The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract
|
|
- Randall Flynn
- 8 years ago
- Views:
Transcription
1 The Gompertz Makeham couplig as a Dyamic Life Table By Abraham Zaks Techio I.I.T. Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 32000, Haifa, Israel Abstract A very famous law of mortality was itroduced by Gompertz i 1825 i [G]. I 1860 Makeham itroduced i [M] a modificatio to obtai aother law of mortality. Both these laws assume that the populatio uder cosideratio is stable. The two laws differ by a costat term i the force of mortality. The updated approach to the study of populatio presume that mortality chages over time. The differece stems from observig that the expectacy of life chages over the years. There were made various attempts to itroduce dyamic life tables, ad the Lee-Carter model has this advatage. The Lee-Carter model i [LC] describes better the mortality uder chages over time as was show recetly by ([MR]). We ited to study the differece arisig from a fixed chage i the force of mortality ad the Gompertz ad Makeham cases may serve to demostrate such a chage. The differece i the expectacy of life i both cases affects directly the correspodig life tables, ad cosequetly the auities (for life, term auitie ad deferred ), as well as the assuraces (whole life, term ad edowmet), ad the premiums ad reserves i the various cases. The chages i a stable life table model may serve to evaluate the chages that arise i terms of a sesitive aalysis of various assurace plas. Cosequetly there results a tool to cope with evaluatig the effect of chage i the force of mortality. Keywords : force of mortality ; Expectacy of life ; Auities, a,, Assuraces, A, A ; Premiums ad Reserves i various cases. 1 1 of 6 Electroic copy available at:
2 1. Auities We let G deote the "Gompertz case", M the "Makeham case" ad GM the "Gompertz ad Makeham cases". We follow the defiitios ad otatios of [Z1], [Z2], ad [Z3]. I G the force of mortality is give by, [, ad i M the force of mortality is give by, [, for some fixed. We may have 0, 0 0, i the last case both models coicide. Note that we may replace by where ad the results that will follow will be symmetric to the results that we obtai whe we replace for (e.g. for ad for ad so o ad for. Recall that the expectacy of a live aged i these models is ad respectively. Let be the force of iterest, ad let (µ,δ) be the PV of the cotiuous auity of 1 p.a. for a live aged ad we have (e.g.[z1]) (µ,δ+ε) ε, δ ad i particular we have the equalities (µ,δ+ ), δ, ad,,. As is well kow, 0 ad sice (e.g.[ Z]) we may deduce :. :, 0, Recall the approximatio (e.g. [Z1] ) : µ, δ ε µ, δ µ, δ where deotes the PV of the cotuuous auity of 1 p.a. for a live aged icreasig by 1 aually. The followig approximatio results :. : The approximatio, µ, 0 holds. The results for geeralizes to, evaluatig i the same table but i differet itersets :. :, δ ) (,δ+ ad a similar geeralizatio hold for the approximatio i the same table ad iterset :. :, δ (,δ), δ The result follows from, δ (,δ+ ) ad the approximatio (,δ+ ) (,δ), δ. It seems that oe ca derive appropriate approximatios for temporary auities a ad deferred auities, as well as for due auities ad i arrear auities. 2 of 6 Electroic copy available at:
3 2. Assuraces Let (µ,δ) be the PV of the cotiuous whole life issurace of 1 for a live aged. The result of comparrig, evaluatig i the same table but i differet itersets yields :. : (µε,δ) µ, δ ε ε (µ,δε). From the well kow equality (µ,δ)δ (µ,δ)=1 there follows the equality (µ,δε)= 1 δ ε (µ,δε). Recall that : (µε,δ)= 1 δ (µ ε, δ) ad we may derive the equality : (µε,δ) µ, δ ε ε (µ,δε) as claimed. Note that sice, δ, 1 we may coclude i a similar way that,,, that is or there follows the equality : A, A, a,. Recall from [Z1] thatwe have the approximatio µ, δ ε µ, δ µ, δ where µ, δ deotes the PV of the cotiuous whole life assurace of 1 for a live aged icreasig aually by 1. ad a similar geeralizatio hold for the approximatio i the same table ad iterset :. : µ, δ ε µ, δ µ, δ ε µ, δ ε. From the approximatio µ, δ ε µ, δ µ, δ (e.g. [Z1]) ad from propositio 3 the result follows as stated. I particular we may derive :. (,δ),δ)δ, δ. The equality,δ)δ (,δ)=1 implies,δ)1 δ (,δ) ad the equality (,δ)δ (,δ)=1 implies (,δ)=1 δ (,δ). Therefore (,δ)=1 δ (,δ ad by [Z1] we may derive the approximatio, δ 1 δ (,δ)δ, δ, ad sice,δ)1 δ (,δ) the approximatio (,δ),δ)δ, δ follows as stated. It seems that oe ca derive appropriate approximatios i a similar way for term assurace 1 A, edowmet assurace various assurace plas paid at the ed of the year of death.. A,ad deferred assurace, as well as for the 3 of 6
4 3. Premiums The premium for the whole life isurace may be derived from. This implies, 1, ad i particular, 1,. Furthermore we have the equality, 1,. Recall the approximatio :,,,, δ. Thus we may state the followig resultig equatio ad approximatio :. :,,..,,, δ, The result follows from the equality,, ad the approximatio,,, δ,. Note that startig with the equalities :, 1, Ad, 1,, it results that,, Ad i particular we may derive the equality : P, P,. It seems that oe ca derive appropriate approximatios i a similar way for premiums for the various assurace plas ad pesio schemes (cosidered as deferred auities).. 4. Reserves The reserve, i time for the whole life isurace for a live aged may be derived from Ad we obtai :, ad i a similar way for M µ, δ (µ,δ ad (µ,δ)= 1δ (µ,δ), 1,,, 1,, ad for G we have, 1,,. Recall that,, ad,, 4 of 6
5 Ad we may derive :. : Ad there results the approximatio :,,.,,,,, δ,, δ, I a similar way if we start with the equalities :, 1,, ad, 1,,, we have, V, holds., ad i particular the equality V, It seems that oe ca derive appropriate approximatios i a similar way for reserves i the various assurace plas ad pesio schemes.. 5. Approximatig higher derivatios Recall [Z2] for relatios of higher derivatives of with ad. A life table satisfies the weak decreasig assumptio e.g. whe we presume ati selectio, i which case the premium decreases whe the force of iterest icreases. I this case the iequality holds. Therefore we have ad we ca derive the iequality 1, that is 1 ad cosequetly :.. I particular followig [Z2] start with the equality ad use iductio, to obtai the equalities A x a x for all 0 where = that is :. 0, 0. The proof of this propositio is a direct cosequece of the observatio : of 6
6 Refereces [FL] Föllmer, H., Leukert, P., Quatile hedgig. Fiace Stoch. 3, [G ] Gompertz, B. " O the Nature of the Fuctio Expressive of the Law of Huma Mortality, ad o a New Mode of Determiig the Value of Life Cotigecies." Phil. Tras. of the Royal Soc. of Lodo, Vol. 115 (1825), pp [M] [LC] Makeham, W. M. "O the Law of Mortality ad the Costructio of Auity Tables." J. Ist. Act. ad Assur. Mag. 8, , Lee, R.D., Carter, L.R., Modellig ad forecastig U.S. mortality. J. Amer. Statist. Assoc. 87 (14), [MR] A.Melikov, Y.Romaiuk "Evaluatig the performace of Gompertz, Makeham ad Lee Carter mortality models for risk maagemet with uit-liked cotracts" IME Volume 39(3)2006, Pp [Z1] Zaks A., Auities uder chages i Life Tables ad chages i the Iterest, Pravartak J. Is. & Risk Ma.2009(8) [Z2] [Z3] Auities uder chages i Life Tables ad chages i the Iterest II, Pravartak J. Is. & Risk Ma.2009(9) Auities uder chages i Life Tables ad chages i the Iterest III, or Premium Icreases whe Iterest Drcreases, Pravartak J. Is. & Risk Ma.2009(10) of 6
Annuities 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 informationSubject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC
TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationChapter 7. V and 10. V (the modified premium reserve using the Full Preliminary Term. V (the modified premium reserves using the Full Preliminary
Chapter 7 1. You are give that Mortality follows the Illustrative Life Table with i 6%. Assume that mortality is uiformly distributed betwee itegral ages. Calculate: a. Calculate 10 V for a whole life
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
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 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 informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
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 informationD I S C U S S I O N P A P E R
I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S ( I S B A UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 2012/27 Worst-case
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
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 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 informationBENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
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 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 informationNr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996
Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi
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 informationEfficient tree methods for pricing digital barrier options
Efficiet tree methods for pricig digital barrier optios arxiv:1401.900v [q-fi.cp] 7 Ja 014 Elisa Appolloi Sapieza Uiversità di Roma MEMOTEF elisa.appolloi@uiroma1.it Abstract Adrea igori Uiversità di Roma
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
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 informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationTransient Behavior of Two-Machine Geometric Production Lines
Trasiet Behavior of Two-Machie Geometric Productio Lies Semyo M. Meerkov Nahum Shimki Liag Zhag Departmet of Electrical Egieerig ad Computer Sciece Uiversity of Michiga, A Arbor, MI 489-222, USA (e-mail:
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 informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationAutomatic Tuning for FOREX Trading System Using Fuzzy Time Series
utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationBOUNDS FOR THE PRICE OF A EUROPEAN-STYLE ASIAN OPTION IN A BINARY TREE MODEL
BOUNDS FOR THE PRICE OF A EUROPEAN-STYLE ASIAN OPTION IN A BINARY TREE MODEL HUGUETTE REYNAERTS, MICHELE VANMAELE, JAN DHAENE ad GRISELDA DEELSTRA,1 Departmet of Applied Mathematics ad Computer Sciece,
More informationInstallment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate
Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate Nia-Nia JI a,*, Yue LI, Dog-Hui WNG College of Sciece, Harbi
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 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 informationEvaluating Model for B2C E- commerce Enterprise Development Based on DEA
, pp.180-184 http://dx.doi.org/10.14257/astl.2014.53.39 Evaluatig Model for B2C E- commerce Eterprise Developmet Based o DEA Weli Geg, Jig Ta Computer ad iformatio egieerig Istitute, Harbi Uiversity of
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
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 informationA unified pricing of variable annuity guarantees under the optimal stochastic control framework
A uified pricig of variable auity guaratees uder the optimal stochastic cotrol framework Pavel V. Shevcheko 1 ad Xiaoli Luo 2 arxiv:1605.00339v1 [q-fi.pr] 2 May 2016 Draft paper: 16 April 2016 1 CSIRO
More informationmodel Poits And Risk Measurement Based Models
Model poits ad Tail-VaR i life isurace Michel Deuit Istitute of Statistics, Biostatistics ad Actuarial Sciece Uiversité Catholique de Louvai (UCL) Louvai-la-Neuve, Belgium Julie Trufi Departmet of Mathematics
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/
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 informationFactors of sums of powers of binomial coefficients
ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRE-CALCULUS 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 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 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 informationData Analysis and Statistical Behaviors of Stock Market Fluctuations
44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Email:
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 informationZ-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about
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 informationCDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest
CDs Bought at a Bak verses CD s Bought from a Brokerage Floyd Vest CDs bought at a bak. CD stads for Certificate of Deposit with the CD origiatig i a FDIC isured bak so that the CD is isured by the Uited
More informationPresent Values, Investment Returns and Discount Rates
Preset Values, Ivestmet Returs ad Discout Rates Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC dmidli@cdiadvisors.com May 2, 203 Copyright 20, CDI Advisors LLC The cocept of preset value lies
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 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 informationOur aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
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 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 informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2
TO: Users of the ACTEX Review Semiar o DVD for SOA Exam FM/CAS Exam FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Exam FM (CAS
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationA Constant-Factor Approximation Algorithm for the Link Building Problem
A Costat-Factor Approximatio Algorithm for the Lik Buildig Problem Marti Olse 1, Aastasios Viglas 2, ad Ilia Zvedeiouk 2 1 Ceter for Iovatio ad Busiess Developmet, Istitute of Busiess ad Techology, Aarhus
More informationThis chapter considers the effect of managerial compensation on the desired
Chapter 4 THE EFFECT OF MANAGERIAL COMPENSATION ON OPTIMAL PRODUCTION AND HEDGING WITH FORWARDS AND PUTS 4.1 INTRODUCTION This chapter cosiders the effect of maagerial compesatio o the desired productio
More informationCapacity Management for Contract Manufacturing
OPERATIONS RESEARCH Vol. 55, No. 2, March April 2007, pp. 367 377 iss 0030-364X eiss 526-5463 07 5502 0367 iforms doi 0.287/opre.060.0359 2007 INFORMS Capacity Maagemet for Cotract Maufacturig Diwakar
More informationTHE problem of fitting a circle to a collection of points
IEEE TRANACTION ON INTRUMENTATION AND MEAUREMENT, VOL. XX, NO. Y, MONTH 000 A Few Methods for Fittig Circles to Data Dale Umbach, Kerry N. Joes Abstract Five methods are discussed to fit circles to data.
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 informationRISK TRANSFER FOR DESIGN-BUILD TEAMS
WILLIS CONSTRUCTION PRACTICE I-BEAM Jauary 2010 www.willis.com RISK TRANSFER FOR DESIGN-BUILD TEAMS Desig-builD work is icreasig each quarter. cosequetly, we are fieldig more iquiries from cliets regardig
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationExploratory Data Analysis
1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios
More informationA RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY
J. Appl. Prob. 45, 060 070 2008 Prited i Eglad Applied Probability Trust 2008 A RANDOM PERMUTATION MODEL ARISING IN CHEMISTRY MARK BROWN, The City College of New York EROL A. PEKÖZ, Bosto Uiversity SHELDON
More informationDegree of Approximation of Continuous Functions by (E, q) (C, δ) Means
Ge. Math. Notes, Vol. 11, No. 2, August 2012, pp. 12-19 ISSN 2219-7184; Copyright ICSRS Publicatio, 2012 www.i-csrs.org Available free olie at http://www.gema.i Degree of Approximatio of Cotiuous Fuctios
More informationPresent Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving
Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig Tax Policy Brach Departmet of Fiace Jue 30, 1998 2 Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig This
More informationADAPTIVE NETWORKS SAFETY CONTROL ON FUZZY LOGIC
8 th Iteratioal Coferece o DEVELOPMENT AND APPLICATION SYSTEMS S u c e a v a, R o m a i a, M a y 25 27, 2 6 ADAPTIVE NETWORKS SAFETY CONTROL ON FUZZY LOGIC Vadim MUKHIN 1, Elea PAVLENKO 2 Natioal Techical
More informationExtreme changes in prices of electricity futures
Isurace Marets ad Compaies: Aalyses ad Actuarial Computatios, Volume 2, Issue, 20 Roald Huisma (The Netherlads), Mehtap Kilic (The Netherlads) Extreme chages i prices of electricity futures Abstract The
More informationNon-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationInference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT - Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationIran. J. Chem. Chem. Eng. Vol. 26, No.1, 2007. Sensitivity Analysis of Water Flooding Optimization by Dynamic Optimization
Ira. J. Chem. Chem. Eg. Vol. 6, No., 007 Sesitivity Aalysis of Water Floodig Optimizatio by Dyamic Optimizatio Gharesheiklou, Ali Asghar* + ; Mousavi-Dehghai, Sayed Ali Research Istitute of Petroleum Idustry
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 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 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 Kolmogorov-Smirov 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 informationApproximate Option Pricing
Approximate Optio Pricig PRASAD CHALASANI Los Alamos Natioal Laboratory chal@lal.gov http://www.c3.lal.gov/chal SOMESH JHA Caregie Mello Uiversity sjha@cs.cmu.edu http://www.cs.cmu.edu/sjha ISAAC SAIAS
More informationA note on the boundary behavior for a modified Green function in the upper-half space
Zhag ad Pisarev Boudary Value Problems (015) 015:114 DOI 10.1186/s13661-015-0363-z RESEARCH Ope Access A ote o the boudary behavior for a modified Gree fuctio i the upper-half space Yulia Zhag1 ad Valery
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
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 informationTerminology for Bonds and Loans
³ ² ± Termiology for Bods ad Loas Pricipal give to borrower whe loa is made Simple loa: pricipal plus iterest repaid at oe date Fixed-paymet loa: series of (ofte equal) repaymets Bod is issued at some
More informationTHE TIME VALUE OF MONEY
QRMC04 9/17/01 4:43 PM Page 51 CHAPTER FOUR THE TIME VALUE OF MONEY 4.1 INTRODUCTION AND FUTURE VALUE The perspective ad the orgaizatio of this chapter differs from that of chapters 2 ad 3 i that topics
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 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 informationIs there employment discrimination against the disabled? Melanie K Jones i. University of Wales, Swansea
Is there employmet discrimiatio agaist the disabled? Melaie K Joes i Uiversity of Wales, Swasea Abstract Whilst cotrollig for uobserved productivity differeces, the gap i employmet probabilities betwee
More informationMEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)
MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:
More informationSavings and Retirement Benefits
60 Baltimore Couty Public Schools offers you several ways to begi savig moey through payroll deductios. Defied Beefit Pesio Pla Tax Sheltered Auities ad Custodial Accouts Defied Beefit Pesio Pla Did you
More informationBond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond
What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal
More informationTradigms of Astundithi and Toyota
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 informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 haupt@ieee.org Abstract:
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 informationInequalities for the surface area of projections of convex bodies
Iequalities for the surface area of projectios of covex bodies Apostolos Giaopoulos, Alexader Koldobsky ad Petros Valettas Abstract We provide geeral iequalities that compare the surface area S(K) of a
More informationTIGHT BOUNDS ON EXPECTED ORDER STATISTICS
Probability i the Egieerig ad Iformatioal Scieces, 20, 2006, 667 686+ Prited i the U+S+A+ TIGHT BOUNDS ON EXPECTED ORDER STATISTICS DIMITRIS BERTSIMAS Sloa School of Maagemet ad Operatios Research Ceter
More informationON THE DENSE TRAJECTORY OF LASOTA EQUATION
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory
More information7. Concepts in Probability, Statistics and Stochastic Modelling
7. Cocepts i Probability, Statistics ad Stochastic Modellig 1. Itroductio 169. Probability Cocepts ad Methods 170.1. Radom Variables ad Distributios 170.. Expectatio 173.3. Quatiles, Momets ad Their Estimators
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More information