. (The calculated sample mean is symbolized by x.)
|
|
- Alicia King
- 5 years ago
- Views:
Transcription
1 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 ad had-i) homework for sectio 5.4 will be from that supplemet. From sectio 5.3: Radom variables,,, form a (simple) radom sample of size if they meet two (importat) requiremets:. The i s are idepedet radom variables.. Every i has the same probability distributio. Data is collected from the sample, i.e., the radom variables,,, each receive values x, x,, x. These values are used to calculate sample statistics. The sample statistics we ll be most iterested i are:. The sample total T0 = The sample mea = T0. (The calculated sample mea is symbolized by x.) 3. The sample variace S = ( i ) i =. (The calculated sample variace is symbolized by s.) Note that T0, ad S are themselves radom variables. Sectios 5.3 ad 5.4 focus o what the probability distributios of these radom variables look like, ad what they ca tell us about the overarchig populatio s distributio. Theory Probability models exist i a theoretical world where everythig is kow. If you costructed every possible sample of a specified size from a give populatio (Example i the supplemet), or were able to toss a coi a ifiite umber of times (Example i the supplemet), you would create what statisticias call a samplig distributio. I the Examples below, we ll use a hypothetical populatio Ψ cosistig of the umbers 0, 0, 30, 40 ad 50. The parameter ad statistic we ll cosider first is the mea. Example A-: Calculate the mea ad stadard deviatio of a populatio Ψ which cosists of elemets from the set {0, 0, 30, 40, 50} with probabilities give i the table below P( = x) Example A-: Costruct a histogram for populatio Ψ. We would get the same histogram if we were to cosider all possible samples of size = that could be take from the populatio Ψ ad calculated each sample s expected value (mea).
2 Example A-3: Costruct all possible samples of size = that ca be made from the elemets of Ψ, desigate the probability of each beig picked, ad calculate each sample s expected value (mea). sample P x sample P x sample P x sample P x 0, 0 0, 0 30, 0 50, 0 0, 0 0, 0 30, 0 50, 0 0, 30 0, 30 30, 30 50, 30 0, 50 0, 50 30, 50 50, 50 Example A-4: Draw the histogram for the samplig distributio from Example A P = x ( ) Example A-5: Calculate mea ad stadard deviatio of the samplig distributio of Ψ for sample size =. Mea of samplig distributio = E ( ) = 0 ( 0.6) + 5( 0.6) + 0( 0.0) + 5( 0.08) + 30( 0.0) + 35( 0.08) + 40( 0.08) + 45( 0.0) + 50( 0. 04) = Variace of samplig distributio = ( ) V = ( 0 4) ( 0.6) + ( 5 4) ( 0.6) + ( 0 4) ( 0.0) + ( 5 4) ( 0.08) + ( 30 4) ( 0.0) + ( 35 4) ( 0.08) + ( 40 4) ( 0.08) + ( 45 4) ( 0.0) + ( 50 4) ( 0.04) = Stadard deviatio of samplig distributio ( ) = σ = Theory: = σ stadard error = (Example A-6 is foud o the ext page.)
3 Example A-6: Draw the histogram ad calculate the mea ad stadard deviatio of the samplig distributio of Ψ for sample size = 4. Mea of samplig distributio = E ( ) = Variace of samplig distributio = Var ( ) = Stadard deviatio of samplig distributio ( ) = σ = = σ stadard error = The histogram for this samplig distributio (sample size = 4) looks like this. Example A-7: O your ow, explore the histogram for samplig distributios of Ψ for various sample sizes. (That is, coduct a series of simulatio experimets usig various values of.) The supplemet provides two sources, alog with illustratios: Notes followig Examples A: (Proofs for coclusios about samples of ay size are o the ext page.)
4 I Example A- we foud E() ad σ (). I Examples A-5 ad A-6, we foud that ( ) E( ) σ ( ) ( ) σ =. Will the same be true for ay sample size? E = ad 4.3 Example D revisited. Distributio of ACT scores is approximately ormal. I 00 mea score for the ACT =.6, with a stadard deviatio of 4.3. What is the probability that a sigle studet chose at radom has a ACT score betwee ad 4? aswer: (source: Usefuless of High School Average ad ACT Scores i Makig College Admissio Decisios, retrieved from Theory: samplig distributio for a ormally distributed populatio
5 4.3 Example D a ew questio. Distributio of ACT scores is approximately ormal. I 00 mea score for the ACT =.6, with a stadard deviatio of 4.3. a) If a radom sample of 50 studets who took the ACT is selected, what is the shape of the resultig samplig distributio? b) What are E ( ) ad σ? c) What is the probability that the sample mea is betwee ad 4? aswers: ormal;.6, 0.608; 0.88 This is fie for a populatio kow to be ormally distributed, but ca we make ay statemets about samplig distributios from a populatio which may ot (or defiitely do ot) have a ormal probability distributio? Recall the various versios of Example A doe earlier. Populatio Ψ did ot have a ormal distributio, ad was ot eve symmetric. However, the shape of the samplig distributio took o a shape close to that of a ormal distributio as icreased. Eter the Cetral Limit Theorem: Give a populatio with mea µ ad stadard deviatio σ : ) As the sample size icreases, or as the umber of trials approaches ifiite, the shape of a samplig distributio becomes icreasigly like a ormal distributio. ) The mea of samplig distributio = the mea of the populatio, E ( ) µ =. σ 3) The stadard deviatio of samplig distributio = stadard error, σ =. The proof of () i the Cetral Limit Theorem requires momet geeratig fuctios which we do ot have yet, ad which we may (but probably wo t) get to before the ed of the semester. For statistics, a sample size of 30 is usually large eough to use the ormal distributio probability table for hypothesis tests ad cofidece itervals. For Lecture examples, ad for homework exercises from the hadout, we ll use the ormal distributio table to fid various probabilities for sample statistics. The Cetral Limit Theorem tells us, i short, that a samplig distributio is ofte close to a ormal distributio. What does this mea for radom samplig? It tells us that 68% of the time, a radom sample will give us a result a statistic withi stadard deviatio of the true parameter. We would expect that 95% of the time, a radom sample will give a statistic withi stadard deviatios of the populatio parameter, ad 99.7% of the time, a radom sample will give a statistic withi 3 stadard deviatios of the populatio parameter. I statistics, the Cetral Limit Theorem is the justificatio for costructig cofidece itervals ad coductig hypothesis tests.
6 Example B. A populatio has mea µ = 50 ad stadard deviatio σ =. For a radom sample of size 47, calculate a) the expected value of the sample mea ad b) the stadard error. Fid the followig probabilities: 45 < < 53 P > 54. c) P ( ) ad d) ( ) aswers: 50,, , WARNING: Example B is ot like those we did i sectio 4.3! That is, it does ot fid probabilities for sigle µ values of for a ormally-distributed populatio, which uses Z =. (I fact, because we do ot kow σ the probability distributio, we caot specify probabilities for idividual subjects.) Rather, Example B looks at oe sample ad fids probabilities ivolvig the mea of that sample,, cosidered as part of all the hypothetical samples which could have bee costructed (the samplig µ µ distributio). Therefore, for example B we used Z = =. σ σ Also, i some homework exercises, you ll eed to use the skills from previous sectios to determie µ ad σ.
7 Example C. A radom variable has probability desity fuctio ( x) ( x) 6x 0 x f =. 0 otherwise a) What are the expected value ad stadard deviatio for a sigle radomly-chose value of? b) You radomly select a sample of size = 00. What is the expected value for the sample mea, E ( ) is the stadard error, σ, for the samplig distributio?? What c) What is the probability that a sigle radomly-chose subject from this populatio will exhibit a value of at least 0.55? d) You radomly select a sample of size = 00. What is the probability that the sample mea will be at least 0.55? e) You radomly select a sample of size = 00. There is a 5% probability that the sample mea will be below what value? aswers:, ; 5, ; 0.455; 0.05; Go to the supplemet for more examples worked out. There are examples similar to each of the homework exercises.
Sampling 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 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 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 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 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 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 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 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 informationpoint 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 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 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 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 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 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 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 informationIntroduction to Probability and Statistics Chapter 7
Itroductio to Probability ad Statistics Chapter 7 Ammar M. Sarha, asarha@mathstat.dal.ca Departmet of Mathematics ad Statistics, Dalhousie Uiversity Fall Semester 008 Chapter 7 Statistical Itervals Based
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 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 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 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 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 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 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 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 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 informationValue (x) probability Example A-2: Construct a histogram for population Ψ.
Calculus 111, section 08.x The Central Limit Theorem notes by Tim Pilachowski If you haven t done it yet, go to the Math 111 page and download the handout: Central Limit Theorem supplement. Today s lecture
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationChapter 17 Sampling Distribution Models
Chapter 17 Samplig Distributio Models 353 Chapter 17 Samplig Distributio Models 1. Sed moey. All of the histograms are cetered aroud p 0.05. As gets larger, the shape of the histograms get more uimodal
More informationSection 3.3 Exercises Part A Simplify the following. 1. (3m 2 ) 5 2. x 7 x 11
123 Sectio 3.3 Exercises Part A Simplify the followig. 1. (3m 2 ) 5 2. x 7 x 11 3. f 12 4. t 8 t 5 f 5 5. 3-4 6. 3x 7 4x 7. 3z 5 12z 3 8. 17 0 9. (g 8 ) -2 10. 14d 3 21d 7 11. (2m 2 5 g 8 ) 7 12. 5x 2
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 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 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 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 informationDepartment of Mathematics, S.R.K.R. Engineering College, Bhimavaram, A.P., India 2
Skewess Corrected Cotrol charts for two Iverted Models R. Subba Rao* 1, Pushpa Latha Mamidi 2, M.S. Ravi Kumar 3 1 Departmet of Mathematics, S.R.K.R. Egieerig College, Bhimavaram, A.P., Idia 2 Departmet
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 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 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 informationTwitter: @Owe134866 www.mathsfreeresourcelibrary.com Prior Kowledge Check 1) State whether each variable is qualitative or quatitative: a) Car colour Qualitative b) Miles travelled by a cyclist c) Favourite
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 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 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 informationSolutions to Problem Sheet 1
Solutios to Problem Sheet ) Use Theorem.4 to prove that p log for all real x 3. This is a versio of Theorem.4 with the iteger N replaced by the real x. Hit Give x 3 let N = [x], the largest iteger x. The,
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 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 information2. Find the annual percentage yield (APY), to the nearest hundredth of a %, for an account with an APR of 12% with daily compounding.
1. Suppose that you ivest $4,000 i a accout that ears iterest at a of 5%, compouded mothly, for 58 years. `Show the formula that you would use to determie the accumulated balace, ad determie the accumulated
More informationIntroduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 2
Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd- Numbered Ed- of- Chapter Exercises: Chapter 2 (This versio August 7, 204) Stock/Watso - Itroductio to
More information= α e ; x 0. Such a random variable is said to have an exponential distribution, with parameter α. [Here, view X as time-to-failure.
1 Homewor 1 AERE 573 Fall 018 DUE 8/9 (W) Name ***NOTE: A wor MUST be placed directly beeath the associated part of a give problem.*** PROBEM 1. (5pts) [Boo 3 rd ed. 1.1 / 4 th ed. 1.13] et ~Uiform[0,].
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 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 informationA New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions
A New Costructive Proof of Graham's Theorem ad More New Classes of Fuctioally Complete Fuctios Azhou Yag, Ph.D. Zhu-qi Lu, Ph.D. Abstract A -valued two-variable truth fuctio is called fuctioally complete,
More informationFOUNDATION ACTED COURSE (FAC)
FOUNDATION ACTED COURSE (FAC) What is the Foudatio ActEd Course (FAC)? FAC is desiged to help studets improve their mathematical skills i preparatio for the Core Techical subjects. It is a referece documet
More informationVariance and Standard Deviation (Tables) Lecture 10
Variace ad Stadard Deviatio (Tables) Lecture 10 Variace ad Stadard Deviatio Theory I this lesso: 1. Calculatig stadard deviatio with ugrouped data.. Calculatig stadard deviatio with grouped data. What
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 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 information0.07. i PV Qa Q Q i n. Chapter 3, Section 2
Chapter 3, Sectio 2 1. (S13HW) Calculate the preset value for a auity that pays 500 at the ed of each year for 20 years. You are give that the aual iterest rate is 7%. 20 1 v 1 1.07 PV Qa Q 500 5297.01
More informationSystematic and Complex Sampling!
Systematic ad Complex Samplig! Professor Ro Fricker! Naval Postgraduate School! Moterey, Califoria! Readig Assigmet:! Scheaffer, Medehall, Ott, & Gerow! Chapter 7.1-7.4! 1 Goals for this Lecture! Defie
More informationChapter 10 Statistical Inference About Means and Proportions With Two Populations. Learning objectives
Chater 0 Statistical Iferece About Meas ad Proortios With Two Poulatios Slide Learig objectives. Uderstad ifereces About the Differece Betwee Two Poulatio Meas: σ ad σ Kow. Uderstad Ifereces About the
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 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 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 informationJust Lucky? A Statistical Test for Option Backdating
Workig Paper arch 27, 2007 Just Lucky? A Statistical Test for Optio Backdatig Richard E. Goldberg James A. Read, Jr. The Brattle Group Abstract The literature i fiacial ecoomics provides covicig evidece
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 informationSTRAND: FINANCE. Unit 3 Loans and Mortgages TEXT. Contents. Section. 3.1 Annual Percentage Rate (APR) 3.2 APR for Repayment of Loans
CMM Subject Support Strad: FINANCE Uit 3 Loas ad Mortgages: Text m e p STRAND: FINANCE Uit 3 Loas ad Mortgages TEXT Cotets Sectio 3.1 Aual Percetage Rate (APR) 3.2 APR for Repaymet of Loas 3.3 Credit Purchases
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 informationAPPLIED STATISTICS Complementary Course of BSc Mathematics - IV Semester CUCBCSS Admn onwards Question Bank
Prepared by: Prof (Dr) K.X. Joseph Multiple Choice Questios 1. Statistical populatio may cosists of (a) a ifiite umber of items (b) a fiite umber of items (c) either of (a) or (b) Module - I (d) oe of
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 informationCAPITAL PROJECT SCREENING AND SELECTION
CAPITAL PROJECT SCREEIG AD SELECTIO Before studyig the three measures of ivestmet attractiveess, we will review a simple method that is commoly used to scree capital ivestmets. Oe of the primary cocers
More informationProbability and statistics
4 Probability ad statistics Basic deitios Statistics is a mathematical disciplie that allows us to uderstad pheomea shaped by may evets that we caot keep track of. Sice we miss iformatio to predict the
More informationThe material in this chapter is motivated by Experiment 9.
Chapter 5 Optimal Auctios The material i this chapter is motivated by Experimet 9. We wish to aalyze the decisio of a seller who sets a reserve price whe auctioig off a item to a group of bidders. We begi
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 informationOnline appendices from The xva Challenge by Jon Gregory. APPENDIX 10A: Exposure and swaption analogy.
APPENDIX 10A: Exposure ad swaptio aalogy. Sorese ad Bollier (1994), effectively calculate the CVA of a swap positio ad show this ca be writte as: CVA swap = LGD V swaptio (t; t i, T) PD(t i 1, t i ). i=1
More informationKernel Density Estimation. Let X be a random variable with continuous distribution F (x) and density f(x) = d
Kerel Desity Estimatio Let X be a radom variable wit cotiuous distributio F (x) ad desity f(x) = d dx F (x). Te goal is to estimate f(x). Wile F (x) ca be estimated by te EDF ˆF (x), we caot set ˆf(x)
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 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 information