Article Certi sufficiet coditios o N, p, k summbility of orthogol series Xhevt Z. Krsiqi Deprtmet of Mthemtics d Iformtics, Fculty of
More informationConvergence rates of approximate sums of Riemann integrals
Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem
More information8.3 Sequences & Series: Convergence & Divergence
8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of
More informationName: A2RCC Midterm Review Unit 1: Functions and Relations Know your parent functions!
Nme: ARCC Midterm Review Uit 1: Fuctios d Reltios Kow your pret fuctios! 1. The ccompyig grph shows the mout of rdio-ctivity over time. Defiitio of fuctio. Defiitio of 1-1. Which digrm represets oe-to-oe
More informationENGINEERING PROBABILITY AND STATISTICS
ENGINEERING PROBABILITY AND STATISTICS DISPERSION, MEAN, MEDIAN, AND MODE VALUES If X, X,, X represet the vlues of rdom smple of items or oservtios, the rithmetic me of these items or oservtios, deoted
More informationExponential and Logarithmic Functions (4.1, 4.2, 4.4, 4.6)
WQ017 MAT16B Lecture : Mrch 8, 017 Aoucemets W -4p Wellm 115-4p Wellm 115 Q4 ue F T 3/1 10:30-1:30 FINAL Expoetil Logrithmic Fuctios (4.1, 4., 4.4, 4.6) Properties of Expoets Let b be positive rel umbers.
More informationEVALUATING DEFINITE INTEGRALS
Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationNumbers (Part I) -- Solutions
Ley College -- For AMATYC SML Mth Competitio Cochig Sessios v.., [/7/00] sme s /6/009 versio, with presettio improvemets Numbers Prt I) -- Solutios. The equtio b c 008 hs solutio i which, b, c re distict
More informationNumerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials
Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,
More information2015/2016 SEMESTER 2 SEMESTRAL EXAMINATION
05/06 SEMESTER SEMESTRA EXAMINATION Course : Diplom i Electroics, Computer & Commuictios Egieerig Diplom i Aerospce Systems & Mgemet Diplom i Electricl Egieerig with Eco-Desig Diplom i Mechtroics Egieerig
More informationCanonical Form and Separability of PPT States on Multiple Quantum Spaces
Coicl Form d Seprbility of PPT Sttes o Multiple Qutum Spces Xio-Hog Wg d Sho-Mig Fei, 2 rxiv:qut-ph/050445v 20 Apr 2005 Deprtmet of Mthemtics, Cpitl Norml Uiversity, Beijig, Chi 2 Istitute of Applied Mthemtics,
More informationContent: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationBasic Limit Theorems
Bsic Limit Theorems The very "cle" proof of L9 usig L8 ws provided to me by Joh Gci d it ws this result which ispired me to write up these otes. Absolute Vlue Properties: For rel umbers x, d y x x if x
More informationIndices and Logarithms
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic
More informationOrthogonality, orthogonalization, least squares
ier Alger for Wireless Commuictios ecture: 3 Orthogolit, orthogoliztio, lest squres Ier products d Cosies he gle etee o-zero vectors d is cosθθ he l of Cosies: + cosθ If the gle etee to vectors is π/ (90º),
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationOptions: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG.
O C. PG.-3 #, 3b, 4, 5ce O C. PG.4 # Optios: Clculus O D PG.8 #, 3, 4, 5, 7 O E PG.3-33 #, 3, 4, 5 O F PG.36-37 #, 3 O G. PG.4 #c, 3c O G. PG.43 #, O H PG.49 #, 4, 5, 6, 7, 8, 9, 0 O I. PG.53-54 #5, 8
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More informationParticle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise
Prticle i Box We must hve me where = 1,,3 Solvig for E, π h E = = where = 1,,3, m 8m d the stte fuctio is x A si for 0 x, d 0 otherwise x ˆ d KE V. m dx I this cse, the Hermiti opertor 0iside the box The
More informationSection 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
More informationCrushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationProbability for mathematicians INDEPENDENCE TAU
Probbility for mthemticis INDEPENDENCE TAU 2013 21 Cotets 2 Cetrl limit theorem 21 2 Itroductio............................ 21 2b Covolutio............................ 22 2c The iitil distributio does
More informationGeneral properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationExponential Families and Bayesian Inference
Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where
More information2a a a 2a 4a. 3a/2 f(x) dx a/2 = 6i) Equation of plane OAB is r = λa + µb. Since C lies on the plane OAB, c can be expressed as c = λa +
-6-5 - - - - 5 6 - - - - - - / GCE A Level H Mths Nov Pper i) z + z 6 5 + z 9 From GC, poit of itersectio ( 8, 9 6, 5 ). z + z 6 5 9 From GC, there is o solutio. So p, q, r hve o commo poits of itersectio.
More informationELEG 3143 Probability & Stochastic Process Ch. 5 Elements of Statistics
Deprtet of Electricl Egieerig Uiversity of Arkss ELEG 3143 Probbility & Stochstic Process Ch. 5 Eleets of Sttistics Dr. Jigxi Wu wuj@urk.edu OUTLINE Itroductio: wht is sttistics? Sple e d sple vrice Cofidece
More information=> PARALLEL INTERCONNECTION. Basic Properties LTI Systems. The Commutative Property. Convolution. The Commutative Property. The Distributive Property
Lier Time-Ivrit Bsic Properties LTI The Commuttive Property The Distributive Property The Associtive Property Ti -6.4 / Chpter Covolutio y ] x ] ] x ]* ] x ] ] y] y ( t ) + x( τ ) h( t τ ) dτ x( t) * h(
More informationMath 140 Introductory Statistics
Sttistics of Exm Mth Itrouctory Sttistics Professor B. Ábrego Lecture Sectios.3,.4 Me 7. SD.7 Mi 3 Q Me 7 Q3 8 Mx 0 0 0 0 0 0 70 80 0 Importt Uses of Coitiol Probbility To compre smplig with or without
More informationSimpson s 1/3 rd Rule of Integration
Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1 Simpso s 1/3 rd Rule o Itegrtio Wht is Itegrtio?
More informationReversing the Arithmetic mean Geometric mean inequality
Reversig the Arithmetic me Geometric me iequlity Tie Lm Nguye Abstrct I this pper we discuss some iequlities which re obtied by ddig o-egtive expressio to oe of the sides of the AM-GM iequlity I this wy
More informationLimits and an Introduction to Calculus
Nme Chpter Limits d Itroductio to Clculus Sectio. Itroductio to Limits Objective: I this lesso ou lered how to estimte limits d use properties d opertios of limits. I. The Limit Cocept d Defiitio of Limit
More informationALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions
Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +
More informationSequence and Series of Functions
6 Sequece d Series of Fuctios 6. Sequece of Fuctios 6.. Poitwise Covergece d Uiform Covergece Let J be itervl i R. Defiitio 6. For ech N, suppose fuctio f : J R is give. The we sy tht sequece (f ) of fuctios
More informationSimilar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication
Next. Covered bsics of simple desig techique (Divided-coquer) Ch. of the text.. Next, Strsse s lgorithm. Lter: more desig d coquer lgorithms: MergeSort. Solvig recurreces d the Mster Theorem. Similr ide
More informationPower Series Solutions to Generalized Abel Integral Equations
Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp. 5-54 http://www.isciece.org/jourl/ijmcs Power Series Solutios to Geerlized Abel Itegrl Equtios Rufi Abdulli * Deprtmet of Physics d Mthemtics,
More information1 Section 8.1: Sequences. 2 Section 8.2: Innite Series. 1.1 Limit Rules. 1.2 Common Sequence Limits. 2.1 Denition. 2.
Clculus II d Alytic Geometry Sectio 8.: Sequeces. Limit Rules Give coverget sequeces f g; fb g with lim = A; lim b = B. Sum Rule: lim( + b ) = A + B, Dierece Rule: lim( b ) = A B, roduct Rule: lim b =
More information* power rule: * fraction raised to negative exponent: * expanded power rule:
Mth 15 Iteredite Alger Stud Guide for E 3 (Chpters 7, 8, d 9) You use 3 5 ote crd (oth sides) d scietific clcultor. You re epected to kow (or hve writte o our ote crd) foruls ou eed. Thik out rules d procedures
More information