Mathematics HL and Further mathematics HL Formula booklet
|
|
- Elmer Ethelbert Stanley
- 5 years ago
- Views:
Transcription
1 Dploma Programme Mathematcs HL ad Further mathematcs HL Formula booklet For use durg the course ad the eamatos Frst eamatos 04 Mathematcal Iteratoal Baccalaureate studes SL: Formula Orgazato booklet 0
2 CONTENTS Formulae Pror learg Topc Core: Algebra Topc Core: Fuctos ad equatos 3 Topc 3 Core: Crcular fuctos ad trgoometry 4 Topc 4 Core: Vectors 5 Topc 5 Core: Statstcs ad probablty 7 Topc 6 Core: Calculus 9 Topc 7 Opto: Statstcs ad probablty (further mathematcs HL topc 3) Topc 8 Opto: Sets, relatos ad groups (further mathematcs HL topc 4) 3 Topc 9 Opto: Calculus (further mathematcs HL topc 5) 4 Topc 0 Opto: Dscrete mathematcs (further mathematcs HL topc 6) 5 Formulae for dstrbutos (topc 5.6, 5.7, 7., further mathematcs HL topc 3.) 6 Dscrete dstrbutos 6 Cotuous dstrbutos 7 Further Mathematcs Topc Lear algebra 8 Iteratoal Baccalaureate Orgazato 0
3 Mathematcal studes SL: Formula booklet 3
4
5 Formulae Pror learg Area of a parallelogram A b h, where b s the base, h s the heght Area of a tragle Area of a trapezum Area of a crcle A ( bh ), where b s the base, h s the heght A ( a b ) h, where a ad b are the parallel sdes, h s the heght A r, where r s the radus Crcumferece of a crcle C r, where r s the radus Volume of a pyramd V (area of base vertcal heght) 3 Volume of a cubod V l w h, where l s the legth, w s the wdth, h s the heght Volume of a cylder V r h, where r s the radus, h s the heght Area of the curved surface of a cylder Volume of a sphere A rh, where r s the radus, h s the heght 4 V 3 3 r, where r s the radus Volume of a coe V 3 r h, where r s the radus, h s the heght Dstace betwee two pots (, y) ad (, y ) Coordates of the mdpot of a le segmet wth edpots (, y ) ad (, y ) d ( ) ( y y ) y y, Solutos of a quadratc equato The solutos of a b c 0 are b b 4ac a Iteratoal Baccalaureate Orgazato 004
6 Topc Core: Algebra. The th term of a arthmetc sequece The sum of terms of a arthmetc sequece The th term of a geometrc sequece u u ( ) d S ( u ( ) d) ( u u ) u u r The sum of terms of a fte geometrc sequece The sum of a fte geometrc sequece S u( r ) u( r ) rr u S, r r, r. Epoets ad logarthms.3 Combatos a b log b, where a0, b 0 a e l a loga log a a a logc a logb a log b! r r!( r)! c a Permutatos Pr! ( r)! Bomal theorem ( a b) a a b a b b r r r.5 Comple umbers z a b r (cos s ) re r cs.7 De Movre s theorem r(cos s ) r (cos s ) r e r cs Iteratoal Baccalaureate Orgazato 004
7 Topc Core: Fuctos ad equatos.5 As of symmetry of the graph of a quadratc fucto y a b c as of symmetry b a.6 Dscrmat b 4ac Iteratoal Baccalaureate Orgazato 004 3
8 Topc 3 Core: Crcular fuctos ad trgoometry 3. Legth of a arc l r, where s the agle measured radas, r s the radus Area of a sector 3. Idettes Pythagorea dettes 3.3 Compoud agle dettes radus A r, where s the agle measured radas, r s the s ta cos sec cos cosec = s cos s ta sec cot csc s( A B) s Acos B cos As B cos( A B) cos Acos B s As B ta A ta B ta( AB) ta Ata B Double agle dettes s s cos cos cos s cos s ta ta ta 3.7 Cose rule Se rule Area of a tragle a b c ab c a b abcos C; cosc a b c s A s B sc A absc 4 Iteratoal Baccalaureate Orgazato 004
9 Topc 4 Core: Vectors 4. Magtude of a vector v 3 v v v, where v v v v3 Dstace betwee two pots (, y, z ) ad (, y, z ) d ( ) ( y y ) ( z z ) Coordates of the mdpot of a le segmet wth edpots (, y, z ), (, y, z ) y, y z, z 4. Scalar product v w v w cos, where s the agle betwee v ad w v w v w v w v w, where 3 3 v v v, v3 w w w w3 Agle betwee two vectors cos v w v w v w vw Vector equato of a le r = a +λb Parametrc form of the equato of a le Cartesa equatos of a le l, y y m, z z y y z z l m Vector product vw3 v3w v w v3w vw3 where vw vw v v v, v3 w w w w3 v w v w s, where s the agle betwee v ad w Area of a tragle 4.6 Vector equato of a plae Equato of a plae (usg the ormal vector) A vw where v ad w form two sdes of a tragle r = a + λb + c r a Cartesa equato of a a by cz d Iteratoal Baccalaureate Orgazato 004 5
10 plae 6 Iteratoal Baccalaureate Orgazato 004
11 Topc 5 Core: Statstcs ad probablty 5. Let k f Populato parameters Mea Varace k f k f f k Stadard devato k f 5. Probablty of a evet A ( ) P( A) A U ( ) Complemetary evets P( A) P( A ) 5.3 Combed evets P( A B) P( A) P( B) P( A B ) Mutually eclusve evets P( A B) P( A) P( B ) Iteratoal Baccalaureate Orgazato 004 7
12 Topc 5 Core: Statstcs ad probablty (cotued) 5.4 Codtoal probablty P AB P( A B) P( B) Idepedet evets P( A B) P( A) P( B ) Bayes Theorem P B A P( B)P A B P( B)P A B P( B )P A B P( A B) P( B) P( B A) P( A B ) P( B ) P( A B ) P( B ) P( A B ) P( B ) 5.5 Epected value of a dscrete radom varable X Epected value of a cotuous radom varable X E( X ) P( X ) E( X ) f ( )d Varace Var( X ) E( X ) E( X ) E( X ) Varace of a dscrete radom varable X Varace of a cotuous radom varable X 5.6 Bomal dstrbuto Mea Varace Posso dstrbuto Mea Varace 5.7 Stadardzed ormal varable Var( X ) ( ) P( X ) P( X ) Var( X ) ( ) f ( )d f ( )d ~ B(, ) P( ) ( ) X p X p p, 0,,, E( X ) p Var( X ) p( p ) m m e X ~ P o ( m) P( X ), 0,,,! E( X) m Var( X) m z 8 Iteratoal Baccalaureate Orgazato 004
13 Topc 6 Core: Calculus 6. Dervatve of f( ) Dervatve of d y f ( h) f ( ) y f ( ) f ( ) lm d h0 h f ( ) f ( ) Dervatve of s f ( ) s f ( ) cos Dervatve of cos f ( ) cos f ( ) s Dervatve of ta f ( ) ta f ( ) sec Dervatve of e f ( ) e f ( ) e Dervatve of l f ( ) l f ( ) Dervatve of sec f ( ) sec f ( ) sec ta Dervatve of csc f ( ) csc f ( ) csc cot Dervatve of cot f ( ) cot f ( ) csc Dervatve of a ( ) f a f ( ) a (l a ) Dervatve of log a f ( ) log a f ( ) l a Dervatve of arcs f ( ) arcs f ( ) Dervatve of arccos f ( ) arccos f ( ) Dervatve of arcta f ( ) arcta f ( ) Cha rule y g( u ), where dy dy du u f ( ) d du d Product rule Quotet rule dy dv du y uv u v d d d du dv v u u dy y d d v d v Iteratoal Baccalaureate Orgazato 004 9
14 Topc 6 Core: Calculus (cotued) 6.4 Stadard tegrals d C, d l C sd cos C cos d s C e de C a d a C l a d arcta C a a a a d arcs C, a a 6.5 Area uder a curve Volume of revoluto (rotato) A b yd or A d a b y a b b π d or π d a a V y V y 6.7 Itegrato by parts dv du u d uv d d v d or udv uv vdu 0 Iteratoal Baccalaureate Orgazato 004
15 Topc 7 Opto: Statstcs ad probablty (further mathematcs HL topc 3) 7. (3.) Probablty geeratg fucto for a dscrete radom varable X G( t) E( t X ) P( X ) t 7. (3.) Lear combatos of two depedet radom varables X, X a X a X a X a X a X a X a X a X E E E Var Var Var 7.3 (3.3) Sample statstcs Mea Varace s s k f k k f ( ) f Stadard devato s s k f ( ) Ubased estmate of populato varace s k k f ( ) f s s 7.5 (3.5) Cofdece tervals Mea, wth kow varace z Mea, wth ukow varace s t 7.6 (3.6) Test statstcs Mea, wth kow varace Mea, wth ukow z / Iteratoal Baccalaureate Orgazato 004
16 Iteratoal Baccalaureate Orgazato 004 varace / t s 7.7 Sample product momet correlato coeffcet Test statstc for H 0 : ρ = 0 Equato of regresso le of o y Equato of regresso le of y o y y y y r r r t ) ( y y y y y y ) ( y y y y
17 Topc 8 Opto: Sets, relatos ad groups (further mathematcs HL topc 4) 8. (4.) De Morga s laws ( A B) A B ( A B) A B Iteratoal Baccalaureate Orgazato 004 3
18 Topc 9 Opto: Calculus (further mathematcs HL topc 5) 9.6 (5.6) Maclaur seres Taylor seres Taylor appromatos (wth error term R () ) Lagrage form f ( ) f (0) f (0) f (0)! ( a) f ( ) f ( a) ( a) f ( a) f ( a )...! ( a) f f a a f a f a R! ( ) ( ) ( ) ( ) ( )... ( ) ( ) ( ) f () c R ( ) ( a) ( )!, where c les betwee a ad Maclaur seres for specal fuctos e...! 3 l( ) s... 3! 5! 4 cos...! 4! 3 5 arcta (5.5) Euler s method y y h f (, y ) ; h, where h s a costat (steplegth) Itegratg factor for yp( ) y Q( ) ( )d e P 4 Iteratoal Baccalaureate Orgazato 004
19 Topc 0 Opto: Dscrete mathematcs (further mathematcs HL topc 6) 0.7 (6.7) Euler s formula for coected plaar graphs Plaar, smple, coected graphs v e f, where v s the umber of vertces, e s the umber of edges, f s the umber of faces e3v6for v 3 ev4f the graph has o tragles Iteratoal Baccalaureate Orgazato 004 5
20 Formulae for dstrbutos (topc 5.6, 5.7, 7., further mathematcs HL topc 3.) Dscrete dstrbutos Dstrbuto Notato Probablty mass fucto Mea Varace Geometrc X ~ Geo p pq for,,... p q p Negatve bomal ~ NB, X r p r pq r r for r, r,... r p rq p 6 Iteratoal Baccalaureate Orgazato 004
21 Cotuous dstrbutos Dstrbuto Notato Probablty desty fucto Mea Varace Normal X ~ N, e π Iteratoal Baccalaureate Orgazato 004 7
22 Further Mathematcs Topc Lear algebra. Determat of a matr Iverse of a matr Determat of a 3 3 matr a b A det A A ad bc c d a b d b A A, ad bc c d det A c a a b c e f d f d e A d e f det A a b c h k g k g h g h k 8 Iteratoal Baccalaureate Orgazato 004
23 Iteratoal Baccalaureate Orgazato 004 9
INTERNATIONAL BACCALAUREATE ORGANIZATION GROUP 5 MATHEMATICS FORMULAE AND STATISTICAL TABLES
INTERNATIONAL BACCALAUREATE ORGANIZATION GROUP 5 MATHEMATICS ORMULAE AND STATISTICAL TABLES To be used the teachg ad eamato of: Mathematcs HL Mathematcal Methods SL Mathematcal Studes SL urther Mathematcs
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 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 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 Cotets Pror lerg Core Topc : Algebr Topc
More informationIFYMB002 Mathematics Business Appendix C Formula Booklet
Iteratoal Foudato Year (IFY IFYMB00 Mathematcs Busess Apped C Formula Booklet Related Documet: IFY Mathematcs Busess Syllabus 07/8 IFYMB00 Maths Busess Apped C Formula Booklet Cotets lease ote that the
More informationSpecial Instructions / Useful Data
JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth
More informationContinuous Distributions
7//3 Cotuous Dstrbutos Radom Varables of the Cotuous Type Desty Curve Percet Desty fucto, f (x) A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Test scores Desty Curve Percet Probablty Desty Fucto, f
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationLinear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan
Lear Regresso Hsao-Lug Cha Dept Electrcal Egeerg Chag Gug Uverst, Tawa chahl@mal.cgu.edu.tw Curve fttg Least-squares regresso Data ehbt a sgfcat degree of error or scatter A curve for the tred of the data
More informationMidterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..
More informationENGI 3423 Simple Linear Regression Page 12-01
ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable
More informationENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections
ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationLinear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan
Lear Regresso Hsao-Lug Cha Dept Electrcal Egeerg Chag Gug Uverst, Tawa chahl@mal.cgu.edu.tw Curve fttg Least-squares regresso Data ehbt a sgfcat degree of error or scatter A curve for the tred of the data
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationArithmetic Mean Suppose there is only a finite number N of items in the system of interest. Then the population arithmetic mean is
Topc : Probablty Theory Module : Descrptve Statstcs Measures of Locato Descrptve statstcs are measures of locato ad shape that perta to probablty dstrbutos The prmary measures of locato are the arthmetc
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More informationChapter 8: Statistical Analysis of Simulated Data
Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by Marquette Uversty MSCS600 Ageda 8. The Sample
More informationModule 7. Lecture 7: Statistical parameter estimation
Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato
More informationLinear Regression with One Regressor
Lear Regresso wth Oe Regressor AIM QA.7. Expla how regresso aalyss ecoometrcs measures the relatoshp betwee depedet ad depedet varables. A regresso aalyss has the goal of measurg how chages oe varable,
More informationSTATISTICAL INFERENCE
(STATISTICS) STATISTICAL INFERENCE COMPLEMENTARY COURSE B.Sc. MATHEMATICS III SEMESTER ( Admsso) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITY P.O., MALAPPURAM, KERALA, INDIA -
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationGenerative classification models
CS 75 Mache Learg Lecture Geeratve classfcato models Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Data: D { d, d,.., d} d, Classfcato represets a dscrete class value Goal: lear f : X Y Bar classfcato
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationMidterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uverst Mchael Bar Sprg 5 Mdterm xam, secto Soluto Thursda, Februar 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes exam.. No calculators of a kd are allowed..
More informationSTA302/1001-Fall 2008 Midterm Test October 21, 2008
STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from
More informationENGI 4421 Propagation of Error Page 8-01
ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationCS5620 Intro to Computer Graphics
CS56 Itro to Computer Graphcs Geometrc Modelg art II Geometrc Modelg II hyscal Sples Curve desg pre-computers Cubc Sples Stadard sple put set of pots { } =, No dervatves specfed as put Iterpolate by cubc
More informationTHE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA
THE ROYAL STATISTICAL SOCIETY 3 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER I STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad
More informationTest Paper-II. 1. If sin θ + cos θ = m and sec θ + cosec θ = n, then (a) 2n = m (n 2 1) (b) 2m = n (m 2 1) (c) 2n = m (m 2 1) (d) none of these
Test Paer-II. If s θ + cos θ = m ad sec θ + cosec θ =, the = m ( ) m = (m ) = m (m ). If a ABC, cos A = s B, the t s C a osceles tragle a eulateral tragle a rght agled tragle. If cos B = cos ( A+ C), the
More informationbest estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best
Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg
More informationRegresso What s a Model? 1. Ofte Descrbe Relatoshp betwee Varables 2. Types - Determstc Models (o radomess) - Probablstc Models (wth radomess) EPI 809/Sprg 2008 9 Determstc Models 1. Hypothesze
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationLaw of Large Numbers
Toss a co tmes. Law of Large Numbers Suppose 0 f f th th toss came up H toss came up T s are Beroull radom varables wth p ½ ad E( ) ½. The proporto of heads s. Itutvely approaches ½ as. week 2 Markov s
More informationSTA 105-M BASIC STATISTICS (This is a multiple choice paper.)
DCDM BUSINESS SCHOOL September Mock Eamatos STA 0-M BASIC STATISTICS (Ths s a multple choce paper.) Tme: hours 0 mutes INSTRUCTIONS TO CANDIDATES Do ot ope ths questo paper utl you have bee told to do
More informationChapter 13 Student Lecture Notes 13-1
Chapter 3 Studet Lecture Notes 3- Basc Busess Statstcs (9 th Edto) Chapter 3 Smple Lear Regresso 4 Pretce-Hall, Ic. Chap 3- Chapter Topcs Types of Regresso Models Determg the Smple Lear Regresso Equato
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More informationIntroduction to Probability
Itroducto to Probablty Nader H Bshouty Departmet of Computer Scece Techo 32000 Israel e-mal: bshouty@cstechoacl 1 Combatorcs 11 Smple Rules I Combatorcs The rule of sum says that the umber of ways to choose
More informationCorrelation and Simple Linear Regression
Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uverst Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationTHE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA
THE ROYAL STATISTICAL SOCIETY EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER II STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for
More informationSimple Linear Regression
Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uversty Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal
More informationLine Fitting and Regression
Marquette Uverst MSCS6 Le Fttg ad Regresso Dael B. Rowe, Ph.D. Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Coprght 8 b Marquette Uverst Least Squares Regresso MSCS6 For LSR we have pots
More informationReview Exam II Complex Analysis
Revew Exam II Complex Aalyss Uderled Propostos or Theorems: Proofs May Be Asked for o Exam Chapter 3. Ifte Seres Defto: Covergece Defto: Absolute Covergece Proposto. Absolute Covergece mples Covergece
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:
More informationSimple Linear Regression and Correlation.
Smple Lear Regresso ad Correlato. Correspods to Chapter 0 Tamhae ad Dulop Sldes prepared b Elzabeth Newto (MIT) wth some sldes b Jacquele Telford (Johs Hopks Uverst) Smple lear regresso aalss estmates
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationPGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation
PGE 30: Formulato ad Soluto Geosystems Egeerg Dr. Balhoff Iterpolato Numercal Methods wth MATLAB, Recktewald, Chapter 0 ad Numercal Methods for Egeers, Chapra ad Caale, 5 th Ed., Part Fve, Chapter 8 ad
More informationChapter 4 Multiple Random Variables
Revew o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for Chapter 4-5 Notes: Although all deftos ad theorems troduced our lectures ad ths ote are mportat ad you should be famlar wth, but I put those
More informationMu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
More informationMMJ 1113 FINITE ELEMENT METHOD Introduction to PART I
MMJ FINITE EEMENT METHOD Cotut requremets Assume that the fuctos appearg uder the tegral the elemet equatos cota up to (r) th order To esure covergece N must satsf Compatblt requremet the fuctos must have
More information( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model
Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch
More informationRandom Variate Generation ENM 307 SIMULATION. Anadolu Üniversitesi, Endüstri Mühendisliği Bölümü. Yrd. Doç. Dr. Gürkan ÖZTÜRK.
adom Varate Geerato ENM 307 SIMULATION Aadolu Üverstes, Edüstr Mühedslğ Bölümü Yrd. Doç. Dr. Gürka ÖZTÜK 0 adom Varate Geerato adom varate geerato s about procedures for samplg from a varety of wdely-used
More informationChapter 4 Multiple Random Variables
Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:
More informationUncertainty, Data, and Judgment
Ucertaty, Data, ad Judgmet Sesso 06 Structure of the Course Topc Sesso Probablty -5 Estmato 6-8 Hypothess Testg 9-10 Regresso 11-16 1 Mcrosoft AND Itel (50-50) You vest $,500 MSFT ad $,500 INTC X = Aual
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More informationSimulation Output Analysis
Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5
More informationCS 2750 Machine Learning Lecture 5. Density estimation. Density estimation
CS 750 Mache Learg Lecture 5 esty estmato Mlos Hausrecht mlos@tt.edu 539 Seott Square esty estmato esty estmato: s a usuervsed learg roblem Goal: Lear a model that rereset the relatos amog attrbutes the
More informationStudy of Correlation using Bayes Approach under bivariate Distributions
Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 Stud of Correlato usg Baes Approach uder bvarate Dstrbutos N.S.Padharkar* ad. M.N.Deshpade** *Govt.Vdarbha Isttute of
More informationParameter, Statistic and Random Samples
Parameter, Statstc ad Radom Samples A parameter s a umber that descrbes the populato. It s a fxed umber, but practce we do ot kow ts value. A statstc s a fucto of the sample data,.e., t s a quatty whose
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More informationBASICS ON DISTRIBUTIONS
BASICS ON DISTRIBUTIONS Hstograms Cosder a epermet whch dfferet outcomes are possble (e. Dce tossg). The probablty of all the outcomes ca be represeted a hstogram Dstrbutos Probabltes are descrbed wth
More informationα1 α2 Simplex and Rectangle Elements Multi-index Notation of polynomials of degree Definition: The set P k will be the set of all functions:
Smplex ad Rectagle Elemets Mult-dex Notato = (,..., ), o-egatve tegers = = β = ( β,..., β ) the + β = ( + β,..., + β ) + x = x x x x = x x β β + D = D = D D x x x β β Defto: The set P of polyomals of degree
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationConstruction and Evaluation of Actuarial Models. Rajapaksha Premarathna
Costructo ad Evaluato of Actuaral Models Raapaksha Premaratha Table of Cotets Modelg Some deftos ad Notatos...4 Case : Polcy Lmtu...4 Case : Wth a Ordary deductble....5 Case 3: Maxmum Covered loss u wth
More informationLecture Notes 2. The ability to manipulate matrices is critical in economics.
Lecture Notes. Revew of Matrces he ablt to mapulate matrces s crtcal ecoomcs.. Matr a rectagular arra of umbers, parameters, or varables placed rows ad colums. Matrces are assocated wth lear equatos. lemets
More informationSection 2 Notes. Elizabeth Stone and Charles Wang. January 15, Expectation and Conditional Expectation of a Random Variable.
Secto Notes Elzabeth Stoe ad Charles Wag Jauar 5, 9 Jot, Margal, ad Codtoal Probablt Useful Rules/Propertes. P ( x) P P ( x; ) or R f (x; ) d. P ( xj ) P (x; ) P ( ) 3. P ( x; ) P ( xj ) P ( ) 4. Baes
More informationLecture Notes Forecasting the process of estimating or predicting unknown situations
Lecture Notes. Ecoomc Forecastg. Forecastg the process of estmatg or predctg ukow stuatos Eample usuall ecoomsts predct future ecoomc varables Forecastg apples to a varet of data () tme seres data predctg
More informationExample: Multiple linear regression. Least squares regression. Repetition: Simple linear regression. Tron Anders Moger
Example: Multple lear regresso 5000,00 4000,00 Tro Aders Moger 0.0.007 brthweght 3000,00 000,00 000,00 0,00 50,00 00,00 50,00 00,00 50,00 weght pouds Repetto: Smple lear regresso We defe a model Y = β0
More informationDr. Shalabh. Indian Institute of Technology Kanpur
Aalyss of Varace ad Desg of Expermets-I MODULE -I LECTURE - SOME RESULTS ON LINEAR ALGEBRA, MATRIX THEORY AND DISTRIBUTIONS Dr. Shalabh Departmet t of Mathematcs t ad Statstcs t t Ida Isttute of Techology
More informationRandom Variables. ECE 313 Probability with Engineering Applications Lecture 8 Professor Ravi K. Iyer University of Illinois
Radom Varables ECE 313 Probablty wth Egeerg Alcatos Lecture 8 Professor Rav K. Iyer Uversty of Illos Iyer - Lecture 8 ECE 313 Fall 013 Today s Tocs Revew o Radom Varables Cumulatve Dstrbuto Fucto (CDF
More informationJAM 2015: General Instructions during Examination
JAM 05 JAM 05: Geeral Istructos durg Examato. Total durato of the JAM 05 examato s 80 mutes.. The clock wll be set at the server. The coutdow tmer at the top rght corer of scree wll dsplay the remag tme
More informationmeans the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.
9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc,
More informationMEASURES OF DISPERSION
MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda
More informationComparison of Dual to Ratio-Cum-Product Estimators of Population Mean
Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationCorrelation and Regression Analysis
Chapter V Correlato ad Regresso Aalss R. 5.. So far we have cosdered ol uvarate dstrbutos. Ma a tme, however, we come across problems whch volve two or more varables. Ths wll be the subject matter of the
More informationCS475 Parallel Programming
CS475 Parallel Programmg Deretato ad Itegrato Wm Bohm Colorado State Uversty Ecept as otherwse oted, the cotet o ths presetato s lcesed uder the Creatve Commos Attrbuto.5 lcese. Pheomea Physcs: heat, low,
More informationMean is only appropriate for interval or ratio scales, not ordinal or nominal.
Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot
More information12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model
1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed
More informationTHE ROYAL STATISTICAL SOCIETY 2010 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULE 2 STATISTICAL INFERENCE
THE ROYAL STATISTICAL SOCIETY 00 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA MODULE STATISTICAL INFERENCE The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the
More informationRandom Variables and Probability Distributions
Radom Varables ad Probablty Dstrbutos * If X : S R s a dscrete radom varable wth rage {x, x, x 3,. } the r = P (X = xr ) = * Let X : S R be a dscrete radom varable wth rage {x, x, x 3,.}.If x r P(X = x
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationChapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn:
Chapter 3 3- Busess Statstcs: A Frst Course Ffth Edto Chapter 2 Correlato ad Smple Lear Regresso Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic. Chap 2- Learg Objectves I ths chapter, you lear:
More informationTHE ROYAL STATISTICAL SOCIETY 2016 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5
THE ROYAL STATISTICAL SOCIETY 06 EAMINATIONS SOLUTIONS HIGHER CERTIFICATE MODULE 5 The Socety s provdg these solutos to assst cadtes preparg for the examatos 07. The solutos are teded as learg ads ad should
More informationProbability and. Lecture 13: and Correlation
933 Probablty ad Statstcs for Software ad Kowledge Egeers Lecture 3: Smple Lear Regresso ad Correlato Mocha Soptkamo, Ph.D. Outle The Smple Lear Regresso Model (.) Fttg the Regresso Le (.) The Aalyss of
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationCHAPTER 6. d. With success = observation greater than 10, x = # of successes = 4, and
CHAPTR 6 Secto 6.. a. We use the samle mea, to estmate the oulato mea µ. Σ 9.80 µ 8.407 7 ~ 7. b. We use the samle meda, 7 (the mddle observato whe arraged ascedg order. c. We use the samle stadard devato,
More informationSTA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1
STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal
More informationChapter Statistics Background of Regression Analysis
Chapter 06.0 Statstcs Backgroud of Regresso Aalyss After readg ths chapter, you should be able to:. revew the statstcs backgroud eeded for learg regresso, ad. kow a bref hstory of regresso. Revew of Statstcal
More informationC. Statistics. X = n geometric the n th root of the product of numerical data ln X GM = or ln GM = X 2. X n X 1
C. Statstcs a. Descrbe the stages the desg of a clcal tral, takg to accout the: research questos ad hypothess, lterature revew, statstcal advce, choce of study protocol, ethcal ssues, data collecto ad
More informationChapter 3 Sampling For Proportions and Percentages
Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys
More informationhp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations
HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several
More informationσ σ r = x i x N Statistics Formulas Sample Mean Population Mean Interquartile Range Population Variance Population Standard Deviation
Stattc Formula Samle Mea Poulato Mea µ Iterquartle Rae IQR Q 3 Q Samle Varace ( ) Samle Stadard Devato ( ) Poulato Varace Poulato Stadard Devato ( µ ) Coecet o Varato Stadard Devato CV 00% Mea ( µ ) -Score
More informationObjectives of Multiple Regression
Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of
More informationLecture Note to Rice Chapter 8
ECON 430 HG revsed Nov 06 Lecture Note to Rce Chapter 8 Radom matrces Let Y, =,,, m, =,,, be radom varables (r.v. s). The matrx Y Y Y Y Y Y Y Y Y Y = m m m s called a radom matrx ( wth a ot m-dmesoal dstrbuto,
More information