University of Texas MD Anderson Cancer Center Department of Biostatistics. Inequality Calculator, Version 3.0 November 25, 2013 User s Guide
|
|
- Beatrice Owen
- 5 years ago
- Views:
Transcription
1 University of Texs MD Anderson Cncer Center Deprtment of Biosttistics Inequlity Clcultor, Version 3.0 November 5, 013 User s Guide 0. Overview The purpose of the softwre is to clculte the probbility tht one rndom vrible is greter thn nother. The two rndom vribles re ssumed to follow the sme stndrd distribution fmily, with different prmeter vlues. The distributions currently supported re: bet, gmm, inverse gmm, norml, log norml nd Weibull. The softwre my be downloded from the web site: This softwre occsionlly sends usge sttistics nd crsh reports to our biosttistics softwre support tem to improve your experience using it. License: This progrm is distributed t no cost to the user. However, redistribution of this progrm is not permitted. Rther thn giving this progrm to someone else, plese hve them obtin their own copy directly from M. D. Anderson Cncer Center. This llows us to keep record of who is using the softwre nd llows us to notify ll users when progrm enhncements become vilble. NO WARRANTY We provide bsolutely no wrrnty of ny kind, either expressed or implied, including but not limited to the implied wrrnties of merchntbility nd fitness for prticulr purpose. The entire risk s to the qulity nd performnce of the progrm is with the user. Should this progrm prove defective, the user ssumes the cost of ll necessry servicing, repir, or correction. In no event shll The University of Texs or ny of its component institutions, including M. D. Anderson Cncer Center, be lible for dmges, including ny lost profits, lost monies, or other specil, incidentl or consequentil dmges rising out of the use of or inbility to use the progrm (including but not limited to loss of dt or its nlysis being rendered inccurte or losses sustined by third prties).
2 1. Description For two generl continuous rndom vribles nd Y, the probbility tht > Y is given by P( > Y ) = f ( t) F ( t dt Y ) where f is the PDF (probbility density function) of nd F Y is the CDF (cumultive distribution function) of Y. The progrm computes this probbility for the cse of nd Y following the sme stndrd distribution fmily. In the exmple of Figure 1, we let ~ Norml(0, 1) nd Y ~ Norml(1, 1) nd click on Clculte to obtin the result Prob( > Y) = , long with the complementry vlue Prob(Y > ) = Figure 1:
3 . Shift Prmeter For the cse of nonzero delt shift prmeter, the probbility inequlity is given by P( > Y + delt ) = f ( t) F ( t delt dt. Y ) In the exmple of Figure, we let ~ bet(6, 6), Y ~ bet(3, 3) nd delt = 0.1, giving the result If delt = 0 then the result would be 0.5 due to symmetry. Figure :
4 3. Clcultions When the rndom vribles nd Y follow one of the specil distributions, there re more efficient methods for computing P(>Y) thn directly pplying the integrtion formul. For exmple, if nd Y follow exponentil distributions (gmm with shpe 1) with mens μ nd μ Y, then µ P( > Y ) = µ + µ Y If nd Y follow the norml distribution with prmeters (μ, σ ) nd (μ Y, σ Y ) respectively, then ( ) µ µ Y P( > Y ) = Φ 1/ σ + σ Y where Φ is the cumultive distribution function of the stndrd norml. More informtion on these clcultions cn be found in the pper Numericl computtion of stochstic inequlity probbilities by John Cook. For nonzero delt nd the norml distribution fmily, Y+delt ~ Norml(μ Y + delt, σ Y ) so tht ( ) µ µ Y delt P( > Y + delt) = Φ 1/ σ + σ Y The other probbility distribution fmilies re not closed under trnsltion (i.e., Y+delt cnnot be prmeterized s stndrd distribution when delt is nonzero), so the probbility inequlity clcultions do use numericl integrtion.
5 4. Prmeteriztions The softwre generlly follows the conventions (with some exception) in the book Sttisticl Distributions by Merrn Evns, Nichols Hstings, nd Brin Pecock. 4.1 Bet The bet distribution with prmeters nd b nd hs PDF Γ( + b) 1 b 1 x (1 x) Γ( ) Γ( b) with men + b nd vrince b ( + b) ( + b + 1) 4. Gmm The gmm distribution with shpe prmeter nd scle prmeter b hs men b, vrince b nd PDF 1 1 x / b x e Γ( ) b 4.3 Inverse Gmm The inverse gmm distribution with shpe prmeter nd scle prmeter b hs PDF b b / x e + 1 x ( ) Γ If > 1 then the men is b 1 If > then the vrince is b ( 1) ( ) If is distributed s gmm distribution with prmeters (, b) then 1/ is distributed s n inverse gmm with prmeters (, 1/b). Note tht the b in our prmeteriztion of the inverse gmm corresponds to 1/b in nother populr convention.
6 4.4 Norml The norml distribution prmeterized by its men m nd vrince s s 1 1 ( x m) s e π hs PDF 4.5 Log Norml The log norml distribution is prmeterized by m nd s. If is log norml with these prmeters, log is N(m,s ). Note tht m nd s re not the men nd stndrd devition of but rther of log. The PDF is given by 1 (log( x) m) exp xs π s with men 1 exp m + s nd vrince exp(m + s )(exp( s 4.6 Weibull ) 1) The Weibull distribution hs shpe prmeter nd scle prmeter b. It hs PDF 1 x exp( ( x / b) ) b with men b Γ (( + 1) / ) nd vrince b Γ (( + ) / ) Γ(( + 1) / ) ( )
7 5. Miscellneous 5.1 Grphic Options The user my customize the PDF plot by right-clicking on the window to bring up menu with the item Grphic Options (Figure 5). Figure 5: Choosing this item will bring up the dilog box in Figure 5b. Any chnge to one of these dilog fields will be reflected immeditely in the grph pne. Clicking the Accept button will sve chnges to the grph, while clicking Cncel will discrd ll chnges.
8 Figure 5b: The Copy menu item in the popup menu of Figure 5 will copy the grph imge to the clipbord where it cn be psted into nother document. The Sve menu item will bring up "Sve As..." dilog box for selecting the nme nd loction of the file to be sved. The qulity of the grph is dependent on the size of the grph when it is copied or sved. Therefore, it is suggested tht the window be mximized before copying or sving the grph. This will produce clerer imge when enlrged. 5. Error Indictor Ech numericl input field is equipped with n error indictor. When the symbol is displyed, its djcent field contins n invlid entry. By moving the mouse cursor over this symbol, tip will be displyed s shown in Figure 5c. Figure 5c:
Tests for the Ratio of Two Poisson Rates
Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More information38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes
The Uniform Distribution 8. Introduction This Section introduces the simplest type of continuous probbility distribution which fetures continuous rndom vrible X with probbility density function f(x) which
More information7 - Continuous random variables
7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationHybrid Group Acceptance Sampling Plan Based on Size Biased Lomax Model
Mthemtics nd Sttistics 2(3): 137-141, 2014 DOI: 10.13189/ms.2014.020305 http://www.hrpub.org Hybrid Group Acceptnce Smpling Pln Bsed on Size Bised Lomx Model R. Subb Ro 1,*, A. Ng Durgmmb 2, R.R.L. Kntm
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationAsymptotic results for Normal-Cauchy model
Asymptotic results for Norml-Cuchy model John D. Cook Deprtment of Biosttistics P. O. Box 342, Unit 49 The University of Texs, M. D. Anderson Cncer Center Houston, Texs 7723-42, USA cook@mdnderson.org
More informationAcceptance Sampling by Attributes
Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationNon-Linear & Logistic Regression
Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationMath 135, Spring 2012: HW 7
Mth 3, Spring : HW 7 Problem (p. 34 #). SOLUTION. Let N the number of risins per cookie. If N is Poisson rndom vrible with prmeter λ, then nd for this to be t lest.99, we need P (N ) P (N ) ep( λ) λ ln(.)
More information4.1. Probability Density Functions
STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationUNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction
Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from
More informationExpectation and Variance
Expecttion nd Vrince : sum of two die rolls P(= P(= = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 P(=2) = 1/36 P(=3) = 1/18 P(=4) = 1/12 P(=5) = 1/9 P(=7) = 1/6 P(=13) =? 2 1/36 3 1/18 4 1/12 5 1/9 6 5/36 7 1/6
More informationTime Truncated Two Stage Group Sampling Plan For Various Distributions
Time Truncted Two Stge Group Smpling Pln For Vrious Distributions Dr. A. R. Sudmni Rmswmy, S.Jysri Associte Professor, Deprtment of Mthemtics, Avinshilingm University, Coimbtore Assistnt professor, Deprtment
More informationMATH20812: PRACTICAL STATISTICS I SEMESTER 2 NOTES ON RANDOM VARIABLES
MATH20812: PRACTICAL STATISTICS I SEMESTER 2 NOTES ON RANDOM VARIABLES Things to Know Rndom Vrible A rndom vrible is function tht ssigns numericl vlue to ech outcome of prticulr experiment. A rndom vrible
More informationContinuous Random Variables
CPSC 53 Systems Modeling nd Simultion Continuous Rndom Vriles Dr. Anirn Mhnti Deprtment of Computer Science University of Clgry mhnti@cpsc.uclgry.c Definitions A rndom vrile is sid to e continuous if there
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More informationUsing QM for Windows. Using QM for Windows. Using QM for Windows LEARNING OBJECTIVES. Solving Flair Furniture s LP Problem
LEARNING OBJECTIVES Vlu%on nd pricing (November 5, 2013) Lecture 11 Liner Progrmming (prt 2) 10/8/16, 2:46 AM Olivier J. de Jong, LL.M., MM., MBA, CFD, CFFA, AA www.olivierdejong.com Solving Flir Furniture
More informationDepartment of Physical Pharmacy and Pharmacokinetics Poznań University of Medical Sciences Pharmacokinetics laboratory
Deprtment of Physicl Phrmcy nd Phrmcoinetics Poznń University of Medicl Sciences Phrmcoinetics lbortory Experiment 1 Phrmcoinetics of ibuprofen s n exmple of the first-order inetics in n open one-comprtment
More informationProblem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume
Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom r vrible. Define new rndom vrible Y = g. Find the pdf of Y. Method: Step : Step : Step 3: Plot Y = g( ). Find F ( y) by mpping
More informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance
Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between
More informationNew Integral Inequalities for n-time Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationStudent Activity 3: Single Factor ANOVA
MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether
More informationModule 11.4: nag quad util Numerical Integration Utilities. Contents
Qudrture Module Contents Module 11.4: ng qud util Numericl Integrtion Utilities ng qud util provides utility procedures for computtion involving integrtion of functions, e.g., the computtion of the weights
More informationLecture 21: Order statistics
Lecture : Order sttistics Suppose we hve N mesurements of sclr, x i =, N Tke ll mesurements nd sort them into scending order x x x 3 x N Define the mesured running integrl S N (x) = 0 for x < x = i/n for
More informationCalculus - Activity 1 Rate of change of a function at a point.
Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More information5 Probability densities
5 Probbility densities 5. Continuous rndom vribles 5. The norml distribution 5.3 The norml pproimtion to the binomil distribution 5.5 The uniorm distribution 5. Joint distribution discrete nd continuous
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationThe use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.
ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion
More informationTHE EXISTENCE OF NEGATIVE MCMENTS OF CONTINUOUS DISTRIBUTIONS WALTER W. PIEGORSCH AND GEORGE CASELLA. Biometrics Unit, Cornell University, Ithaca, NY
THE EXISTENCE OF NEGATIVE MCMENTS OF CONTINUOUS DISTRIBUTIONS WALTER W. PIEGORSCH AND GEORGE CASELLA. Biometrics Unit, Cornell University, Ithc, NY 14853 BU-771-M * December 1982 Abstrct The question of
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics MOMENTS INEQUALITIES OF A RANDOM VARIABLE DEFINED OVER A FINITE INTERVAL PRANESH KUMAR Deprtment of Mthemtics & Computer Science University of Northern
More informationSession Trimester 2. Module Code: MATH08001 MATHEMATICS FOR DESIGN
School of Science & Sport Pisley Cmpus Session 05-6 Trimester Module Code: MATH0800 MATHEMATICS FOR DESIGN Dte: 0 th My 06 Time: 0.00.00 Instructions to Cndidtes:. Answer ALL questions in Section A. Section
More informationMonte Carlo method in solving numerical integration and differential equation
Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationMethod: Step 1: Step 2: Find f. Step 3: = Y dy. Solution: 0, ( ) 0, y. Assume
Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom vrible. Define new rndom vrible Y g( ) ). Find the pdf of Y. Method: Step : Step : Step 3: Plot Y g( ). Find F ( ) b mpping
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationMarkscheme May 2016 Mathematics Standard level Paper 1
M6/5/MATME/SP/ENG/TZ/XX/M Mrkscheme My 06 Mthemtics Stndrd level Pper 7 pges M6/5/MATME/SP/ENG/TZ/XX/M This mrkscheme is the property of the Interntionl Bcclurete nd must not be reproduced or distributed
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information2008 Mathematical Methods (CAS) GA 3: Examination 2
Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed
More informationSolution for Assignment 1 : Intro to Probability and Statistics, PAC learning
Solution for Assignment 1 : Intro to Probbility nd Sttistics, PAC lerning 10-701/15-781: Mchine Lerning (Fll 004) Due: Sept. 30th 004, Thursdy, Strt of clss Question 1. Bsic Probbility ( 18 pts) 1.1 (
More informationSection 11.5 Estimation of difference of two proportions
ection.5 Estimtion of difference of two proportions As seen in estimtion of difference of two mens for nonnorml popultion bsed on lrge smple sizes, one cn use CLT in the pproximtion of the distribution
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationThe steps of the hypothesis test
ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of
More informationSongklanakarin Journal of Science and Technology SJST R1 Thongchan. A Modified Hyperbolic Secant Distribution
A Modified Hyperbolic Secnt Distribution Journl: Songklnkrin Journl of Science nd Technology Mnuscript ID SJST-0-0.R Mnuscript Type: Originl Article Dte Submitted by the Author: 0-Mr-0 Complete List of
More informationA Matrix Algebra Primer
A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,
More informationDecision Science Letters
Decision Science Letters 8 (09) 37 3 Contents lists vilble t GrowingScience Decision Science Letters homepge: www.growingscience.com/dsl The negtive binomil-weighted Lindley distribution Sunthree Denthet
More informationA SHORT NOTE ON THE MONOTONICITY OF THE ERLANG C FORMULA IN THE HALFIN-WHITT REGIME. Bernardo D Auria 1
Working Pper 11-42 (31) Sttistics nd Econometrics Series December, 2011 Deprtmento de Estdístic Universidd Crlos III de Mdrid Clle Mdrid, 126 28903 Getfe (Spin) Fx (34) 91 624-98-49 A SHORT NOTE ON THE
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationS. S. Dragomir. 2, we have the inequality. b a
Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely
More informationTO: Next Year s AP Calculus Students
TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC
More informationLab 11 Approximate Integration
Nme Student ID # Instructor L Period Dte Due L 11 Approximte Integrtion Ojectives 1. To ecome fmilir with the right endpoint rule, the trpezoidl rule, nd Simpson's rule. 2. To compre nd contrst the properties
More informationDriving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d
Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,
More informationEstimation of Binomial Distribution in the Light of Future Data
British Journl of Mthemtics & Computer Science 102: 1-7, 2015, Article no.bjmcs.19191 ISSN: 2231-0851 SCIENCEDOMAIN interntionl www.sciencedomin.org Estimtion of Binomil Distribution in the Light of Future
More informationChapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses
Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationThe Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases.
Section 1 Section 2 The module clcultes nd plots isotherml 1-, 2- nd 3-metl predominnce re digrms. ccesses only compound dtbses. Tble of Contents Tble of Contents Opening the module Section 3 Stoichiometric
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationMultivariate problems and matrix algebra
University of Ferrr Stefno Bonnini Multivrite problems nd mtrix lgebr Multivrite problems Multivrite sttisticl nlysis dels with dt contining observtions on two or more chrcteristics (vribles) ech mesured
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationNormal Distribution. Lecture 6: More Binomial Distribution. Properties of the Unit Normal Distribution. Unit Normal Distribution
Norml Distribution Lecture 6: More Binomil Distribution If X is rndom vrible with norml distribution with men µ nd vrince σ 2, X N (µ, σ 2, then P(X = x = f (x = 1 e 1 (x µ 2 2 σ 2 σ Sttistics 104 Colin
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationObjectives. Materials
Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationProbability Distributions for Gradient Directions in Uncertain 3D Scalar Fields
Technicl Report 7.8. Technische Universität München Probbility Distributions for Grdient Directions in Uncertin 3D Sclr Fields Tobis Pfffelmoser, Mihel Mihi, nd Rüdiger Westermnn Computer Grphics nd Visuliztion
More informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationThe Shortest Confidence Interval for the Mean of a Normal Distribution
Interntionl Journl of Sttistics nd Proility; Vol. 7, No. 2; Mrch 208 ISSN 927-7032 E-ISSN 927-7040 Pulished y Cndin Center of Science nd Eduction The Shortest Confidence Intervl for the Men of Norml Distriution
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationThe EbayesThresh Package
The EbyesThresh Pckge Mrch 26, 2005 Title Empiricl Byes thresholding nd relted methods Version 1.3.0 Dte 2005-03-24 Author Bernrd W. Silvermn Mintiner Bernrd W. Silvermn
More informationX Z Y Table 1: Possibles values for Y = XZ. 1, p
ECE 534: Elements of Informtion Theory, Fll 00 Homework 7 Solutions ll by Kenneth Plcio Bus October 4, 00. Problem 7.3. Binry multiplier chnnel () Consider the chnnel Y = XZ, where X nd Z re independent
More informationCS 109 Lecture 11 April 20th, 2016
CS 09 Lecture April 0th, 06 Four Prototypicl Trjectories Review The Norml Distribution is Norml Rndom Vrible: ~ Nµ, σ Probbility Density Function PDF: f x e σ π E[ ] µ Vr σ x µ / σ Also clled Gussin Note:
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More informationLecture INF4350 October 12008
Biosttistics ti ti Lecture INF4350 October 12008 Anj Bråthen Kristoffersen Biomedicl Reserch Group Deprtment of informtics, UiO Gol Presenttion of dt descriptive tbles nd grphs Sensitivity, specificity,
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationNumerical Analysis. 10th ed. R L Burden, J D Faires, and A M Burden
Numericl Anlysis 10th ed R L Burden, J D Fires, nd A M Burden Bemer Presenttion Slides Prepred by Dr. Annette M. Burden Youngstown Stte University July 9, 2015 Chpter 4.1: Numericl Differentition 1 Three-Point
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationConstruction of Gauss Quadrature Rules
Jim Lmbers MAT 772 Fll Semester 2010-11 Lecture 15 Notes These notes correspond to Sections 10.2 nd 10.3 in the text. Construction of Guss Qudrture Rules Previously, we lerned tht Newton-Cotes qudrture
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
More informationGeneralized Fano and non-fano networks
Generlized Fno nd non-fno networks Nildri Ds nd Brijesh Kumr Ri Deprtment of Electronics nd Electricl Engineering Indin Institute of Technology Guwhti, Guwhti, Assm, Indi Emil: {d.nildri, bkri}@iitg.ernet.in
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More information