ON DIFFERENTIATION AND HARMONIC NUMBERS
|
|
|
- Lesley Shields
- 5 years ago
- Views:
Transcription
1 ON DIFFERENTIATION AND HARMONIC NUMBERS ERIC MORTENSON Abstract. I a paper of Adrews ad Uchimura [AU, it is show how differetiatio applied to hypergeometric idetities produces formulas for harmoic ad q-harmoic umbers. Here we recall two biomial coefficiet sums that appear i [M, ad reprove them usig the techiques of [AU. 1. Itroductio ad Goal It is kow that may biomial coefficiet formulas follow from hypergeometric series idetities [A1. However, as is demostrated i [AU there are biomial coefficiet idetities which are ot ecessarily of this type. Here we recall two biomial coefficiet sums from [M ad reprove them usig the methods of [AU. We first recall idetities from (6.21 ad (5.28 i [M: (1.1 (1.2 ( 1 (H+ + H m+ 2H 0, ( 1 (1 + (H + + H m+ 2H ( 1 (1 + + m, where H : For both idetities we have the coditio m 1. I [M, these idetities were crucial i provig several Beukers like supercogrueces that had bee observed umerically by Ferado Rodriguez-Villegas [FRV. I [M, these idetities were broke up ito smaller pieces, ad each part was evaluated usig Wilf-Zeilberger [PWZ theory. Although these were evaluated exactly, for the goal of [M oe oly eeded to show that the two sums were cogruet to zero modulo p, where + m p 1 ad p 5 was prime. Our obective here is two-fold: to show that this ca be 2000 Mathematics Subect Classificatio. 05A19. The author thaks the Natioal Sciece Foudatio for their geerous support. 1
2 2 ERIC MORTENSON accomplished from differetiatio of classical hypergeometric idetities ad to brig attetio to this useful techique foud i [AU. We recall some basic defiitios, ( x + + (1.3 δ[f(x : f (0 ad where (1.4 (a : a(a + 1 (a + 1. Thus we ca write [ ( x + + ( + (1.5 δ (H + H, ad : (x (1, ( ( 1. We also recall the stadard otatio for hypergeometric series [A2, (1.6 2F 1 ( a, b c x 0 as well as the Chu-Vadermode sum [A2: (1.7 2F 1 (, a c 1 (a (b (c x, (c a (c. 2. The First Idetity Here we break the secod idetity up ito two parts. For the first part, we proceed as follows: ( 1 (H + H [ [ (x + 1 [ (x + 1 (x [ (x + 1 2F 1 (, x x ( (x + 1 [( 1 0 ( + (H + H (x (x + 1
3 ON DIFFERENTIATION AND HARMONIC NUMBERS 3 The secod part is aalogous, but we replace with p m 1, ad the cosider it mod p. ( 1 ( + 1 (H m+ H (H m+ H m (m p + 1 ( m (p m (H m+ H (m + 1 (H m+ H (mod p 0 Puttig both parts together, it follows that the first idetity is cogruet to zero mod p. 3. The Secod Idetity We break the first idetity up ito three sums, ad aalyze each separately. The first is well kow, but we iclude it for the sake of completeess. ( + ( 1 ( ( + 1! 2 F 1 (, For the secod sum, we proceed as follows: ( 1 ((H + H [ (x [ (x + 1 (x + + 1( (x + 1 [ (x ( ( 1. (1 1 ( + 1 ( 1 ( 1! (x ( 1! (x + 1 (x ( 1 (x + 2 ( 1 [ ( (x + 1 (x + + 1( + 1, x F 1 1 (x + 1 x + 2 [ (x + 1 (x + + 1( ( 1 (x + 1 (x [ (x + 1 (x + + 1( ( 1 1 (x + 1 [ (x + + 1( ( 1 1 ( 1.
4 4 ERIC MORTENSON for the third ad fial sum, we recall that +m p 1 ad argue as before ( 1 ((H m+ H [ (x + 1m δ m! m 1 ( 1 m m ( 1 m ( m (x + m + 1 ( 1! (x + 1 (mod p The last equality follows because + m p 1 is eve, thus ad m have the same parity. Therefore the sum is cogruet mod p to ( 1 (1 + + m ( 1 p, i.e. zero mod p. ( Remark The referee speculated as to the truth of the followig geeral idetity ( ( x + m ( 1 ( k + 1 k1 ( (x + 1 m, m which would shorte the proofs i the previous two sectios. Ideed, replace x with x + ad the let let m 0, 1 i sectios 2 ad 3 respectively. Provig this idetity is a opportuity to emphasize this ote s uderlyig theme: write it as a hypergeometric series ad the use Chu-Vadermode to trasform it. Replacig x with x + for coveiece, we have ( ( x + + m ( 1 ( k + 1 m (x + 1 (x + 1 (x + 1 k1 (x (x + 1 ( m! (x m ( 1 ( m (x m ( m (x + 1 m m (x m ( m (x + 1 m 2F 1 m (x m ( m ( m (x + 1 m (x + m + 1 m ( 1 m ( 1 m ( m! m! (x m. (x ( m! (x + 1 ( + m ( m (x + + m + 1 ( m ( m! (x + m + 1 ( m ( + m, x + + m x + m + 1
5 ON DIFFERENTIATION AND HARMONIC NUMBERS 5 5. Ackowledgemets The author would like to thak the referee for the helpful commets ad suggestios which improved this paper. Refereces [A1 G. Adrews, Applicatios of Hypergeometric Fuctios, S.I.A.M. Review, 16 (1974, pages [A2 G. Adrews, The Theory of Partitios, Vol. 2, Ecycl. of Math. ad Its Appl., Addiso-Wesley, Readig, (reprited: Cambridge Uiversity Press, [AU G. Adrews, K. Uchimura, Idetities i Combiatorics IV: Differetio ad Harmoic Numbers, Utilitas Mathematica 28 (1985, pages [FRV F. Rodriguez-Villegas, Hypergeometric Families of Calabi-Yau Maifolds. Calabi-Yau varieties ad mirror symmerty, Fields Ist. Commu., 38, Amer. Math. Soc., Providece, RI, [M E. Morteso, Supercogrueces Betwee Trucated 2 F 1 Hypergeometric Fuctios ad their Gaussia Aalogs, Tras. Amer. Math. Soc., 335 (2003, pages [PWZ M. Petkovsek, H. Wilf, D. Zeilberger, AB, A. K. Peters, Ltd., Wellesley, Ma, Departmet of Mathematics, Pe State Uiversity, Uiversity Park, Pesylvaia address: mort@math.psu.edu
A New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions
A New Costructive Proof of Graham's Theorem ad More New Classes of Fuctioally Complete Fuctios Azhou Yag, Ph.D. Zhu-qi Lu, Ph.D. Abstract A -valued two-variable truth fuctio is called fuctioally complete,
Binomial Theorem. Combinations with repetition.
Biomial.. Permutatios ad combiatios Give a set with elemets. The umber of permutatios of the elemets the set: P() =! = ( 1) ( 2)... 1 The umber of r-permutatios of the set: P(, r) =! = ( 1) ( 2)... ( r
FOUNDATION ACTED COURSE (FAC)
FOUNDATION ACTED COURSE (FAC) What is the Foudatio ActEd Course (FAC)? FAC is desiged to help studets improve their mathematical skills i preparatio for the Core Techical subjects. It is a referece documet
Hopscotch and Explicit difference method for solving Black-Scholes PDE
Mälardale iversity Fiacial Egieerig Program Aalytical Fiace Semiar Report Hopscotch ad Explicit differece method for solvig Blac-Scholes PDE Istructor: Ja Röma Team members: A Gog HaiLog Zhao Hog Cui 0
Estimation of Population Variance Utilizing Auxiliary Information
Iteratioal Joural of Statistics ad Systems ISSN 0973-675 Volume 1, Number (017), pp. 303-309 Research Idia Publicatios http://www.ripublicatio.com Estimatio of Populatio Variace Utilizig Auxiliary Iformatio
by Jim Pitman Department of Statistics University of California March 31,1999 Abstract
A lattice path model for the Bessel polyomials by Jim Pitma Techical Report No. 551 Departmet of Statistics Uiversity of Califoria 367 Evas Hall # 3860 Berkeley, CA94720-3860 March 31,1999 Abstract The
When you click on Unit V in your course, you will see a TO DO LIST to assist you in starting your course.
UNIT V STUDY GUIDE Percet Notatio Course Learig Outcomes for Uit V Upo completio of this uit, studets should be able to: 1. Write three kids of otatio for a percet. 2. Covert betwee percet otatio ad decimal
Sequences and Series
Sequeces ad Series Matt Rosezweig Cotets Sequeces ad Series. Sequeces.................................................. Series....................................................3 Rudi Chapter 3 Exercises........................................
A random variable is a variable whose value is a numerical outcome of a random phenomenon.
The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss
Parametric Density Estimation: Maximum Likelihood Estimation
Parametric Desity stimatio: Maimum Likelihood stimatio C6 Today Itroductio to desity estimatio Maimum Likelihood stimatio Itroducto Bayesia Decisio Theory i previous lectures tells us how to desig a optimal
A New Approach to Obtain an Optimal Solution for the Assignment Problem
Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): 231-7064 Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 A New Approach to Obtai a Optimal Solutio for the Assigmet Problem A. Seethalakshmy
Game Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012
Game Theory Lecture Notes By Y. Narahari Departmet of Computer Sciece ad Automatio Idia Istitute of Sciece Bagalore, Idia July 01 Chapter 4: Domiat Strategy Equilibria Note: This is a oly a draft versio,
BOUNDS FOR TAIL PROBABILITIES OF MARTINGALES USING SKEWNESS AND KURTOSIS. January 2008
BOUNDS FOR TAIL PROBABILITIES OF MARTINGALES USING SKEWNESS AND KURTOSIS V. Betkus 1,2 ad T. Juškevičius 1 Jauary 2008 Abstract. Let M = X 1 + + X be a sum of idepedet radom variables such that X k 1,
EXERCISE - BINOMIAL THEOREM
BINOMIAL THOEREM / EXERCISE - BINOMIAL THEOREM LEVEL I SUBJECTIVE QUESTIONS. Expad the followig expressios ad fid the umber of term i the expasio of the expressios. (a) (x + y) 99 (b) ( + a) 9 + ( a) 9
Random Sequences Using the Divisor Pairs Function
Radom Sequeces Usig the Divisor Pairs Fuctio Subhash Kak Abstract. This paper ivestigates the radomess properties of a fuctio of the divisor pairs of a atural umber. This fuctio, the atecedets of which
NOTES ON ESTIMATION AND CONFIDENCE INTERVALS. 1. Estimation
NOTES ON ESTIMATION AND CONFIDENCE INTERVALS MICHAEL N. KATEHAKIS 1. Estimatio Estimatio is a brach of statistics that deals with estimatig the values of parameters of a uderlyig distributio based o observed/empirical
An Improved Estimator of Population Variance using known Coefficient of Variation
J. Stat. Appl. Pro. Lett. 4, No. 1, 11-16 (017) 11 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.18576/jsapl/04010 A Improved Estimator of Populatio Variace
Estimating Proportions with Confidence
Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter
Annual compounding, revisited
Sectio 1.: No-aual compouded iterest MATH 105: Cotemporary Mathematics Uiversity of Louisville August 2, 2017 Compoudig geeralized 2 / 15 Aual compoudig, revisited The idea behid aual compoudig is that
An Empirical Study of the Behaviour of the Sample Kurtosis in Samples from Symmetric Stable Distributions
A Empirical Study of the Behaviour of the Sample Kurtosis i Samples from Symmetric Stable Distributios J. Marti va Zyl Departmet of Actuarial Sciece ad Mathematical Statistics, Uiversity of the Free State,
Solutions to Problem Sheet 1
Solutios to Problem Sheet ) Use Theorem.4 to prove that p log for all real x 3. This is a versio of Theorem.4 with the iteger N replaced by the real x. Hit Give x 3 let N = [x], the largest iteger x. The,
x satisfying all regularity conditions. Then
AMS570.01 Practice Midterm Exam Sprig, 018 Name: ID: Sigature: Istructio: This is a close book exam. You are allowed oe-page 8x11 formula sheet (-sided). No cellphoe or calculator or computer is allowed.
Solution to Tutorial 6
Solutio to Tutorial 6 2012/2013 Semester I MA4264 Game Theory Tutor: Xiag Su October 12, 2012 1 Review Static game of icomplete iformatio The ormal-form represetatio of a -player static Bayesia game: {A
Combining imperfect data, and an introduction to data assimilation Ross Bannister, NCEO, September 2010
Combiig imperfect data, ad a itroductio to data assimilatio Ross Baister, NCEO, September 00 rbaister@readigacuk The probability desity fuctio (PDF prob that x lies betwee x ad x + dx p (x restrictio o
Lecture 5: Sampling Distribution
Lecture 5: Samplig Distributio Readigs: Sectios 5.5, 5.6 Itroductio Parameter: describes populatio Statistic: describes the sample; samplig variability Samplig distributio of a statistic: A probability
Section Mathematical Induction and Section Strong Induction and Well-Ordering
Sectio 4.1 - Mathematical Iductio ad Sectio 4. - Strog Iductio ad Well-Orderig A very special rule of iferece! Defiitio: A set S is well ordered if every subset has a least elemet. Note: [0, 1] is ot well
Rafa l Kulik and Marc Raimondo. University of Ottawa and University of Sydney. Supplementary material
Statistica Siica 009: Supplemet 1 L p -WAVELET REGRESSION WITH CORRELATED ERRORS AND INVERSE PROBLEMS Rafa l Kulik ad Marc Raimodo Uiversity of Ottawa ad Uiversity of Sydey Supplemetary material This ote
Calculation of the Annual Equivalent Rate (AER)
Appedix to Code of Coduct for the Advertisig of Iterest Bearig Accouts. (31/1/0) Calculatio of the Aual Equivalet Rate (AER) a) The most geeral case of the calculatio is the rate of iterest which, if applied
Statistics for Economics & Business
Statistics for Ecoomics & Busiess Cofidece Iterval Estimatio Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for the mea ad the proportio How to determie
We learned: $100 cash today is preferred over $100 a year from now
Recap from Last Week Time Value of Moey We leared: $ cash today is preferred over $ a year from ow there is time value of moey i the form of willigess of baks, busiesses, ad people to pay iterest for its
Lecture 9: The law of large numbers and central limit theorem
Lecture 9: The law of large umbers ad cetral limit theorem Theorem.4 Let X,X 2,... be idepedet radom variables with fiite expectatios. (i) (The SLLN). If there is a costat p [,2] such that E X i p i i=
5. Best Unbiased Estimators
Best Ubiased Estimators http://www.math.uah.edu/stat/poit/ubiased.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 5. Best Ubiased Estimators Basic Theory Cosider agai
NORMALIZATION OF BEURLING GENERALIZED PRIMES WITH RIEMANN HYPOTHESIS
Aales Uiv. Sci. Budapest., Sect. Comp. 39 2013) 459 469 NORMALIZATION OF BEURLING GENERALIZED PRIMES WITH RIEMANN HYPOTHESIS We-Bi Zhag Chug Ma Pig) Guagzhou, People s Republic of Chia) Dedicated to Professor
. (The calculated sample mean is symbolized by x.)
Stat 40, sectio 5.4 The Cetral Limit Theorem otes by Tim Pilachowski If you have t doe it yet, go to the Stat 40 page ad dowload the hadout 5.4 supplemet Cetral Limit Theorem. The homework (both practice
Department of Mathematics, S.R.K.R. Engineering College, Bhimavaram, A.P., India 2
Skewess Corrected Cotrol charts for two Iverted Models R. Subba Rao* 1, Pushpa Latha Mamidi 2, M.S. Ravi Kumar 3 1 Departmet of Mathematics, S.R.K.R. Egieerig College, Bhimavaram, A.P., Idia 2 Departmet
ON THE RATE OF CONVERGENCE
ON THE RATE OF CONVERGENCE OF BINOMIAL GREEKS SAN-LIN CHUNG WEIFENG HUNG HAN-HSING LEE* PAI-TA SHIH This study ivestigates the covergece patters ad the rates of covergece of biomial Greeks for the CRR
MA Lesson 11 Section 1.3. Solving Applied Problems with Linear Equations of one Variable
MA 15200 Lesso 11 Sectio 1. I Solvig Applied Problems with Liear Equatios of oe Variable 1. After readig the problem, let a variable represet the ukow (or oe of the ukows). Represet ay other ukow usig
A Note About Maximum Likelihood Estimator in Hypergeometric Distribution
Comuicacioes e Estadística Juio 2009, Vol. 2, No. 1 A Note About Maximum Likelihood Estimator i Hypergeometric Distributio Ua ota sobre los estimadores de máxima verosimilitud e la distribució hipergeométrica
Lecture 4: Probability (continued)
Lecture 4: Probability (cotiued) Desity Curves We ve defied probabilities for discrete variables (such as coi tossig). Probabilities for cotiuous or measuremet variables also are evaluated usig relative
ST 305: Exam 2 Fall 2014
ST 305: Exam Fall 014 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad
Models of Asset Pricing
APPENDIX 1 TO CHAPTER 4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
Models of Asset Pricing
APPENDIX 1 TO CHAPTER4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
Today: Finish Chapter 9 (Sections 9.6 to 9.8 and 9.9 Lesson 3)
Today: Fiish Chapter 9 (Sectios 9.6 to 9.8 ad 9.9 Lesso 3) ANNOUNCEMENTS: Quiz #7 begis after class today, eds Moday at 3pm. Quiz #8 will begi ext Friday ad ed at 10am Moday (day of fial). There will be
0.1 Valuation Formula:
0. Valuatio Formula: 0.. Case of Geeral Trees: q = er S S S 3 S q = er S S 4 S 5 S 4 q 3 = er S 3 S 6 S 7 S 6 Therefore, f (3) = e r [q 3 f (7) + ( q 3 ) f (6)] f () = e r [q f (5) + ( q ) f (4)] = f ()
ii. Interval estimation:
1 Types of estimatio: i. Poit estimatio: Example (1) Cosider the sample observatios 17,3,5,1,18,6,16,10 X 8 X i i1 8 17 3 5 118 6 16 10 8 116 8 14.5 14.5 is a poit estimate for usig the estimator X ad
Diener and Diener and Walsh follow as special cases. In addition, by making. smooth, as numerically observed by Tian. Moreover, we propose the center
Smooth Covergece i the Biomial Model Lo-Bi Chag ad Ke Palmer Departmet of Mathematics, Natioal Taiwa Uiversity Abstract Various authors have studied the covergece of the biomial optio price to the Black-Scholes
Math 312, Intro. to Real Analysis: Homework #4 Solutions
Math 3, Itro. to Real Aalysis: Homework #4 Solutios Stephe G. Simpso Moday, March, 009 The assigmet cosists of Exercises 0.6, 0.8, 0.0,.,.3,.6,.0,.,. i the Ross textbook. Each problem couts 0 poits. 0.6.
Bayes Estimator for Coefficient of Variation and Inverse Coefficient of Variation for the Normal Distribution
Iteratioal Joural of Statistics ad Systems ISSN 0973-675 Volume, Number 4 (07, pp. 7-73 Research Idia Publicatios http://www.ripublicatio.com Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet
Journal of Statistical Software
JSS Joural of Statistical Software Jue 2007, Volume 19, Issue 6. http://www.jstatsoft.org/ Ratioal Arithmetic Mathematica Fuctios to Evaluate the Oe-sided Oe-sample K-S Cumulative Samplig Distributio J.
14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpeCourseWare http://ocwmitedu 430 Itroductio to Statistical Methods i Ecoomics Sprig 009 For iformatio about citig these materials or our Terms of Use, visit: http://ocwmitedu/terms 430 Itroductio
Research Article The Average Lower Connectivity of Graphs
Applied Mathematics, Article ID 807834, 4 pages http://dx.doi.org/10.1155/2014/807834 Research Article The Average Lower Coectivity of Graphs Ersi Asla Turgutlu Vocatioal Traiig School, Celal Bayar Uiversity,
Minhyun Yoo, Darae Jeong, Seungsuk Seo, and Junseok Kim
Hoam Mathematical J. 37 (15), No. 4, pp. 441 455 http://dx.doi.org/1.5831/hmj.15.37.4.441 A COMPARISON STUDY OF EXPLICIT AND IMPLICIT NUMERICAL METHODS FOR THE EQUITY-LINKED SECURITIES Mihyu Yoo, Darae
Monetary Economics: Problem Set #5 Solutions
Moetary Ecoomics oblem Set #5 Moetary Ecoomics: oblem Set #5 Solutios This problem set is marked out of 1 poits. The weight give to each part is idicated below. Please cotact me asap if you have ay questios.
0.07. i PV Qa Q Q i n. Chapter 3, Section 2
Chapter 3, Sectio 2 1. (S13HW) Calculate the preset value for a auity that pays 500 at the ed of each year for 20 years. You are give that the aual iterest rate is 7%. 20 1 v 1 1.07 PV Qa Q 500 5297.01
AUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY
AUTOMATIC GENERATION OF FUZZY PAYOFF MATRIX IN GAME THEORY Dr. Farha I. D. Al Ai * ad Dr. Muhaed Alfarras ** * College of Egieerig ** College of Coputer Egieerig ad scieces Gulf Uiversity * Dr.farha@gulfuiversity.et;
SCHOOL OF ACCOUNTING AND BUSINESS BSc. (APPLIED ACCOUNTING) GENERAL / SPECIAL DEGREE PROGRAMME
All Right Reserved No. of Pages - 10 No of Questios - 08 SCHOOL OF ACCOUNTING AND BUSINESS BSc. (APPLIED ACCOUNTING) GENERAL / SPECIAL DEGREE PROGRAMME YEAR I SEMESTER I (Group B) END SEMESTER EXAMINATION
REVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS. Guangwu Liu L. Jeff Hong
Proceedigs of the 2008 Witer Simulatio Coferece S. J. Maso, R. R. Hill, L. Möch, O. Rose, T. Jefferso, J. W. Fowler eds. REVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS Guagwu Liu L. Jeff
TENS Unit Prior Authorization Process
TENS Uit Prior Authorizatio Process Objectives Uderstad the HUSKY Health program s prior authorizatio process for TENS uits (Trascutaeous Electrical Nerve Stimulatio) Access the DSS Fee Schedule Reduce
Overlapping Generations
Eco. 53a all 996 C. Sims. troductio Overlappig Geeratios We wat to study how asset markets allow idividuals, motivated by the eed to provide icome for their retiremet years, to fiace capital accumulatio
Optimal Risk Classification and Underwriting Risk for Substandard Annuities
1 Optimal Risk Classificatio ad Uderwritig Risk for Substadard Auities Nadie Gatzert, Uiversity of Erlage-Nürberg Gudru Hoerma, Muich Hato Schmeiser, Istitute of Isurace Ecoomics, Uiversity of St. Galle
Models of Asset Pricing
4 Appedix 1 to Chapter Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
The Time Value of Money in Financial Management
The Time Value of Moey i Fiacial Maagemet Muteau Irea Ovidius Uiversity of Costata irea.muteau@yahoo.com Bacula Mariaa Traia Theoretical High School, Costata baculamariaa@yahoo.com Abstract The Time Value
Using Math to Understand Our World Project 5 Building Up Savings And Debt
Usig Math to Uderstad Our World Project 5 Buildig Up Savigs Ad Debt Note: You will have to had i aswers to all umbered questios i the Project Descriptio See the What to Had I sheet for additioal materials
Research Article The Probability That a Measurement Falls within a Range of n Standard Deviations from an Estimate of the Mean
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 70806, 8 pages doi:0.540/0/70806 Research Article The Probability That a Measuremet Falls withi a Rage of Stadard Deviatios
Chapter 8: Estimation of Mean & Proportion. Introduction
Chapter 8: Estimatio of Mea & Proportio 8.1 Estimatio, Poit Estimate, ad Iterval Estimate 8.2 Estimatio of a Populatio Mea: σ Kow 8.3 Estimatio of a Populatio Mea: σ Not Kow 8.4 Estimatio of a Populatio
11.7 (TAYLOR SERIES) NAME: SOLUTIONS 31 July 2018
.7 (TAYLOR SERIES NAME: SOLUTIONS 3 July 08 TAYLOR SERIES ( The power series T(x f ( (c (x c is called the Taylor Series for f(x cetered at x c. If c 0, this is called a Maclauri series. ( The N-th partial
4.5 Generalized likelihood ratio test
4.5 Geeralized likelihood ratio test A assumptio that is used i the Athlete Biological Passport is that haemoglobi varies equally i all athletes. We wish to test this assumptio o a sample of k athletes.
Exam 1 Spring 2015 Statistics for Applications 3/5/2015
8.443 Exam Sprig 05 Statistics for Applicatios 3/5/05. Log Normal Distributio: A radom variable X follows a Logormal(θ, σ ) distributio if l(x) follows a Normal(θ, σ ) distributio. For the ormal radom
Economic Computation and Economic Cybernetics Studies and Research, Issue 2/2016, Vol. 50
Ecoomic Computatio ad Ecoomic Cyberetics Studies ad Research, Issue 2/216, Vol. 5 Kyoug-Sook Moo Departmet of Mathematical Fiace Gacho Uiversity, Gyeoggi-Do, Korea Yuu Jeog Departmet of Mathematics Korea
Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1
Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for
Anomaly Correction by Optimal Trading Frequency
Aomaly Correctio by Optimal Tradig Frequecy Yiqiao Yi Columbia Uiversity September 9, 206 Abstract Uder the assumptio that security prices follow radom walk, we look at price versus differet movig averages.
Monopoly vs. Competition in Light of Extraction Norms. Abstract
Moopoly vs. Competitio i Light of Extractio Norms By Arkadi Koziashvili, Shmuel Nitza ad Yossef Tobol Abstract This ote demostrates that whether the market is competitive or moopolistic eed ot be the result
Subject CT1 Financial Mathematics Core Technical Syllabus
Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig
Chapter 10 - Lecture 2 The independent two sample t-test and. confidence interval
Assumptios Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Upooled case Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Pooled case Idepedet samples - Pooled variace - Large samples Chapter 10 - Lecture The
Fourier Transform in L p (R) Spaces, p 1
Ge. Math. Notes, Vol. 3, No., March 20, pp.4-25 ISSN 229-784; Copyright c ICSS Publicatio, 200 www.i-csrs.org Available free olie at http://www.gema.i Fourier Trasform i L p () Spaces, p Devedra Kumar
Pricing 50ETF in the Way of American Options Based on Least Squares Monte Carlo Simulation
Pricig 50ETF i the Way of America Optios Based o Least Squares Mote Carlo Simulatio Shuai Gao 1, Ju Zhao 1 Applied Fiace ad Accoutig Vol., No., August 016 ISSN 374-410 E-ISSN 374-49 Published by Redfame
Math 124: Lecture for Week 10 of 17
What we will do toight 1 Lecture for of 17 David Meredith Departmet of Mathematics Sa Fracisco State Uiversity 2 3 4 April 8, 2008 5 6 II Take the midterm. At the ed aswer the followig questio: To be revealed
The Likelihood Ratio Test
LM 05 Likelihood Ratio Test 1 The Likelihood Ratio Test The likelihood ratio test is a geeral purpose test desiged evaluate ested statistical models i a way that is strictly aalogous to the F-test for
MATH : EXAM 2 REVIEW. A = P 1 + AP R ) ny
MATH 1030-008: EXAM 2 REVIEW Origially, I was havig you all memorize the basic compoud iterest formula. I ow wat you to memorize the geeral compoud iterest formula. This formula, whe = 1, is the same as
Control Charts for Mean under Shrinkage Technique
Helderma Verlag Ecoomic Quality Cotrol ISSN 0940-5151 Vol 24 (2009), No. 2, 255 261 Cotrol Charts for Mea uder Shrikage Techique J. R. Sigh ad Mujahida Sayyed Abstract: I this paper a attempt is made to
This article is part of a series providing
feature Bryce Millard ad Adrew Machi Characteristics of public sector workers SUMMARY This article presets aalysis of public sector employmet, ad makes comparisos with the private sector, usig data from
CreditRisk + Download document from CSFB web site:
CreditRis + Dowload documet from CSFB web site: http://www.csfb.com/creditris/ Features of CreditRis+ pplies a actuarial sciece framewor to the derivatio of the loss distributio of a bod/loa portfolio.
Binomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge
Biomial Model Stock Price Dyamics The value of a optio at maturity depeds o the price of the uderlyig stock at maturity. The value of the optio today depeds o the expected value of the optio at maturity
Combinatorial Proofs of Fibonomial Identities
Clareot Colleges Scholarship @ Clareot All HMC aculty Publicatios ad Research HMC aculty Scholarship 12-1-2014 Cobiatorial Proofs of ibooial Idetities Arthur Bejai Harvey Mudd College Elizabeth Reilad
Chapter 5: Sequences and Series
Chapter 5: Sequeces ad Series 1. Sequeces 2. Arithmetic ad Geometric Sequeces 3. Summatio Notatio 4. Arithmetic Series 5. Geometric Series 6. Mortgage Paymets LESSON 1 SEQUENCES I Commo Core Algebra I,
Building a Dynamic Two Dimensional Heat Transfer Model part #1
Buildig a Dyamic Two Dimesioal Heat Trasfer Model part #1 - Tis is te first alf of a tutorial wic sows ow to build a basic dyamic eat coductio model of a square plate. Te same priciple could be used to
ISBN Copyright 2015 The Continental Press, Inc.
TABLE OF CONTENTS Itroductio 3 Format of Books 4 Suggestios for Use 7 Aotated Aswer Key ad Extesio Activities 9 Reproducible Tool Set 183 ISBN 978-0-8454-7897-4 Copyright 2015 The Cotietal Press, Ic. Exceptig
Sampling Distributions and Estimation
Cotets 40 Samplig Distributios ad Estimatio 40.1 Samplig Distributios 40. Iterval Estimatio for the Variace 13 Learig outcomes You will lear about the distributios which are created whe a populatio is
How the Default Probability is Defined by the CreditRisk+Model?
Iteratioal Joural of Global Eergy Marets ad Fiace, 28, Vol, No, 2-25 vailable olie at http://pubssciepubcom/igefm///4 Sciece ad Educatio Publishig DOI:269/igefm---4 How the Default Probability is Defied
Appendix 1 to Chapter 5
Appedix 1 to Chapter 5 Models of Asset Pricig I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy a asset, we are
Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
APPLICATION OF GEOMETRIC SEQUENCES AND SERIES: COMPOUND INTEREST AND ANNUITIES
APPLICATION OF GEOMETRIC SEQUENCES AND SERIES: COMPOUND INTEREST AND ANNUITIES Example: Brado s Problem Brado, who is ow sixtee, would like to be a poker champio some day. At the age of twety-oe, he would
Notes on Expected Revenue from Auctions
Notes o Epected Reveue from Auctios Professor Bergstrom These otes spell out some of the mathematical details about first ad secod price sealed bid auctios that were discussed i Thursday s lecture You
BASIC STATISTICS ECOE 1323
BASIC STATISTICS ECOE 33 SPRING 007 FINAL EXAM NAME: ID NUMBER: INSTRUCTIONS:. Write your ame ad studet ID.. You have hours 3. This eam must be your ow work etirely. You caot talk to or share iformatio
The self-assessment will test the following six major areas, relevant to studies in the Real Estate Division's credit-based courses:
Math Self-Assessmet This self-assessmet tool has bee created to assist studets review their ow math kowledge ad idetify areas where they may require more assistace. We hope that studets will complete this
Stochastic Processes and their Applications in Financial Pricing
Stochastic Processes ad their Applicatios i Fiacial Pricig Adrew Shi Jue 3, 1 Cotets 1 Itroductio Termiology.1 Fiacial.............................................. Stochastics............................................
5 Statistical Inference
5 Statistical Iferece 5.1 Trasitio from Probability Theory to Statistical Iferece 1. We have ow more or less fiished the probability sectio of the course - we ow tur attetio to statistical iferece. I statistical
Limits of sequences. Contents 1. Introduction 2 2. Some notation for sequences The behaviour of infinite sequences 3
Limits of sequeces I this uit, we recall what is meat by a simple sequece, ad itroduce ifiite sequeces. We explai what it meas for two sequeces to be the same, ad what is meat by the -th term of a sequece.
Intercensal life tables consistent with population projections
Itercesal life tables cosistet with populatio projectios Populatio Associatio of America 2011 Aual Meetig Program Washigto, DC March 31- April 2 Author iformatio: Jeroimo Oliveira Muiz is a Assistat Professor
Introduction to Probability and Statistics Chapter 7
Itroductio to Probability ad Statistics Chapter 7 Ammar M. Sarha, asarha@mathstat.dal.ca Departmet of Mathematics ad Statistics, Dalhousie Uiversity Fall Semester 008 Chapter 7 Statistical Itervals Based