Incremental calculation of weighted mean and variance


 Cleopatra Pierce
 4 years ago
 Views:
Transcription
1 Icremetal calculatio of weighted mea ad variace Toy Fich Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically stable calculatio of the mea ad stadard deviatio, which are also suitable for icremetal olie calculatio. I the geeralize these formulae to weighted meas ad stadard deviatios. I upick the difficulties that arise whe geeralizig further to ormalized weights. Fially I show that the epoetially weighted movig average is a special case of the icremetal ormalized weighted mea formula, ad derive a formula for the epoetially weighted movig stadard deviatio. Simple mea Straightforward traslatio of equatio ito code ca suffer from loss of precisio because of the differece i magitude betwee a sample ad the sum of all samples. Equatio 4 calculates the mea i a way that is more umerically stable because it avoids accumulatig large sums. µ i ) + i ) ) This formula also provides us with some useful idetities. Simple variace + )µ ) 3) µ + µ ) 4) µ µ µ ) 5) µ µ µ ) µ + µ )µ µ ) 6) The defiitio of the stadard deviatio i equatio 7 below requires us to already kow the mea, which implies two passes over the data. This is t feasible for olie algorithms that eed to produce icremetal results after each sample becomes available. Equatio solves this problem sice it allows us to calculate the stadard deviatio from two ruig sums. σ i µ) 7) i i µ + µ ) 8)
2 3 Icremetal variace i µ i + µ 9) i µµ + µ 0) i µ ) ) ) i i ) Kuth otes [] that equatio is proe to loss of precisio because it takes the differece betwee two large sums of similar size, ad suggests equatio 4 as a alterative that avoids this problem. However he does ot say how it is derived. I the followig, equatio 0 is derived from the previous step usig equatio 5. Let S σ 3) i µ ) 4) i µ 5) S S i µ i + )µ 6) µ + )µ 7) µ + µ µ ) 8) µ + µ µ )µ + µ ) 9) µ + µ )µ + µ ) 0) µ + µ µ µ + µ µ ) µ µ + µ µ ) µ ) µ ) 3) S S + µ ) µ ) 4) σ S / 5) Mathworld [] has a alterative derivatio of a similar formula, which i our otatio is as follows. S i µ ) 6) i µ ) µ µ )) 7) i µ ) + µ µ ) i µ )µ µ ) 8) Simplify the first summatio: i µ ) µ ) + i µ ) 9) µ ) + S 30) S + µ µ ) 3)
3 Simplify the secod summatio: Simplify the third summatio: µ µ ) µ µ ) 3) i µ )µ µ ) µ µ ) i µ ) 33) ) µ µ ) µ + i µ ) ) µ µ ) µ )µ + i µ µ ) µ )µ + )µ ) 36) µ µ ) µ ) 37) µ µ ) 38) Back to the complete formula: S S + µ µ ) + µ µ ) µ µ ) 39) S + µ µ ) µ µ ) 40) S + )µ µ ) 4) We ca use equatios 6 ad 5 to show this is equivalet to equatio 4. S S + )µ µ ) 4) S + µ µ ) µ ) 43) S + µ ) µ ) 44) 4 Weighted mea The weighted mea is defied as follows. 34) 35) µ w i i w i 45) It is equivalet to the simple mea whe all the weights are equal, sice µ w i w w i i 46) w If the samples are all differet, the weights ca be thought of as sample frequecies, or they ca be used to calculate probabilities where p i w i / w i. The followig derivatio of the icremetal formula equatio 53) follows the same patter as the derivatio of equatio 4. For brevity we also defie as the sum of the weights. µ w i 47) w i i 48) ) w + w i i 3 49)
4 Useful idetities derived from this formula are: 5 Weighted variace w + W µ ) 50) w + w )µ ) 5) µ + w w µ ) 5) µ + w µ ) 53) µ µ ) w µ ) 54) µ µ ) w µ 55) µ w µ µ ) µ + µ w w µ µ ) 56) w µ ) 57) Similarly, we derive a umerically stable formula for calculatig the weighted variace equatio 68) usig the same patter as the derivatio of the uweighed wersio equatio 4). σ w i i µ) w i i µ 58) Let S σ 59) w i i µ 60) S S 6 Variable weights w i i µ w i i + W µ 6) w µ + W µ 6) w µ + w )µ 63) w µ ) + W µ µ ) 64) w µ ) + W µ µ )µ + µ ) 65) w µ + µ )µ + µ ) ) 66) w µ ) µ ) 67) S S + w µ ) µ ) 68) σ S / 69) I the previous three sectios, I have assumed that weights are costat oce assiged. However a commo requiremet is to ormalize the weights, such that W w i 70) If we are repeatedly addig ew data to our workig set, the we ca t have both costat weights ad ormalized weights. To allow us to keep weights ormalized, we eed to allow the weight of each 4
5 sample to chage as the set of samples chages. To idicate this we will give weights two idices, the first idetifyig the set of samples usig the sample cout as we have bee doig for µ etc.) ad the secod beig the ide of the sample i that set. We will ot make ay assumptios about the sum of the weights, that is we will ot require them to be ormalized. For eample, w,i, µ w,i i 7) Havig doe this we eed to reeamie some of the logical steps i the previous sectios to esure they are still valid. I equatios 49 5, we used the fact that i the fiedweight settig, w i i W µ w )µ 7) I the ew settig, this equality is fairly obviously o loger true. For eample, if we are keepig weights ormalized the W.) Fortuately there is a differet middle step which justifies equatio 7 whe weights vary, so the results of sectio 4 remai valid. w,i i w, )µ 73) w,i i w,i i w,i w,i w i,i w,j w,i w w,i i,i w,i i w,i w,i 74) 75) w,i w i 76),i w,j w,i where j 77) This says that for the weighted mea formulae to remai valid the ew ad old weights should be cosistet. Equatio 75 says that we get the same result whe we calculate the mea of the previous workig set whether we use the old weights or the ew weights. Equatio 77 says that whe we ormalize the weights across the previous set up to ) we get the same set of weights whether we start from the old weights or the ew oes. This requiremet is t eough by itself to make the weighted variace formulae work, so we will eamie it agai below. 7 The epectatio fuctio At this poit it is worth defiig some better otatio to reduce the umber of summatios we eed to write. The epectatio fuctio is a geeralized versio of the mea, whose argumet is some arbitrary fuctio of each sample. E f)) w,i f i ) 78) E k) k 79) E af)) ae f)) 80) E f) + g)) E f)) + E g)) 8) µ E ) 8) σ E µ ) ) 83) E + µ µ ) 84) E ) + µ µ E ) 85) 5
6 E ) µ 86) E ) E ) 87) The icremetal formula is derived i the usual way. Equatio 9 is particularly useful. E f)) w,i f i ) 88) 8 Variableweight variace w, f ) + w,i f i ) 89) w, f ) + w, ) w,if i ) w,i 90) w, f ) + w, ),if i ) w,i 9) w, f ) + w, )E f)) 9) E f)) E f)) + w, f ) E f))) 93) I equatios 6 63 we made the followig assumptios which are ot true whe weights ca vary. w,i i µ w,i i + W µ w, µ + W µ w, µ + w, )µ If we try to redo the short derivatio of the icremetal stadard deviatio formula startig from S S the we soo get stuck. Fortuately the loger derivatio shows how to made it work. S σ 94) E µ ) ) 95) Simplify the first term: E [ µ ] [µ µ ]) ) 96) E [ µ ] + [µ µ ] [ µ ][µ µ ] ) 97) E [ µ ] ) + E [µ µ ] ) E [ µ ][µ µ ]) 98) E [ µ ] ) w, [ µ ] + w, )E [ µ ] ) 99) Simplify the secod term: Simplify the third term: w, [ µ ] + w, ) S W 00) w, S + [µ µ ] 0) W w, E [µ µ ] ) [µ µ ] 0) E [ µ ][µ µ ]) [µ µ ] E [ µ ] 03) 6
7 Back to the complete formula: [µ µ ] w, [ µ ] + w, )E [ µ ]) 04) [µ µ ] w, [ µ ] + w, )[E ) E µ )]) 05) [µ µ ] w, [ µ ] + w, )[µ µ ]) 06) [µ µ ]w, [ µ ] 07) [µ µ ] 08) S w, S + [µ µ ] + [µ µ ] [µ µ ] 09) W w, w, S + [µ µ ] w, [µ µ ] 0) W w, w, w, S + w, ) [µ µ ] ) W w, w, W S + w, )[µ µ ] µ ) ) S w, W S + w, µ ) µ ) 3) This is the same as equatio 68, ecept for the multiplier W w, W which captures the chage i weights betwee the old ad ew sets. w,,i W w w,j,i w,j where j 4) Now that we kow the rescalig trick which makes it work, we ca write dow the short versio. S w, S W E ) µ ) W w, ) E ) µ ) 5) E ) µ ) W E ) + w, + w, )µ 6) w, µ + w, )µ 7) w, µ ) + W µ µ ) 8) w, µ ) + W µ µ )µ + µ ) 9) w, µ + µ )µ + µ ) ) 0) w, µ ) µ ) ) 9 Epoetiallyweighted mea ad variace Startig from equatio 53, let s set w, / to a costat 0 < α < ad let a α. This produces the stadard formula for the epoetially weighted movig average. µ µ + α µ ) ) α)µ + α 3) aµ + a) 4) I the followig it s more coveiet to use a lower boud of 0 istead of, i.e. 0 i. We are goig to show that the weights are reormalized each time a datum is added. First, we epad out the iductive defiitio of the mea. µ aµ + a) 5) a µ + a a) + a) 6) 7
8 a 3 µ 3 + a a) + a a) + a) 7) µ a 0 + a i a) i 8) This allows us to write dow the weights directly. Note that w, is idepedet of. w,0 a 9) w,i a i a), for i 30) w, a α 3) Sice w, α w, / we ca see that., that is, the weights are always ormalized. We ca get the same result by summig the geometric series. a i a j a a w,i j0 3) a i a) a 33) w,0 + These weights satisfy the cosistecy requiremet because w,j aw,j w,j w,i aw,i w,i w,i a + a ) 34) We ca use the epectatio fuctio to write dow the aïve formula for the variace. 35) E f)) E f)) + w, f ) E f))) 36) E f)) + αf ) E f))) 37) E ) E ) + α E )) 38) σ E ) µ 39) So usig the formula from the previous sectio we ca write the icremetal versio: S w, W S + w, µ ) µ ) 40) S α S + α µ ) µ ) 4) σ S S as + a) µ ) µ ) 4) This latter form is slightly more coveiet for code: diff :  mea icr : alpha * diff mea : mea + icr variace :  alpha) * variace + diff * icr) Refereces α)s + α µ ) ) 43) [] Doald E. Kuth. Semiumerical Algorithms, volume of The Art of Computer Programmig, chapter 4.., page 3. AddisoWesley, Bosto, third editio, 998. [] Eric W. Weisstei. Sample variace computatio. From Mathworld, a Wolfram web resource, 8
I. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationFactoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>
(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationhp calculators HP 12C Statistics  average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics  average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationOnesample test of proportions
Oesample test of proportios The Settig: Idividuals i some populatio ca be classified ito oe of two categories. You wat to make iferece about the proportio i each category, so you draw a sample. Examples:
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationSwaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps
Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationLecture 2: Karger s Min Cut Algorithm
priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationMannWhitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)
NoParametric ivariate Statistics: WilcoxoMaWhitey 2 Sample Test 1 MaWhitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo) MaWhitey (WMW) test is the oparametric equivalet of a pooled
More informationTime Value of Money. First some technical stuff. HP10B II users
Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle
More informationLecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.
18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: CouratFischer formula ad Rayleigh quotiets The
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationSolutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork
Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationNow here is the important step
LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"
More informationLaws of Exponents Learning Strategies
Laws of Epoets Learig Strategies What should studets be able to do withi this iteractive? Studets should be able to uderstad ad use of the laws of epoets. Studets should be able to simplify epressios that
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationThis document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.
SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More informationA GUIDE TO LEVEL 3 VALUE ADDED IN 2013 SCHOOL AND COLLEGE PERFORMANCE TABLES
A GUIDE TO LEVEL 3 VALUE ADDED IN 2013 SCHOOL AND COLLEGE PERFORMANCE TABLES Cotets Page No. Summary Iterpretig School ad College Value Added Scores 2 What is Value Added? 3 The Learer Achievemet Tracker
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationGCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4
GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook Alevel Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5
More informationA Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as:
A Test of Normality Textbook Referece: Chapter. (eighth editio, pages 59 ; seveth editio, pages 6 6). The calculatio of p values for hypothesis testig typically is based o the assumptio that the populatio
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationFM4 CREDIT AND BORROWING
FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 6985295 Email: bcm1@cec.wustl.edu Supervised
More informationTHIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK
THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for
More informationProject Deliverables. CS 361, Lecture 28. Outline. Project Deliverables. Administrative. Project Comments
Project Deliverables CS 361, Lecture 28 Jared Saia Uiversity of New Mexico Each Group should tur i oe group project cosistig of: About 612 pages of text (ca be loger with appedix) 612 figures (please
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationOverview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals
Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of
More informationElementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette iterestig patters of fractios Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationBiology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships
Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the
More informationNotes on exponential generating functions and structures.
Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a elemet set, (2) to fid for each the
More informationSampling Distribution And Central Limit Theorem
() Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationhttp://www.webassign.net/v4cgijeff.downs@wnc/control.pl
Assigmet Previewer http://www.webassig.et/vcgijeff.dows@wc/cotrol.pl of // : PM Practice Eam () Questio Descriptio Eam over chapter.. Questio DetailsLarCalc... [] Fid the geeral solutio of the differetial
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationInteger Factorization Algorithms
Iteger Factorizatio Algorithms Coelly Bares Departmet of Physics, Orego State Uiversity December 7, 004 This documet has bee placed i the public domai. Cotets I. Itroductio 3 1. Termiology 3. Fudametal
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationA Faster ClauseShortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 4960 A Faster ClauseShorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More information