Chapter 2: Probability and Statistics
|
|
- Kathleen Stokes
- 5 years ago
- Views:
Transcription
1 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Itroucto to sttstcs... 7 Cotuous Dstrbutos... 9 Guss Dstrbuto (D)... Coutg evets to etere probbltes... Bol Coeffcets (Dstrbuto)... 3 Strlg s Appoto... 4 Guss Approto to bol strbuto for lrge... 5 Dervto of the Guss Dstrbuto... 6 Chpter : Probblty Sttstcs The essetl rguet sttstcl echc epes o probbltes. A prtculr cofgurto s fou wth cert probblty, we f propertes of sple by vergg proceure. Becuse the uber of olecules s so lrge ( ~ ) verges re er certtes, evtos fro the verge re eceegly sll (e.g. reltve error / N ~ ). I ths set of lectures I wll scuss: - Itroucto to sttstcs: verges str evtos - Cotuous strbutos, the orl or Guss strbuto - Coutg possbltes to rrve t probbltes - A relevt stt- ech eple: the bol strbuto Itroucto to sttstcs To trouce bsc ssues let us scuss sple eple: Throwg ce If you throw e, you wll get the result of,, 3, 4, 5, 6 ech throw. Suppose we throw the ce y tes (6) You get Possble results: =,,3, 4,5, 6 For fr ce, the rw of ech uber hs equl chce, the probbltes to throw 4 s / Totl 6 Chpter : Probblty Sttstcs 7
2 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs We c efe the frctol occurrece s f 98 f = e. f =, f = etc tot I the lt of lrge uber of throws, ths uber wll pproch the probblty P of 6 Hece l f P = tot 6 We woul epect totp But the ctul ubers woul fluctute rou the epecte Averge: If the possble outcoes for vul eperet re, the uber of evets s the b = = = f tot = tot l f = P = tot A For ce: verge = ( ) = = Note tht the verge vlue y ot be possble result! = = ll ottos of verge Vrce: We re lso tereste the spre rou the e.. let us efe Vrce: = tot ( ) l = f P tot = P ( ) = P + P P P = + Use P = = P + = = P = = tot P = = = = tot tot tot Chpter : Probblty Sttstcs 8
3 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Ths verge of fro us the verge of squre s lwys greter th. Ths s esly see P, the se vlue P ( ). The spre s oly f every eperet yels You c esly verfy (for the eple gve) tht both wys of clcultg, ( ) P Str Devto:, yels the se result. Here we hve prove tht they re lwys the se. = The bove results cosere screte outcoes of cert eperet, =,,3..., but the lyss c be geerlze to cotuous strbutos. Cotuous Dstrbutos Let s coser cotuous strbuto p( ). Ths ght represet for eple ss strbuto log D le. Mss esty of leves ρ = ss betwee + b ρ ( ) = M the totl ss ρ s the ss betwee b For our curret purposes t s ore coveet to orlze ( ) P( ) M ρ = = b Such tht P s the frcto of the totl ss lyg the tervl [,b] P( ) s esoless qutty s clle the strbuto fucto over the vrble, t s logous to P = P( ). Here serves s our vrble. = P logous to P = P logous to P Chpter : Probblty Sttstcs 9
4 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs P = = = P Eple: ( ) = posto = = p p Guss Dstrbuto (D) A fous strbuto tht we wll ecouter ore ofte s the Guss or orl strbuto / G = Ce : the wth of the strbuto (wll be show to be ) C : orlzto costt / G C e = = Also C = = (see below for ervto) π π / = e π = g () g( ) g s o fucto = / = e = π π / e π π = = ( show below). = = = s cle before Useful Guss Itegrl forul (ws use bove) k α y ( k ) π y e y = k =,,3... ( k + ) k α α I ths cse k =, α = Chpter : Probblty Sttstcs
5 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs (Plese ote the tegrto rge. For eve tegrs (w.r.t. ) oe c tke twce the result for the full tegrto rge.) Let us evlute crefully: = π e / = π e / = π π = Nothg essetl chges by shftg the u the strbuto wy fro ( y ) k e ( k ) y =. α y Coutg evets to etere probbltes k+ α k A bsc strtegy to etere probbltes s s follows # of evets of terest P = totl # of possble evets Here we ssue ech evet tself to be eqully lkely. Eg. Throw co or ce or rwg cr fro eck To llustrte I wll use eples usg pck of crs: 4 suts: clubs, os, herts, spes 3 crs:, 3, 4, 5, 6, 7, 8, 9,, J, Q, K, A 5 crs totl ) Drw sequece of 5 crs ( poker h) where the orer oes tter # of st cr possbltes 5 # of cr possbltes 5 3 r 5 : : Hece the uber of possbltes of poker h where the orer oes tter s 5! = 47! ) Wht bout the uber of peruttos the 5 crs where the orer oes tter # of st cr possbltes 5 # of cr possbltes 4 3 r 3 4 th 5 th # of peruttos (fferet cobtos) = = 5! 3) Fro the prevous results: rwg sequece of 5 crs, where orer oes ot tter π α Chpter : Probblty Sttstcs
6 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs ! 5! # of sequeces = = = ! 5! 47!5! 5 Ths c lso be wrtte s or C (5,5) 5 choose 5 5 So geerl, f we re to choose objects fro N totl objects, the orer of the cobtos oes tter N! N! Nuber of cobtos = The ore portt cse for us: If the orer of the cobto oes ot tter the N N! Nuber of cobtos = =! ( N )! Soe ore vce eples ) How y cobtos of 3 Quees + o Quee re there? There s Qs, Qh, Qc, Q. 4 4! 4 4 possbltes choose 3, = = = 4 3 3!! So there re 4 wys to rw 3 quees out of 4 A the o quees? There re 48 other crs (f you ot the lst quee) we pck, so we 48 get - ow wht s the probblty to rw 3 quees o quees? Prob = 5 5 = (# of rw 5) # of 3 Q's (# of other) b) wht bout y trple ( + z + y)? ote tht z=y s clue but =y =z s eclue Prob = full house? ( + yy) Chpter : Probblty Sttstcs
7 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Prob = 5 5 You c check these results o goo poker wk pge! Bol Coeffcets (Dstrbuto) ( + b) = ( + b)( + b)( + b)...( + b) ( + b) = b = The bol / choose coeffcets for so clle Pscl trgle + b + b + b = + b+ b b = + 3 b+ 3b + b You c scer the ptter strtg fro the top row Rtolzto: = b C b + = Drw tes out of, the orer oes ot tter C = Specl cse = p, p, b= q= p ( p+ q) = = = p q = P = p q P = the probblty to rw p tes whe rwg tes totl Chpter : Probblty Sttstcs 3
8 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Applcto of the bol Strlg s Appoto = # of prtcles o p se ( V L ) N = # of prtcles o q se ( V R ) = Np whch hppes to lso hve the hghest probblty 5! quckly becoes very lrge uber. I stt- ech ght be! For lrge eough uber Strlg s pproto s ccurte where s teger (screte vlue) l! l for screte vlues l! l for cotuous vlues l! ( l ) = l + = l Dervg Strlg s Approto ( ) ( ) ( ) l! = l... = l + l + l... + l = l for screte = If we go to cotuous we c replce the su wth tegrl Chpter : Probblty Sttstcs 4
9 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs ( l ) = l I sury l = l = Δ l = l = l = = = = l l+ l (f s bg, ths pproto works very well) l! l l! l Guss Approto to bol strbuto for lrge (ot so portt, teous. Wll show usg Mtlb). For p+ q= p, we h the bol strbuto P p q = It s frly esy to show (see et pge) tht for lrge N, ths pproches the Guss strbuto where = Np = Npq PN e = e π π ( ) / ( Np) / Ths strbuto becoes cresgly rrow or hghly peke s creses. By ths we e tht N = Npq Np = q p N For eple f boes p= q=, we hve = prtcles whe strbutg the prtcles over verge uber of prtcles the left bo : Str evto: Npq = = So we epect the uber of prtcles to be ± Chpter : Probblty Sttstcs 5
10 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs To be precse, for Guss strbuto to f result to be wth the e ± s 66.6 %, whle there s 99.9 % probblty to f the result betwee the e ±3. The portt pot s tht =!!! We woul ke very sll errors ssug ectly prtcles ech bo. Fluctutos re of orer ~ Dervto of the Guss Dstrbuto fro the bol strbuto N Let us ssue Tylor seres epso of l P rou ts u l PN = l PN + l PN ( ) l P N ( ) N N N! N sce PN = p q = p q! ( N )! l P N () = l N! -l! l( N )! + l p + ( N )lq We kow fro Strlg s l! = l l! = l ( ) l ( N )! = N l N! = N So the st ervtves re l PN = l + l ( N ) + l p l q= t the u Note: ( l N!) = l P N N + = + = = N N N N At the u of the strbuto the frst ervtve goes to l P () = l + l N N + l p lq = N p l = q N p = q Np p = q Chpter : Probblty Sttstcs 6
11 Wter 4 Che 35: Sttstcl Mechcs Checl Ketcs Np ( p q) = + = = = Np So the u s fou to be = Np. Ths s the epecte verge. Lookg t the seco ervtve N Usg = = Np = = Np N Np Np p Npq ( ) ( ) P N () = = Npq Gog bck to the tylor epso P l PN = l PN + l PN ( ) l P N ( ) l P N = l P N ( Npq ) +... P N e l P N () e ( ) / Npq ( ) /Npq PN = Ce where C = (orlzto costt) Npqπ Averge = = Np, vrce = = Npq Note e: Ths proof (wely quote tet books) s pretty b. You coul show ths wy (gog to seco orer oly) tht y strbuto wth u s Guss strbuto, whch s osese. Oe relly hs to show tht the hgher ervtves re eglgble. We t. I the coputer lb, we wll ke the coprso o coputer, you wll see tht ths pproto s ecellet. As so ofte the result s correct, the correct ervto s lckg. I t f t y otes. Perhps, f you uerst why ths proof s b, ths tself s useful thg to ler! Chpter : Probblty Sttstcs 7
Chapter 2 Intro to Math Techniques for Quantum Mechanics
Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
More information24 Concept of wave function. x 2. Ae is finite everywhere in space.
4 Cocept of wve fucto Chpter Cocept of Wve Fucto. Itroucto : There s lwys qutty sscocte wth y type of wves, whch vres peroclly wth spce te. I wter wves, the qutty tht vres peroclly s the heght of the wter
More informationthis is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]
Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow
More informationChapter 2 Intro to Math Techniques for Quantum Mechanics
Fll 4 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
More informationCURVE FITTING LEAST SQUARES METHOD
Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued
More informationMTH 146 Class 7 Notes
7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg
More informationChapter 3 Supplemental Text Material
S3-. The Defto of Fctor Effects Chpter 3 Supplemetl Text Mterl As oted Sectos 3- d 3-3, there re two wys to wrte the model for sglefctor expermet, the mes model d the effects model. We wll geerlly use
More informationSequences and summations
Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course
More informationArea and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
More informationOptimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek
Optmlt of Strteges for Collpsg Expe Rom Vrles Smple Rom Smple E Stek troucto We revew the propertes of prectors of ler comtos of rom vrles se o rom vrles su-spce of the orgl rom vrles prtculr, we ttempt
More informationICS141: Discrete Mathematics for Computer Science I
Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll
More informationRandom variables and sampling theory
Revew Rdom vrbles d smplg theory [Note: Beg your study of ths chpter by redg the Overvew secto below. The red the correspodg chpter the textbook, vew the correspodg sldeshows o the webste, d do the strred
More informationunder the curve in the first quadrant.
NOTES 5: INTEGRALS Nme: Dte: Perod: LESSON 5. AREAS AND DISTANCES Are uder the curve Are uder f( ), ove the -s, o the dom., Prctce Prolems:. f ( ). Fd the re uder the fucto, ove the - s, etwee,.. f ( )
More informationPubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS
PubH 745: REGRESSION ANALSIS REGRESSION IN MATRIX TERMS A mtr s dspl of umbers or umercl quttes ld out rectgulr rr of rows d colums. The rr, or two-w tble of umbers, could be rectgulr or squre could be
More informationDATA FITTING. Intensive Computation 2013/2014. Annalisa Massini
DATA FITTING Itesve Computto 3/4 Als Mss Dt fttg Dt fttg cocers the problem of fttg dscrete dt to obt termedte estmtes. There re two geerl pproches two curve fttg: Iterpolto Dt s ver precse. The strteg
More informationThe z-transform. LTI System description. Prof. Siripong Potisuk
The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put
More informationChapter 7. Bounds for weighted sums of Random Variables
Chpter 7. Bouds for weghted sums of Rdom Vrbles 7. Itroducto Let d 2 be two depedet rdom vrbles hvg commo dstrbuto fucto. Htczeko (998 d Hu d L (2000 vestgted the Rylegh dstrbuto d obted some results bout
More informationExponents and Powers
EXPONENTS AND POWERS 9 Exponents nd Powers CHAPTER. Introduction Do you know? Mss of erth is 5,970,000,000,000, 000, 000, 000, 000 kg. We hve lredy lernt in erlier clss how to write such lrge nubers ore
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 information6.6 Moments and Centers of Mass
th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder
More informationA MULTISET-VALUED FIBONACCI-TYPE SEQUENCE
Avces Applctos Dscrete Mthemtcs Volume Number 008 Pes 85-95 Publshe Ole: Mrch 3 009 Ths pper s vlble ole t http://wwwpphmcom 008 Pushp Publsh House A MULTISET-VALUED IBONACCI-TYPE SEQUENCE TAMÁS KALMÁR-NAGY
More informationRoberto s Notes on Integral Calculus Chapter 4: Definite integrals and the FTC Section 2. Riemann sums
Roerto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2 Rem sums Wht you eed to kow lredy: The defto of re for rectgle. Rememer tht our curret prolem s how to compute the re of ple rego
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 informationChapter 4: Distributions
Chpter 4: Dstrbutos Prerequste: Chpter 4. The Algebr of Expecttos d Vrces I ths secto we wll mke use of the followg symbols: s rdom vrble b s rdom vrble c s costt vector md s costt mtrx, d F m s costt
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 informationAvailable online through
Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo
More information10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n
0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke
More informationAnswer: First, I ll show how to find the terms analytically then I ll show how to use the TI to find them.
. CHAPTER 0 SEQUENCE, SERIES, d INDUCTION Secto 0. Seqece A lst of mers specfc order. E / Fd the frst terms : of the gve seqece: Aswer: Frst, I ll show how to fd the terms ltcll the I ll show how to se
More informationIntroduction to mathematical Statistics
Itroducto to mthemtcl ttstcs Fl oluto. A grou of bbes ll of whom weghed romtely the sme t brth re rdomly dvded to two grous. The bbes smle were fed formul A; those smle were fed formul B. The weght gs
More informationSt John s College. UPPER V Mathematics: Paper 1 Learning Outcome 1 and 2. Examiner: GE Marks: 150 Moderator: BT / SLS INSTRUCTIONS AND INFORMATION
St Joh s College UPPER V Mthemtcs: Pper Lerg Outcome d ugust 00 Tme: 3 hours Emer: GE Mrks: 50 Modertor: BT / SLS INSTRUCTIONS ND INFORMTION Red the followg structos crefull. Ths questo pper cossts of
More information12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions
HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Itertve Methods er Systems: Guss-Sedel Noler Systems Cse Study: Chemcl Rectos Itertve or ppromte methods or systems o equtos cosst o guessg vlue d the
More informationMore Regression Lecture Notes CE 311K - McKinney Introduction to Computer Methods Department of Civil Engineering The University of Texas at Austin
More Regresso Lecture Notes CE K - McKe Itroducto to Coputer Methods Deprtet of Cvl Egeerg The Uverst of Tes t Aust Polol Regresso Prevousl, we ft strght le to os dt (, ), (, ), (, ) usg the lest-squres
More informationITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS
Numercl Alyss for Egeers Germ Jord Uversty ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS Numercl soluto of lrge systems of ler lgerc equtos usg drect methods such s Mtr Iverse, Guss
More informationIn Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is
Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I
More informationMethods for solving the radiative transfer equation. Part 3: Discreteordinate. 1. Discrete-ordinate method for the case of isotropic scattering.
ecture Metos for sov te rtve trsfer equto. rt 3: Dscreteorte eto. Obectves:. Dscrete-orte eto for te cse of sotropc sctter..geerzto of te screte-orte eto for ooeeous tospere. 3. uerc peetto of te screte-orte
More informationFibonacci and Lucas Numbers as Tridiagonal Matrix Determinants
Rochester Isttute of echology RI Scholr Wors Artcles 8-00 bocc d ucs Nubers s rdgol trx Deterts Nth D. Chll Est Kod Copy Drre Nry Rochester Isttute of echology ollow ths d ddtol wors t: http://scholrwors.rt.edu/rtcle
More informationCS321. Numerical Analysis
CS3 Nuercl Alss Lecture 7 Lest Sures d Curve Fttg Professor Ju Zhg Deprtet of Coputer Scece Uverst of Ketuc Legto KY 456 633 Deceer 4 Method of Lest Sures Coputer ded dt collectos hve produced treedous
More informationRegression. By Jugal Kalita Based on Chapter 17 of Chapra and Canale, Numerical Methods for Engineers
Regresso By Jugl Klt Bsed o Chpter 7 of Chpr d Cle, Numercl Methods for Egeers Regresso Descrbes techques to ft curves (curve fttg) to dscrete dt to obt termedte estmtes. There re two geerl pproches two
More informationC.11 Bang-bang Control
Itroucto to Cotrol heory Iclug Optmal Cotrol Nguye a e -.5 C. Bag-bag Cotrol. Itroucto hs chapter eals wth the cotrol wth restrctos: s boue a mght well be possble to have scotutes. o llustrate some of
More information, we would have a series, designated as + j 1
Clculus sectio 9. Ifiite Series otes by Ti Pilchowski A sequece { } cosists of ordered set of ubers. If we were to begi ddig the ubers of sequece together s we would hve series desigted s. Ech iteredite
More informationA Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares
Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- A Techque for Costructg Odd-order Mgc Squres Usg Bsc Lt Squres Tomb I. Deprtmet of Mthemtcs, Mpur Uversty, Imphl, Mpur (INDIA) tombrom@gml.com
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 informationStrategies for the AP Calculus Exam
Strteges for the AP Clculus Em Strteges for the AP Clculus Em Strtegy : Kow Your Stuff Ths my seem ovous ut t ees to e metoe. No mout of cochg wll help you o the em f you o t kow the mterl. Here s lst
More informationChapter Unary Matrix Operations
Chpter 04.04 Ury trx Opertos After redg ths chpter, you should be ble to:. kow wht ury opertos mes,. fd the trspose of squre mtrx d t s reltoshp to symmetrc mtrces,. fd the trce of mtrx, d 4. fd the ermt
More informationAnalytical Approach for the Solution of Thermodynamic Identities with Relativistic General Equation of State in a Mixture of Gases
Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Volume, Issue 5, September 204, PP 6-0 ISSN 2349-7874 (Prt) & ISSN 2349-7882 (Ole) www.rcourls.org Alytcl Approch for the Soluto of Thermodymc Idettes
More informationDifferential Entropy 吳家麟教授
Deretl Etropy 吳家麟教授 Deto Let be rdom vrble wt cumultve dstrbuto ucto I F s cotuous te r.v. s sd to be cotuous. Let = F we te dervtve s deed. I te s clled te pd or. Te set were > 0 s clled te support set
More informationMathematics HL and further mathematics HL formula booklet
Dplom Progrmme Mthemtcs HL d further mthemtcs HL formul boolet For use durg the course d the emtos Frst emtos 04 Publshed Jue 0 Itertol Bcclurete Orgzto 0 5048 Mthemtcs HL d further mthemtcs formul boolet
More informationChapter Gauss-Seidel Method
Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos
More informationAPPLICATION OF THE CHEBYSHEV POLYNOMIALS TO APPROXIMATION AND CONSTRUCTION OF MAP PROJECTIONS
APPLICATION OF THE CHEBYSHEV POLYNOMIALS TO APPROXIMATION AND CONSTRUCTION OF MAP PROJECTIONS Pweł Pędzch Jerzy Blcerz Wrsw Uversty of Techology Fculty of Geodesy d Crtogrphy Astrct Usully to pproto of
More informationr = cos θ + 1. dt ) dt. (1)
MTHE 7 Proble Set 5 Solutions (A Crdioid). Let C be the closed curve in R whose polr coordintes (r, θ) stisfy () Sketch the curve C. r = cos θ +. (b) Find pretriztion t (r(t), θ(t)), t [, b], of C in polr
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 informationQuantum Mechanics Homework Solutions. 1.5 (page 16) Show that the z component of angular momentum for a point particle
Qutu Mechcs Hoework Solutos.5 (pge 6) Show tht the z copoet of gulr oetu for pot prtcle Lz p p whe epressed sphercl coordtes becoes Lz p r s. rcoss rss r(cos s ) r( s ) s r cos cos r(s s ) r(cos ) s r
More informationNumerical Differentiation and Integration
Numerl Deretto d Itegrto Overvew Numerl Deretto Newto-Cotes Itegrto Formuls Trpezodl rule Smpso s Rules Guss Qudrture Cheyshev s ormul Numerl Deretto Forwrd te dvded deree Bkwrd te dvded deree Ceter te
More informationTaylor 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 information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More information1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.
SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,
More informationUNIT 7 RANK CORRELATION
UNIT 7 RANK CORRELATION Rak Correlato Structure 7. Itroucto Objectves 7. Cocept of Rak Correlato 7.3 Dervato of Rak Correlato Coeffcet Formula 7.4 Te or Repeate Raks 7.5 Cocurret Devato 7.6 Summar 7.7
More information14.2 Line Integrals. determines a partition P of the curve by points Pi ( xi, y
4. Le Itegrls I ths secto we defe tegrl tht s smlr to sgle tegrl except tht sted of tegrtg over tervl [ ] we tegrte over curve. Such tegrls re clled le tegrls lthough curve tegrls would e etter termology.
More informationTopic 4 Fourier Series. Today
Topic 4 Fourier Series Toy Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz) st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will
More informationReview of Linear Algebra
PGE 30: Forulto d Soluto Geosstes Egeerg Dr. Blhoff Sprg 0 Revew of Ler Alger Chpter 7 of Nuercl Methods wth MATLAB, Gerld Recktewld Vector s ordered set of rel (or cople) uers rrged s row or colu sclr
More informationCS473-Algorithms I. Lecture 3. Solving Recurrences. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorthms I Lecture 3 Solvg Recurreces Cevdet Aykt - Blket Uversty Computer Egeerg Deprtmet Solvg Recurreces The lyss of merge sort Lecture requred us to solve recurrece. Recurreces re lke solvg
More informationThe definite Riemann integral
Roberto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4 The defte Rem tegrl Wht you eed to kow lredy: How to ppromte the re uder curve by usg Rem sums. Wht you c ler here: How to use
More informationME 501A Seminar in Engineering Analysis Page 1
Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt
More informationAsymptotic Statistical Analysis on Special Manifolds (Stiefel and Grassmann manifolds) Yasuko Chikuse Faculty of Engineering Kagawa University Japan
Asyptotc Sttstcl Alyss o Specl Mfols Stefel Grss fols Ysuo Chuse Fculty of Egeerg Kgw Uversty Jp [ I ] Stefel fol V ef { -fres R ; -fre set of orthoorl vectors R } ; ' I }. { Deso of V +. Ex: O V : orthogol
More informationMathematics HL and further mathematics HL formula booklet
Dplom Progrmme Mthemtcs HL d further mthemtcs HL formul boolet For use durg the course d the emtos Frst emtos 04 Edted 05 (verso ) Itertol Bcclurete Orgzto 0 5048 Cotets Pror lerg Core 3 Topc : Algebr
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 informationIntegration by Parts for D K
Itertol OPEN ACCESS Jourl Of Moder Egeerg Reserc IJMER Itegrto y Prts for D K Itegrl T K Gr, S Ry 2 Deprtmet of Mtemtcs, Rgutpur College, Rgutpur-72333, Purul, West Begl, Id 2 Deprtmet of Mtemtcs, Ss Bv,
More informationMath 153: Lecture Notes For Chapter 1
Mth : Lecture Notes For Chpter Sectio.: Rel Nubers Additio d subtrctios : Se Sigs: Add Eples: = - - = - Diff. Sigs: Subtrct d put the sig of the uber with lrger bsolute vlue Eples: - = - = - Multiplictio
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 informationA Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics by Walter Gottschalk
The Sgm Summto Notto #8 of Gottschlk's Gestlts A Seres Illustrtg Iovtve Forms of the Orgzto & Exposto of Mthemtcs by Wlter Gottschlk Ifte Vsts Press PVD RI 00 GG8- (8) 00 Wlter Gottschlk 500 Agell St #44
More informationMATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
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 informationDstrbuto Boltzm he gves d dstrbuto Boltzm s hs bth het wth cotct system tht probblty stte prtculr be should temperture t. low At system. sttes ll lbel
Dr Roger Beett R.A.Beett@Redg.c.uk Rm. 3 Xt. 8559 Lecture 19 Dstrbuto Boltzm he gves d dstrbuto Boltzm s hs bth het wth cotct system tht probblty stte prtculr be should temperture t. low At system. sttes
More informationZ = = = = X np n. n n. npq. npq pq
Stt 4, secto 4 Goodess of Ft Ctegory Probbltes Specfed otes by Tm Plchowsk Recll bck to Lectures 6c, 84 (83 the 8 th edto d 94 whe we delt wth populto proportos Vocbulry from 6c: The pot estmte for populto
More informationA Brief Introduction to Olympiad Inequalities
Ev Che Aprl 0, 04 The gol of ths documet s to provde eser troducto to olympd equltes th the stdrd exposto Olympd Iequltes, by Thoms Mldorf I ws motvted to wrte t by feelg gulty for gettg free 7 s o problems
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 informationSUM PROPERTIES FOR THE K-LUCAS NUMBERS WITH ARITHMETIC INDEXES
Avlble ole t http://sc.org J. Mth. Comput. Sc. 4 (04) No. 05-7 ISSN: 97-507 SUM PROPERTIES OR THE K-UCAS NUMBERS WITH ARITHMETIC INDEXES BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA * School of
More informationThe limit comparison test
Roerto s Notes o Ifiite Series Chpter : Covergece tests Sectio 4 The limit compriso test Wht you eed to kow lredy: Bsics of series d direct compriso test. Wht you c ler here: Aother compriso test tht does
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 informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationMathematically, integration is just finding the area under a curve from one point to another. It is b
Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom
More information6. Chemical Potential and the Grand Partition Function
6. Chemcl Potetl d the Grd Prtto Fucto ome Mth Fcts (see ppedx E for detls) If F() s lytc fucto of stte vrles d such tht df d pd the t follows: F F p lso sce F p F we c coclude: p I other words cross dervtves
More informationAn Alternative Method to Find the Solution of Zero One Integer Linear Fractional Programming Problem with the Help of -Matrix
Itertol Jourl of Scetfc d Reserch Pulctos, Volue 3, Issue 6, Jue 3 ISSN 5-353 A Altertve Method to Fd the Soluto of Zero Oe Iteger Ler Frctol Progrg Prole wth the Help of -Mtr VSeeregsy *, DrKJeyr ** *
More informationA Level Mathematics Transition Work. Summer 2018
A Level Mthetics Trsitio Work Suer 08 A Level Mthetics Trsitio A level thetics uses y of the skills you developed t GCSE. The big differece is tht you will be expected to recogise where you use these skills
More informationCOMPLEX NUMBERS AND DE MOIVRE S THEOREM
COMPLEX NUMBERS AND DE MOIVRE S THEOREM OBJECTIVE PROBLEMS. s equl to b d. 9 9 b 9 9 d. The mgr prt of s 5 5 b 5. If m, the the lest tegrl vlue of m s b 8 5. The vlue of 5... s f s eve, f s odd b f s eve,
More informationCS321. Numerical Analysis
CS Numercl Alyss Lecture 4 Numercl Itegrto Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 456 6 Octoer 6, 5 Dete Itegrl A dete tegrl s tervl or tegrto. For ed tegrto tervl, te result
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 informationPhys101 Lecture 4,5 Dynamics: Newton s Laws of Motion
Phys101 Lecture 4,5 Dynics: ewton s Lws of Motion Key points: ewton s second lw is vector eqution ction nd rection re cting on different objects ree-ody Digrs riction Inclines Ref: 4-1,2,3,4,5,6,7,8,9.
More informationSTRAND B: NUMBER THEORY
Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,
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 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 informationChapter 2. LOGARITHMS
Chpter. LOGARITHMS Dte: - 009 A. INTRODUCTION At the lst hpter, you hve studied bout Idies d Surds. Now you re omig to Logrithms. Logrithm is ivers of idies form. So Logrithms, Idies, d Surds hve strog
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 informationFourier Series. Topic 4 Fourier Series. sin. sin. Fourier Series. Fourier Series. Fourier Series. sin. b n. a n. sin
Topic Fourier Series si Fourier Series Music is more th just pitch mplitue it s lso out timre. The richess o sou or ote prouce y musicl istrumet is escrie i terms o sum o umer o istict requecies clle hrmoics.
More informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
More informationStat 6863-Handout 5 Fundamentals of Interest July 2010, Maurice A. Geraghty
S 6863-Hou 5 Fuels of Ieres July 00, Murce A. Gerghy The pror hous resse beef cl occurreces, ous, ol cls e-ulero s ro rbles. The fl copoe of he curl oel oles he ecooc ssupos such s re of reur o sses flo.
More informationCS321. Introduction to Numerical Methods
CS Itroducto to Numercl Metods Lecture Revew Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 6 6 Mrc 7, Number Coverso A geerl umber sould be coverted teger prt d rctol prt seprtely
More informationMATRIX AND VECTOR NORMS
Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the
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 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 information