Introduction to Probability and Statistics Chapter 7
|
|
- Clinton Harrell
- 5 years ago
- Views:
Transcription
1 Itroductio to Probability ad Statistics Chapter 7 Ammar M. Sarha, asarha@mathstat.dal.ca Departmet of Mathematics ad Statistics, Dalhousie Uiversity Fall Semester 008
2 Chapter 7 Statistical Itervals Based o a Sigle Sample Dr. Ammar Sarha
3 Cofidece Itervals A alterative to reportig a sigle value for the parameter beig estimated is to calculate ad report a etire iterval of plausible values a cofidece iterval (CI). A cofidece level is a measure of the degree of reliability of the iterval. Dr. Ammar Sarha 3
4 7.1 Basic Properties of Cofidece Itervals If after observig 1 x 1,, x, we compute the observed sample mea x, the a 95% cofidece iterval for μ ca be expressed as σ σ x 1.96 x 1.96 The populatio has a N(μ, σ ) ad σ is kow., Dr. Ammar Sarha 4
5 Other Levels of Cofidece A (1- α)100% cofidece iterval for the mea μ of a ormal populatio whe the value of is kow α is give by x σ x, σ Other Levels of Cofidece Dr. Ammar Sarha 5
6 Sample Sie The geeral formula for the sample sie ecessary to esure a iterval width w is σ w Dr. Ammar Sarha 6
7 Derivig a Cofidece Iterval Let 1,, deote the sample o which the CI for the parameter θ is to be based. Suppose a radom variable satisfyig the followig properties ca be foud: 1. The variable depeds fuctioally o both 1,, ad θ.. The probability distributio of the variable does ot deped o θ or ay other ukow parameters. Let h( 1,, ; θ) deote this radom variable. I geeral, the form of h is usually suggested by examiig the distributio of a appropriate estimator θˆ. For ay α, 0 < α < 1, costats a ad b ca be foud to satisfy P ( a < h ( 1, L, ; θ ) < b ) 1 α Dr. Ammar Sarha 7
8 Now suppose that the iequalities ca be maipulated to isolate θ P ( l, L, ) < θ < u (, L, )) 1 α ( 1 1 lower cofidece limit upper cofidece limit For a 100(1- α)% CI. Dr. Ammar Sarha 8
9 Examples: 7.5 p. 60. will be give i the class. Dr. Ammar Sarha 9
10 7. Large-Sample Cofidece Itervals for a Populatio Mea ad Proportio Large-Sample Cofidece Iterval Let 1,, deote the sample from a populatio havig a mea μ ad stadard deviatio σ. If is sufficietly large, the This implies that σ σ (0,1) is a large-sample cofidece iterval for μ with level 100(1- α)%. Z, This formula is valid regardless of the shape of the populatio distributio. For practice: > 40. ( μ, ) σ ~ N σ μ ~ N Dr. Ammar Sarha 10
11 Notice, if σ is ukow, replace it with the sample stadard deviatio s. That is, Examples: 7.6 p. 64. Suppose a radom sample with sie 48 from a populatio with ukow mea μ ad ukow variace σ with the followig iformatio: s, x i 66, ad x x 54.7 ad s 5.3 i , ,950 Fid the 95% cofidece iterval of μ? Solutio: From the iformatio give i the problem, we have: The, the 95% cofidece iterval of μ is ( 53., 56. ) That is, with a cofidece level of approximatio 95%, 53. < μ < 56. s Dr. Ammar Sarha 11
12 Dr. Ammar Sarha 1 Cofidece Iterval for a Populatio Proportio Let p deote the proportio of successes i a populatio, where success idetifies a idividual or object that has a specified property. A radom sample of idividuals is to be selected, ad is the umber of successes i the sample. A cofidece iterval for a populatio proportio p with level 100(1- α)% is: ( ) ( ) q p p q p p 1 4 ˆ ˆ ˆ, 1 4 ˆ ˆ ˆ α α α α α α α α where,. ˆ 1 ˆ, ˆ p q p
13 Notice, if the sample sie is quite large, the approximate CI limits become pˆ qˆ pˆ pˆ, Sice α () is egligible compared to pˆ. pˆ qˆ Sample Sie The geeral formula for the sample sie ecessary to esure a iterval width w is 4 pˆ qˆ w Dr. Ammar Sarha 13
14 Examples: 7.8 p. 67. Suppose that i 48 trials i a particular laboratory, 16 resulted i igitio of a particular type of substrate by a lighted cigarette. Fid the 95% cofidece iterval of the log-ru proportio of all such trails that would result i igitio? Solutio: 48 Let p deote the log-ru proportio of all such trails that would result i igitio pˆ 0.333, q 48 3 The, the approximately 95% CI of p is (1.96 ) ( 48 ) ± ± (0.333 )( ) 48 ( 1.96 ) 48 (0.17, 0.474) (1.96 ) 4 ( 48 ) Dr. Ammar Sarha 14
15 Notice, the traditioal 95%CI is (0.333 )( ) ± (0.00, ) Examples: 7.9 p. 67. Fid the sample sie ecessary to esure a width of 0.10 for the 95% cofidece iterval of the log-ru proportio of all such trails that would result i igitio? 4 pˆ qˆ w 4*(1.96) * 0.333* Dr. Ammar Sarha 15
16 7.3 Itervals Based o a Normal Populatio Distributio The populatio of iterest is ormal, so that 1,, costitutes a radom sample from a ormal distributio with both μ ad σ ukow. t Distributio Let 1,, ~ N(μ, σ ), the the rv T S μ has a probability distributio called a t distributio with -1 degrees of freedom (df). Dr. Ammar Sarha 16
17 Properties of t Distributios Let t v deote the desity fuctio curve for v df. 1. Each t v curve is bell-shaped ad cetered at 0.. Each t v curve is spread out more tha the stadard ormal () curve. 3. As v icreases, the spread of the correspodig t v curve decreases. 4. As v, the sequece of t v curves approaches the stadard ormal curve (the curve is called a t curve with df ) curve t 5 curve t 5 curve 0 Dr. Ammar Sarha 17
18 t Critical Value Let t α,v the umber o the measuremet axis for which the area uder the t curve with v df to the right of t α,v is α. t α,v is called t critical value. t critical value Table A.5, p. 671, gives the t critical value for give α, v. t 0.05,15.131, t 0.05, 1.717, t 0.01,.508. Dr. Ammar Sarha 18
19 t Cofidece Iterval Now, let 1,, be a radom sample from a ormal distributio with both μ ad σ ukow, the the (1- α)100% CI of μ is or t S, 1,, 1 t ±, 1 Here, ad S are the sample mea ad sample variace. s t S Dr. Ammar Sarha 19
20 Examples: 7.1 p. 74. Cosider the followig sample of fat cotet (i percetage) o 10 radomly selected hot dogs: Fid the 95% cofidece iterval of the populatio mea fat cotet, assumig the populatio is ormal.? Solutio: We have: 10, x 1.90 ad s 4.134, The, the 95% cofidece iterval of μ is t 0.05,10 1, 1.90 t 0.05, ( 18.94, 4.86 ) , Dr. Ammar Sarha 0
21 Summary The radom sample 1,, ~ N(μ, σ ) σ is kow (1- α)100% CI of μ ± σ 1,, ~ N(μ, σ ) σ is ukow t ±, 1 S 1,, ~ ay populatio with mea μ ad variace σ σ is kow ( 30) ± σ σ is ukow ( 30) ± S Dr. Ammar Sarha 1
point estimator a random variable (like P or X) whose values are used to estimate a population parameter
Estimatio We have oted that the pollig problem which attempts to estimate the proportio p of Successes i some populatio ad the measuremet problem which attempts to estimate the mea value µ of some quatity
More informationChapter 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
More informationChapter 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
More informationInferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty,
Iferetial Statistics ad Probability a Holistic Approach Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike 4.0
More informationSampling Distributions & Estimators
API-209 TF Sessio 2 Teddy Svoroos September 18, 2015 Samplig Distributios & Estimators I. Estimators The Importace of Samplig Radomly Three Properties of Estimators 1. Ubiased 2. Cosistet 3. Efficiet I
More information5. 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
More informationLecture 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
More informationStatistics 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
More informationConfidence Intervals Introduction
Cofidece Itervals Itroductio A poit estimate provides o iformatio about the precisio ad reliability of estimatio. For example, the sample mea X is a poit estimate of the populatio mea μ but because of
More informationA point estimate is the value of a statistic that estimates the value of a parameter.
Chapter 9 Estimatig the Value of a Parameter Chapter 9.1 Estimatig a Populatio Proportio Objective A : Poit Estimate A poit estimate is the value of a statistic that estimates the value of a parameter.
More informationSampling Distributions and Estimation
Samplig Distributios ad Estimatio T O P I C # Populatio Proportios, π π the proportio of the populatio havig some characteristic Sample proportio ( p ) provides a estimate of π : x p umber of successes
More informationToday: 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
More informationTopic-7. Large Sample Estimation
Topic-7 Large Sample Estimatio TYPES OF INFERENCE Ò Estimatio: É Estimatig or predictig the value of the parameter É What is (are) the most likely values of m or p? Ò Hypothesis Testig: É Decidig about
More informationEstimating 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
More informationii. 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
More informationExam 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
More informationChapter 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
More information. (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
More informationBASIC 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
More informationLecture 5 Point Es/mator and Sampling Distribu/on
Lecture 5 Poit Es/mator ad Samplig Distribu/o Fall 03 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech Road map Poit Es/ma/o Cofidece Iterval
More informationExam 2. Instructor: Cynthia Rudin TA: Dimitrios Bisias. October 25, 2011
15.075 Exam 2 Istructor: Cythia Rudi TA: Dimitrios Bisias October 25, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 You are i charge of a study
More informationA 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
More informationLecture 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
More informationLecture 4: Parameter Estimation and Confidence Intervals. GENOME 560 Doug Fowler, GS
Lecture 4: Parameter Estimatio ad Cofidece Itervals GENOME 560 Doug Fowler, GS (dfowler@uw.edu) 1 Review: Probability Distributios Discrete: Biomial distributio Hypergeometric distributio Poisso distributio
More informationConfidence 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).
More information14.30 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
More information18.S096 Problem Set 5 Fall 2013 Volatility Modeling Due Date: 10/29/2013
18.S096 Problem Set 5 Fall 2013 Volatility Modelig Due Date: 10/29/2013 1. Sample Estimators of Diffusio Process Volatility ad Drift Let {X t } be the price of a fiacial security that follows a geometric
More informationChapter 8 Interval Estimation. Estimation Concepts. General Form of a Confidence Interval
Chapter 8 Iterval Estimatio Estimatio Cocepts Usually ca't take a cesus, so we must make decisios based o sample data It imperative that we take the risk of samplig error ito accout whe we iterpret sample
More informationCHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions
CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: I this chapter we wat to fid out the value of a parameter for a populatio. We do t kow the value of this parameter for the etire
More informationB = A x z
114 Block 3 Erdeky == Begi 6.3 ============================================================== 1 / 8 / 2008 1 Correspodig Areas uder a ormal curve ad the stadard ormal curve are equal. Below: Area B = Area
More informationStandard Deviations for Normal Sampling Distributions are: For proportions For means _
Sectio 9.2 Cofidece Itervals for Proportios We will lear to use a sample to say somethig about the world at large. This process (statistical iferece) is based o our uderstadig of samplig models, ad will
More informationCHAPTER 8 Estimating with Confidence
CHAPTER 8 Estimatig with Cofidece 8.2 Estimatig a Populatio Proportio The Practice of Statistics, 5th Editio Stares, Tabor, Yates, Moore Bedford Freema Worth Publishers Estimatig a Populatio Proportio
More information1. Find the area under the standard normal curve between z = 0 and z = 3. (a) (b) (c) (d)
STA 2023 Practice 3 You may receive assistace from the Math Ceter. These problems are iteded to provide supplemetary problems i preparatio for test 3. This packet does ot ecessarily reflect the umber,
More informationST 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
More informationCHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions
CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: We wat to kow the value of a parameter for a populatio. We do t kow the value of this parameter for the etire populatio because
More informationStatistics for Business and Economics
Statistics for Busiess ad Ecoomics Chapter 8 Estimatio: Additioal Topics Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-1 8. Differece Betwee Two Meas: Idepedet Samples Populatio meas,
More informationCombining 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
More informationx 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.
More information4.5 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.
More informationParametric 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
More informationOnline appendices from Counterparty Risk and Credit Value Adjustment a continuing challenge for global financial markets by Jon Gregory
Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8A: Formulas for EE, PFE ad EPE for a ormal distributio Cosider a ormal distributio with mea (expected future value) ad stadard
More informationMath 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
More information1. Suppose X is a variable that follows the normal distribution with known standard deviation σ = 0.3 but unknown mean µ.
Chapter 9 Exercises Suppose X is a variable that follows the ormal distributio with kow stadard deviatio σ = 03 but ukow mea µ (a) Costruct a 95% cofidece iterval for µ if a radom sample of = 6 observatios
More informationNOTES 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
More information5 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
More information1 Random Variables and Key Statistics
Review of Statistics 1 Radom Variables ad Key Statistics Radom Variable: A radom variable is a variable that takes o differet umerical values from a sample space determied by chace (probability distributio,
More informationThese characteristics are expressed in terms of statistical properties which are estimated from the sample data.
0. Key Statistical Measures of Data Four pricipal features which characterize a set of observatios o a radom variable are: (i) the cetral tedecy or the value aroud which all other values are buched, (ii)
More informationResearch 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
More informationr i = a i + b i f b i = Cov[r i, f] The only parameters to be estimated for this model are a i 's, b i 's, σe 2 i
The iformatio required by the mea-variace approach is substatial whe the umber of assets is large; there are mea values, variaces, ad )/2 covariaces - a total of 2 + )/2 parameters. Sigle-factor model:
More informationBasic formula for confidence intervals. Formulas for estimating population variance Normal Uniform Proportion
Basic formula for the Chi-square test (Observed - Expected ) Expected Basic formula for cofidece itervals sˆ x ± Z ' Sample size adjustmet for fiite populatio (N * ) (N + - 1) Formulas for estimatig populatio
More informationDr. Maddah ENMG 624 Financial Eng g I 03/22/06. Chapter 6 Mean-Variance Portfolio Theory
Dr Maddah ENMG 64 Fiacial Eg g I 03//06 Chapter 6 Mea-Variace Portfolio Theory Sigle Period Ivestmets Typically, i a ivestmet the iitial outlay of capital is kow but the retur is ucertai A sigle-period
More informationSTAT 135 Solutions to Homework 3: 30 points
STAT 35 Solutios to Homework 3: 30 poits Sprig 205 The objective of this Problem Set is to study the Stei Pheomeo 955. Suppose that θ θ, θ 2,..., θ cosists of ukow parameters, with 3. We wish to estimate
More informationChpt 5. Discrete Probability Distributions. 5-3 Mean, Variance, Standard Deviation, and Expectation
Chpt 5 Discrete Probability Distributios 5-3 Mea, Variace, Stadard Deviatio, ad Expectatio 1/23 Homework p252 Applyig the Cocepts Exercises p253 1-19 2/23 Objective Fid the mea, variace, stadard deviatio,
More informationBIOSTATS 540 Fall Estimation Page 1 of 72. Unit 6. Estimation. Use at least twelve observations in constructing a confidence interval
BIOSTATS 540 Fall 015 6. Estimatio Page 1 of 7 Uit 6. Estimatio Use at least twelve observatios i costructig a cofidece iterval - Gerald va Belle What is the mea of the blood pressures of all the studets
More informationTopic 14: Maximum Likelihood Estimation
Toic 4: November, 009 As before, we begi with a samle X = (X,, X of radom variables chose accordig to oe of a family of robabilities P θ I additio, f(x θ, x = (x,, x will be used to deote the desity fuctio
More informationCHAPTER 8 CONFIDENCE INTERVALS
CHAPTER 8 CONFIDENCE INTERVALS Cofidece Itervals is our first topic i iferetial statistics. I this chapter, we use sample data to estimate a ukow populatio parameter: either populatio mea (µ) or populatio
More informationLecture 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=
More informationThe Idea of a Confidence Interval
AP Statistics Ch. 8 Notes Estimatig with Cofidece I the last chapter, we aswered questios about what samples should look like assumig that we kew the true values of populatio parameters (like μ, σ, ad
More informationOutline. Populations. Defs: A (finite) population is a (finite) set P of elements e. A variable is a function v : P IR. Population and Characteristics
Outlie Populatio Characteristics Types of Samples Sample Characterstics Sample Aalogue Estimatio Populatios Defs: A (fiite) populatio is a (fiite) set P of elemets e. A variable is a fuctio v : P IR. Examples
More informationUnbiased estimators Estimators
19 Ubiased estimators I Chapter 17 we saw that a dataset ca be modeled as a realizatio of a radom sample from a probability distributio ad that quatities of iterest correspod to features of the model distributio.
More informationSampling 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
More informationAY Term 2 Mock Examination
AY 206-7 Term 2 Mock Examiatio Date / Start Time Course Group Istructor 24 March 207 / 2 PM to 3:00 PM QF302 Ivestmet ad Fiacial Data Aalysis G Christopher Tig INSTRUCTIONS TO STUDENTS. This mock examiatio
More informationFINM6900 Finance Theory How Is Asymmetric Information Reflected in Asset Prices?
FINM6900 Fiace Theory How Is Asymmetric Iformatio Reflected i Asset Prices? February 3, 2012 Referece S. Grossma, O the Efficiecy of Competitive Stock Markets where Traders Have Diverse iformatio, Joural
More informationSCHOOL 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
More informationControl 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
More informationMonetary 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.
More informationA Bayesian perspective on estimating mean, variance, and standard-deviation from data
Brigham Youg Uiversity BYU ScholarsArchive All Faculty Publicatios 006--05 A Bayesia perspective o estimatig mea, variace, ad stadard-deviatio from data Travis E. Oliphat Follow this ad additioal works
More informationSequences and Series
Sequeces ad Series Matt Rosezweig Cotets Sequeces ad Series. Sequeces.................................................. Series....................................................3 Rudi Chapter 3 Exercises........................................
More information1 Estimating sensitivities
Copyright c 27 by Karl Sigma 1 Estimatig sesitivities Whe estimatig the Greeks, such as the, the geeral problem ivolves a radom variable Y = Y (α) (such as a discouted payoff) that depeds o a parameter
More information1 Basic Growth Models
UCLA Aderso MGMT37B: Fudametals i Fiace Fall 015) Week #1 rofessor Eduardo Schwartz November 9, 015 Hadout writte by Sheje Hshieh 1 Basic Growth Models 1.1 Cotiuous Compoudig roof: lim 1 + i m = expi)
More informationMaximum Empirical Likelihood Estimation (MELE)
Maximum Empirical Likelihood Estimatio (MELE Natha Smooha Abstract Estimatio of Stadard Liear Model - Maximum Empirical Likelihood Estimator: Combiatio of the idea of imum likelihood method of momets,
More informationAn 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,
More information0.1 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 ()
More informationRafa 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
More informationECON 5350 Class Notes Maximum Likelihood Estimation
ECON 5350 Class Notes Maximum Likelihood Estimatio 1 Maximum Likelihood Estimatio Example #1. Cosider the radom sample {X 1 = 0.5, X 2 = 2.0, X 3 = 10.0, X 4 = 1.5, X 5 = 7.0} geerated from a expoetial
More informationPoint Estimation by MLE Lesson 5
Poit Estimatio b MLE Lesso 5 Review Defied Likelihood Maximum Likelihood Estimatio Step : Costruct Likelihood Step : Maximize fuctio Take Log of likelihood fuctio Take derivative of fuctio Set derivative
More informationPoint Estimation by MLE Lesson 5
Poit Estimatio b MLE Lesso 5 Review Defied Likelihood Maximum Likelihood Estimatio Step : Costruct Likelihood Step : Maximize fuctio Take Log of likelihood fuctio Take derivative of fuctio Set derivative
More information43. A 000 par value 5-year bod with 8.0% semiaual coupos was bought to yield 7.5% covertible semiaually. Determie the amout of premium amortized i the 6 th coupo paymet. (A).00 (B).08 (C).5 (D).5 (E).34
More information. The firm makes different types of furniture. Let x ( x1,..., x n. If the firm produces nothing it rents out the entire space and so has a profit of
Joh Riley F Maimizatio with a sigle costrait F3 The Ecoomic approach - - shadow prices Suppose that a firm has a log term retal of uits of factory space The firm ca ret additioal space at a retal rate
More informationAsymptotics: Consistency and Delta Method
ad Delta Method MIT 18.655 Dr. Kempthore Sprig 2016 1 MIT 18.655 ad Delta Method Outlie Asymptotics 1 Asymptotics 2 MIT 18.655 ad Delta Method Cosistecy Asymptotics Statistical Estimatio Problem X 1,...,
More informationData Analysis and Statistical Methods Statistics 651
Data Aalyi ad Statitical Method Statitic 65 http://www.tat.tamu.edu/~uhaii/teachig.html Lecture 9 Suhaii Subba Rao Tetig o far We have looked at oe ample hypothei tet of the form H 0 : µ = µ 0 agait H
More information1 Estimating the uncertainty attached to a sample mean: s 2 vs.
Political Sciece 100a/200a Fall 2001 Cofidece itervals ad hypothesis testig, Part I 1 1 Estimatig the ucertaity attached to a sample mea: s 2 vs. σ 2 Recall the problem of descriptive iferece: We wat to
More informationBayes 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
More informationProblem Set 1a - Oligopoly
Advaced Idustrial Ecoomics Sprig 2014 Joha Steek 6 may 2014 Problem Set 1a - Oligopoly 1 Table of Cotets 2 Price Competitio... 3 2.1 Courot Oligopoly with Homogeous Goods ad Differet Costs... 3 2.2 Bertrad
More informationProbability and Statistical Methods. Chapter 8 Fundamental Sampling Distributions
Math 3 Probablty ad Statstcal Methods Chapter 8 Fudametal Samplg Dstrbutos Samplg Dstrbutos I the process of makg a ferece from a sample to a populato we usually calculate oe or more statstcs, such as
More informationProbability and Statistical Methods. Chapter 8 Fundamental Sampling Distributions
Math 3 Probablty ad Statstcal Methods Chapter 8 Fudametal Samplg Dstrbutos Samplg Dstrbutos I the process of makg a ferece from a sample to a populato we usually calculate oe or more statstcs, such as
More informationBinomial 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
More informationSubject CT5 Contingencies Core Technical. Syllabus. for the 2011 Examinations. The Faculty of Actuaries and Institute of Actuaries.
Subject CT5 Cotigecies Core Techical Syllabus for the 2011 Examiatios 1 Jue 2010 The Faculty of Actuaries ad Istitute of Actuaries Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical
More informationConfidence Intervals based on Absolute Deviation for Population Mean of a Positively Skewed Distribution
Iteratioal Joural of Computatioal ad Theoretical Statistics ISSN (220-59) It. J. Comp. Theo. Stat. 5, No. (May-208) http://dx.doi.org/0.2785/ijcts/0500 Cofidece Itervals based o Absolute Deviatio for Populatio
More informationCAPITAL ASSET PRICING MODEL
CAPITAL ASSET PRICING MODEL RETURN. Retur i respect of a observatio is give by the followig formula R = (P P 0 ) + D P 0 Where R = Retur from the ivestmet durig this period P 0 = Curret market price P
More informationSummary. Recap. Last Lecture. .1 If you know MLE of θ, can you also know MLE of τ(θ) for any function τ?
Last Lecture Biostatistics 60 - Statistical Iferece Lecture Cramer-Rao Theorem Hyu Mi Kag February 9th, 03 If you kow MLE of, ca you also kow MLE of τ() for ay fuctio τ? What are plausible ways to compare
More informationIntroduction to Statistical Inference
Itroductio to Statistical Iferece Fial Review CH1: Picturig Distributios With Graphs 1. Types of Variable -Categorical -Quatitative 2. Represetatios of Distributios (a) Categorical -Pie Chart -Bar Graph
More informationSimulation Efficiency and an Introduction to Variance Reduction Methods
Mote Carlo Simulatio: IEOR E4703 Columbia Uiversity c 2017 by Marti Haugh Simulatio Efficiecy ad a Itroductio to Variace Reductio Methods I these otes we discuss the efficiecy of a Mote-Carlo estimator.
More informationQuantitative Analysis
EduPristie www.edupristie.com Modellig Mea Variace Skewess Kurtosis Mea: X i = i Mode: Value that occurs most frequetly Media: Midpoit of data arraged i ascedig/ descedig order s Avg. of squared deviatios
More informationNotes 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
More informationThe Valuation of the Catastrophe Equity Puts with Jump Risks
The Valuatio of the Catastrophe Equity Puts with Jump Risks Shih-Kuei Li Natioal Uiversity of Kaohsiug Joit work with Chia-Chie Chag Outlie Catastrophe Isurace Products Literatures ad Motivatios Jump Risk
More informationAn 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
More informationAnnual 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
More informationSOLUTION QUANTITATIVE TOOLS IN BUSINESS NOV 2011
SOLUTION QUANTITATIVE TOOLS IN BUSINESS NOV 0 (i) Populatio: Collectio of all possible idividual uits (persos, objects, experimetal outcome whose characteristics are to be studied) Sample: A part of populatio
More informationSELECTING THE NUMBER OF CHANGE-POINTS IN SEGMENTED LINE REGRESSION
1 SELECTING THE NUMBER OF CHANGE-POINTS IN SEGMENTED LINE REGRESSION Hyue-Ju Kim 1,, Bibig Yu 2, ad Eric J. Feuer 3 1 Syracuse Uiversity, 2 Natioal Istitute of Agig, ad 3 Natioal Cacer Istitute Supplemetary
More information