Random Variables, Sampling and Estimation
|
|
- Marsha Byrd
- 5 years ago
- Views:
Transcription
1 Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig i the ext chapters. The key topics that we will review are the followig: Descriptive statistics. e.g. mea ad variace. Probability. e.g. evets, relative frequecy, margial ad coditioal probability distributios. Radom variables, probability distributios, ad expectatios. Samplig. e.g. simple radom samplig. Estimatio. e.g. the distictio betwee ad estimator ad a estimate. Statistical iferece. t ad F tests. 1.2 Probabilities Evets Radom experimet. Process leadig to two or more possible outcomes, with ucertaity as to which outcome will occur. Flip of a coi, toss of a die, a studets takes a class ad either obtais a A or ot. Sample space. Set of all basic outcomes of a radom experimet. 1
2 2 1 Radom Variables, Samplig ad Estimatio Whe flippig a coi, S = [head, tail]. Whe takig a class, S = [A, B, C, D, F, drop]. Whe tossig a die, S = [1, 2, 3, 4, 5, 6]. No two outcomes ca occur simultaeously. Evet. Subset of basic outcomes i the sample space. Evet E 1 : Pass the class the the subset of basic outcomes is A, B, C. Itersectio of evet. Whe two evets E 1 ad E 2 have some basic outcomes i commo. It is deoted by E 1 E 2. Evet E 1 : Idividuals with college degree. Evet E 2 : Idividuals who are married. E 1 E 2 : Idividuals who have college degree ad are married. Joit probability. Probability that the itersectio occurs. Mutually exclusive evets. E 1 ad E 2 are mutually exclusive if E 1 E 2 is empty. Uio of evets. Deoted by E 1 E 2. At least oe of these evets occurs. Either E 1, E 2, or both. Complemet. The complemet of E is deoted by Ē ad it is the set of basic outcomes of a radom experimet that belogs to S, but ot to E 1. E 1 is the complemet of Ē 1 Evet E 2 : Idividuals who are married. E 1 ad Ē are mutually exclusive evets Probability postulates Give a radom experimet, we wat to determie the probability that a particular evet will occur. A probability is a measure from 0 to 1.
3 1.3 Discrete radom variables ad expectatios 3 0 the evet will ot occur. 1 the evet is certai. Whe the outcomes are equally likely to occur, the probability of a evet E is: P(E)=N E /N N E : Number of outcomes i evet E. N: Total umber of outcomes i the sample space S. Example 1: Flip of a coi, Evet E is head the P(E) = 1/2. N E = 1 ad N = 2. Example 2: Evet E is wiig the lottery the if there are 1000 lottery tickets ad you bought, 2 P(E) = 2/1000 = Some probability rules P(E Ē) = P(E) + P(Ē) = 1. P(Ē) = 1 - P(E). Coditioal probability P(E 1 E 2 ): Probability that E 1 occurs, give that E 2 has already occurred. P(E 1 E 2 ) = P(E 1 E 2 ) / P(E 2 ) give that P(E 2 ) > 0. Additio rule P(E 1 E 2 ) = P(E 1 ) + P(E 2 ) - P(E 1 E 2 ). Statistically idepedet evets P(E 1 E 2 ) = P(E 1 )P(E 2 ). P(E 1 E 2 ) = P(E 1 )P(E 2 ) / P(E 2 ) = P(E 1 ). 1.3 Discrete radom variables ad expectatios Discrete radom variables Radom variable. Variable that takes umerical values determied by the outcome of a radom experimet. Examples: Hourly wage, GDP, iflatio, the umber whe tossig a die. Notatio: Radom variable X ca take possible values x 1,x 2, x. Discrete radom variable. A radom variable that takes a coutable umber of values.
4 4 1 Radom Variables, Samplig ad Estimatio Examples: Number of years of educatio. Cotiuous radom variable. A radom variable that ca take ay value o a iteral. Examples: Wage, GDP, exact weight. Cosider tossig two dies (gree ad red). This will yield 36 possible outcomes because the gree ca take 6 possible values ad the red ca take also 6 values, 6 6=36. The possible outcomes. Let s defie the radom variable X to be the sum of two dice. Therefore X ca take 11 possible values, from 2 to 12. This iformatio is summarized i the followig tables. Table 1.1 Outcomes with two dies red / gree Table 1.2 Frequecies ad probability distributios Value of X Frequecy Probability (p) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/ Expected value of radom variables Let E(X) be the expected value of the radom variable X. The expected value of a discrete radom variable is the weighted average of all its possible values, takig the probability of each outcome as its weight. Radom variable X ca take particular values x 1,x 2,...,x ad the probability of x i is give by p i. The we have that the expected value is give by: E(X)=x 1 p 1 + x 2 p 2 + +x p = i=1 x i p i. (1.1) We ca also write the expected value as: E(X)= µ X. For the previous example we ca calculate that the expected value is:
5 1.3 Discrete radom variables ad expectatios 5 E(X)=2 1/36+3 2/ /36=252/36=7 (1.2) Table 1.3 Expected value of X, two dice example X p X p 2 1/36 2/36 3 2/36 6/36 4 3/36 12/36 5 4/36 20/36 6 5/36 30/36 7 6/36 42/36 8 5/36 40/36 9 4/36 36/ /36 30/ /36 22/ /36 12/36 Total E(X)= i=1 x i p x 252/36 = Expected value rules E(X+Y + Z)=E(X)+E(Y)+E(Z) (1.3) E(bX)=bE(X) for a costat b (1.4) E(b) = b (1.5) For the example where Y = b 1 + b 2 X, b 1 ad b 2 are costats we wat to calculate E(X). E(Y) = E(b 1 + b 2 X) (1.6) = E(b 1 )+E(b 2 X) = b 1 + b 2 E(X) Variace of a discrete radom variable Let var(x) be the variace of the radom variable X. var(x) is a useful measure of the dispersio of its probability distributio. It is defied as the expected value of the square of the differece betwee X ad its mea. That is, (X µ X ) 2, where µ X is the populatio mea of X. var(x) = σ 2 X = E[(X µ X ) 2 ] (1.7)
6 6 1 Radom Variables, Samplig ad Estimatio = (x 1 µ X ) 2 p 1 +(x 2 µ X ) 2 p 2 + +(x µ X ) 2 p (1.8) = i=1 (x i µ X ) 2 p i Takig the square root of the variace (σx 2 ) oe ca obtai the stadard deviatio, σ X. The stadard deviatio also serves as a measure of dispersio of the probability distributio. A useful way to write the variace is: σ 2 X = E(X 2 ) µ 2 X. (1.9) From the previous example of tossig two dies, we have that the populatio variace ca be calculated as follows: Table 1.4 Populatio variace, X from the two dice example X p X µ X (X µ X ) 2 (X µ X ) 2 p 2 1/ / / / / / / / / / / Total Probability desity Because discrete radom variables, by defiitio, ca oly take a fiite umber of values, they are easy to summarize graphically. The probability distributio is the graph that liks all the values that a radom variable ca take with its correspodig probabilities. For the two dice example above, see Figure 1.1.
7 1.4 Cotiuous radom variables 7 Fig. 1.1 Discrete probabilities, X from the two dice example 1.4 Cotiuous radom variables Probability desity Cotiuous radom variables ca take ay value o a iterval. This meas that it ca take a ifiite umber of differet values, hece it is ot possible to obtai a graph like the oe preseted i Figure 1.1 for a cotiuous radom variable. Istead, we will defie the probability of a radom variable lyig withi a give iterval. For example, the probability that the height of a idividual is betwee 5.5 ad 6 feet. This is depicted i Figure 1.2 as the shaded area below the probability desity curve for the values of X betwee 5.5 ad 6. The probability of the radom variable X writte as a fuctio of the radom variable is kow as the probability desity fuctio. We ca write this oes as f(x). The, if we use a little math we ca easily fid the area uder the curve. Recall that the are uder a curve ca be obtaied by takig the itegral. Probability desity fuctio. Is a fuctio that describes the relative likelihood for a radom variable to occur at a give poit f(x) = 0.18 (1.10) f(x) = 1
8 8 1 Radom Variables, Samplig ad Estimatio Fig. 1.2 Cotiuous probabilities, X from the height example The first lie i the equatio above just calculates the itegral uder the curve f(x) betwee the poits 5.5 ad 6. The secod lie shows that the whole area uder the curve preseted i Figure 1.2 is equal to oe. This is for the same reaso why the summatio of all the bars i Figure 1.1 are also equal to oe; the total probability is always equal to oe Normal distributio The ormal distributio is the most widely kow cotiuous probability distributio. The graph associated with its probability desity fuctio has a bell-shape ad its is kow as the Gaussia fuctio or bell curve. Its probability desity fuctio is give by: f(x)= 1 2πσ 2 e (x µ2 ) 2σ 2 (1.11) where µ is the mea ad σ 2 is the variace. Figure 1.1 is a example of this distributio.
9 1.5 Covariace ad correlatio Expected value ad variace of a cotiuous radom variable The basic differece betwee a discrete ad a cotiuous radom variable is that the secod ca take o ifiite possible values, hece the summatios sigs that are used to calculate the expected value ad the variace of a discrete radom variable caot be used for a cotiuous radom variable. Istead, we use itegral sigs. For the expected value we have: E(X) = X f(x)dx (1.12) where the itegratio is performed over the iterval for which f(x) is defied. For the variace we have: σx 2 = E[(X µ X ) 2 ]= (X µ X ) 2 f(x)dx (1.13) 1.5 Covariace ad correlatio Covariace Whe dealig with two variables, the first questio you wat to aswer is whether these variables move together or whether they move i opposite directios. The covariace will help us aswer that questio. For two radom variables X ad Y, the covariace is defied as: cov(x,y) = σ XY = E[(X µ X )(Y µ Y )] (1.14) where µ X ad µ Y are the populatio meas of X ad Y, respectively. Whe to radom variables are idepedet, their covariace is equal to zero. Whe σ XY > 0 we say that the variables move together. Whe σ XY < 0 they move i opposite directios Correlatio Oe cocer whe usig the cov(x,y) as a measure of associatio is that the result is measured i the uits of X times the uits of Y. The correlatio coefficiet, that is dimesioless, overcomes this difficulty. For variables X ad Y the correlatio coefficiet is defied as:
10 10 1 Radom Variables, Samplig ad Estimatio corr(x,y) = ρ Y X = σ Y X σx 2σ Y 2 (1.15) The correlatio coefficiet is a umber betwee 1 ad 1. Whe it is positive, we say that there is a positive correlatio betwee X ad Y ad that these two variables move i the same directio. Whe it is egative, we say that they move i opposite directios. 1.6 Samplig ad estimators Notice that i the two dice example we kow the populatio characteristics, that is, the probability distributio. From this probability distributio it is easy to obtai the populatio mea a variace. However, what happes most of the time is that we eed to rely o a data set to get estimates of the populatio parameters (e.g the mea ad the variace). I that case the estimates of the populatio parameters are obtaied usig estimators, ad the sample eeds to have certai characteristics. The estimators ad the samplig are the subject of this sectio Samplig The most commo way to obtai a sample from the populatio is through simple radom samplig. Simple radom samplig. It is a procedure to obtai a sample from the populatio, where each of the observatios is chose radomly ad etirely by chace. This meas that each observatio i the populatio has the same probability of beig chose. Oce the sample of the radom variable X has be geerated, each of the observatios ca be deoted by{x 1,x 2,,x }. 1 1 The textbook Dougherty (2007) makes the distictio betwee the specific values of the radom variable X before ad after they are kow, ad emphasizes this distictio by usig uppercase ad lowercase letter. This distictio is useful oly i some cases ad that is why most textbooks do ot make this distictio. We will follow emphasize the distictio ad we will use oly lowercase letters.
11 1.7 Ubiasedess ad efficiecy Estimators Estimator. It is a geeral rule (mathematical formula) for estimatig a ukow populatio parameter give a sample of data. For example, a estimator for the populatio mea is the sample mea: X = 1 (x 1+ x 2 + +x )= 1 i=1 x i. (1.16) A iterestig feature of this estimator is that the variace of X is 1/ times the variace of X. The derivatio is the followig: σ 2 X = var( X) (1.17) σ 2 X = var{1 (x 1+ x 2 + +x )} (1.18) σ 2 X = 1 2 var{1 (x 1+ x 2 + +x )} (1.19) σ 2 X = 1 2{var(x 1)+var(x 2 )+ +var(x )} (1.20) σ 2 X = 1 2{σ 2 X + σ 2 X + +σ 2 X} (1.21) σ 2 X = 1 2{σ 2 X}= σ 2 X (1.22) Graphically, this result is show i Figure 1.3. The distributio of X has a higher variace (it is more dispersed) tha the distributio of X. 1.7 Ubiasedess ad efficiecy Ubiasedess Because estimators are radom variables, we ca take expectatios of the estimators. If the expectatio of the estimator is equal to the true populatio parameter, the we say that this estimator is ubiased. Let θ be the populatio parameter ad let ˆθ be a poit estimator of θ. The, ˆθ is ubiased if: E( ˆθ)=θ (1.23) Example. The sample mea of X is a ubiased estimator of the populatio mea µ X :
12 12 1 Radom Variables, Samplig ad Estimatio X µ X Fig. 1.3 Probability desity fuctios of X ad X. E( X) = E( 1 i )= i=1x 1 E( x i ) (1.24) i=1 = 1 i ))= i=1(e(x 1 i=1 µ X = 1 µ X = µ X Ubiased estimator. A estimator is ubiased if its expected value is equal to the true populatio parameter. The bias of a estimator is just the differece betwee its expected value ad the true populatio parameter: Bias( ˆθ)=E( ˆθ) θ (1.25) Efficiecy It is ot oly importat that a estimator is o average correct (ubiased), but also that it has a high probability of beig close to the true parameter. Whe comparig two estimators, ˆθ 1 ad ˆθ 2, we say that ˆθ 1 is more efficiet if var( ˆθ 1 ) < var( ˆθ 2 ). A compariso of the efficiecy betwee these two estimators i preseted i Figure 1.4. The estimator with higher variace,( ˆθ 2 ), is more dispersed.
13 1.8 Estimators for the variace, covariace, ad correlatio 13 θˆ 1 θˆ 2 µ X Fig. 1.4 Efficiecy of estimators ˆθ 1 ad ˆθ 2, with var( ˆθ 1 )<var( ˆθ 2 ). Most efficiet estimator. The estimator with the smallest variace from all ubiased estimators Ubiasedess versus efficiecy Both, ubiasedess ad efficiecy, are desired properties of a estimator. However, there may be coflicts i the selectio betwee two estimators ˆθ 1 ad ˆθ 2, if, for example, ˆθ 1 is more efficiets, but it is also biased. This case is preseted i Figure 1.5. The simplest way to select betwee these two estimators is to pick the oe that yields the smallest mea square error (MSE): MSE( ˆθ)=var( ˆθ)+bias( ˆθ) 2 (1.26) 1.8 Estimators for the variace, covariace, ad correlatio While we have already see the populatios formulas for the variace, covariace ad correlatio, it is importat to keep i mid that we do ot have the whole populatio. The data sets we will be workig with are just samples of the populatios. The formula for the sample variace is:
14 14 1 Radom Variables, Samplig ad Estimatio θˆ 2 θˆ 1 µ X Fig. 1.5 ˆθ 2 is ubiased, but ˆθ 1 is more efficiet. s 2 X = 1 1 i=1 (x i X) 2 (1.27) Notice how we chaged the otatio from σ 2 to s 2. The first oe deotes the populatio variace, while the secod oe refers to the sample variace. A estimator for the populatio covariace is give by: s XY = 1 1 Fially, the formula for the correlatio coefficiet, r XY, is: r XY = i=1 (x i X)(y i Ȳ). (1.28) i=1 (x i X)(y i Ȳ) i=1 (x i X) 2 i=1 (y (1.29) i Ȳ) Asymptotic properties of estimators Asymptotic properties of estimators just refers to their properties whe the umber of observatios i the sample grows large ad approached to ifiity.
15 1.9 Asymptotic properties of estimators 15 = 1000 = 250 = 40 θ Fig. 1.6 The estimator is biased for small samples, but cosistet Cosistecy A estimator ˆθ is said to be cosistet if its bias becomes smaller as the sample size grows large. Cosistecy is importat because may of the most commo estimators used i ecoometrics are biased, the the miimum we should expect from these estimators is that the bias becomes small as we are able to obtai larger data sets. Figure 1.6 illustrates the cocept of cosistecy by showig how a estimator of the populatio parameter θ becomes ubiased as Cetral limit theorem Havig ormally distributed radom variables is importat because we ca the costruct, for example, cofidece itervals for its mea. However, what if a radom variable does ot follow a ormal distributio? The cetral limit theorem gives us the aswer. Cetral limit theorem. States the coditios uder which the mea of a sufficietly large umber of idepedet radom variables (with fiite mea ad variace) will be approximate a ormal distributio. Hece, eve if we do ot kow the uderlyig distributio of a radom variable, we will still be able to costruct cofidece itervals that will be approximately valid. I a umerical example, let s assume that the radom variable X follows a
16 16 1 Radom Variables, Samplig ad Estimatio = 100 = 20 = 10 Fig. 1.7 Distributio of the sample mea of a uiform distributio. uiform distributio [-0.5,0.5]. Hece, it is equally likely that this radom variable takes ay value withi this rage. Figure 1.7 shows the distributio of the average of this radom variable for = 10, 20, ad 100. All of these three distributios look very close to a ormal distributio.
Expectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationHypothesis Testing. Evaluation of Performance of Learned h. Issues. Trade-off Between Bias and Variance
Hypothesis Testig Empirically evaluatig accuracy of hypotheses: importat activity i ML. Three questios: Give observed accuracy over a sample set, how well does this estimate apply over additioal samples?
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter & Teachig Material.
More informationLecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett
Lecture Note 8 Poit Estimators ad Poit Estimatio Methods MIT 14.30 Sprig 2006 Herma Beett Give a parameter with ukow value, the goal of poit estimatio is to use a sample to compute a umber that represets
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter 2 & Teachig
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationNANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS
NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationEE 4TM4: Digital Communications II Probability Theory
1 EE 4TM4: Digital Commuicatios II Probability Theory I. RANDOM VARIABLES A radom variable is a real-valued fuctio defied o the sample space. Example: Suppose that our experimet cosists of tossig two fair
More informationMathematical Statistics - MS
Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationf X (12) = Pr(X = 12) = Pr({(6, 6)}) = 1/36
Probability Distributios A Example With Dice If X is a radom variable o sample space S, the the probablity that X takes o the value c is Similarly, Pr(X = c) = Pr({s S X(s) = c} Pr(X c) = Pr({s S X(s)
More informationComputing Confidence Intervals for Sample Data
Computig Cofidece Itervals for Sample Data Topics Use of Statistics Sources of errors Accuracy, precisio, resolutio A mathematical model of errors Cofidece itervals For meas For variaces For proportios
More informationMBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationJoint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationUnbiased Estimation. February 7-12, 2008
Ubiased Estimatio February 7-2, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationSome Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables
Some Basic Probability Cocepts 2. Experimets, Outcomes ad Radom Variables A radom variable is a variable whose value is ukow util it is observed. The value of a radom variable results from a experimet;
More informationAMS570 Lecture Notes #2
AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3)
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationFinal Review for MATH 3510
Fial Review for MATH 50 Calculatio 5 Give a fairly simple probability mass fuctio or probability desity fuctio of a radom variable, you should be able to compute the expected value ad variace of the variable
More informationEcon 325: Introduction to Empirical Economics
Eco 35: Itroductio to Empirical Ecoomics Lecture 3 Discrete Radom Variables ad Probability Distributios Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-1 4.1 Itroductio to Probability
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationIE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.
IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationCEU Department of Economics Econometrics 1, Problem Set 1 - Solutions
CEU Departmet of Ecoomics Ecoometrics, Problem Set - Solutios Part A. Exogeeity - edogeeity The liear coditioal expectatio (CE) model has the followig form: We would like to estimate the effect of some
More informationIn this section we derive some finite-sample properties of the OLS estimator. b is an estimator of β. It is a function of the random sample data.
17 3. OLS Part III I this sectio we derive some fiite-sample properties of the OLS estimator. 3.1 The Samplig Distributio of the OLS Estimator y = Xβ + ε ; ε ~ N[0, σ 2 I ] b = (X X) 1 X y = f(y) ε is
More informationJanuary 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationCS 330 Discussion - Probability
CS 330 Discussio - Probability March 24 2017 1 Fudametals of Probability 11 Radom Variables ad Evets A radom variable X is oe whose value is o-determiistic For example, suppose we flip a coi ad set X =
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22
CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first
More informationTopic 8: Expected Values
Topic 8: Jue 6, 20 The simplest summary of quatitative data is the sample mea. Give a radom variable, the correspodig cocept is called the distributioal mea, the epectatio or the epected value. We begi
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More information1.010 Uncertainty in Engineering Fall 2008
MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval
More informationUNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY
UNIT 2 DIFFERENT APPROACHES TO PROBABILITY THEORY Structure 2.1 Itroductio Objectives 2.2 Relative Frequecy Approach ad Statistical Probability 2. Problems Based o Relative Frequecy 2.4 Subjective Approach
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationCSE 527, Additional notes on MLE & EM
CSE 57 Lecture Notes: MLE & EM CSE 57, Additioal otes o MLE & EM Based o earlier otes by C. Grat & M. Narasimha Itroductio Last lecture we bega a examiatio of model based clusterig. This lecture will be
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationECE 8527: Introduction to Machine Learning and Pattern Recognition Midterm # 1. Vaishali Amin Fall, 2015
ECE 8527: Itroductio to Machie Learig ad Patter Recogitio Midterm # 1 Vaishali Ami Fall, 2015 tue39624@temple.edu Problem No. 1: Cosider a two-class discrete distributio problem: ω 1 :{[0,0], [2,0], [2,2],
More informationApril 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE
April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece
More informationFirst Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >
More informationTopic 10: Introduction to Estimation
Topic 0: Itroductio to Estimatio Jue, 0 Itroductio I the simplest possible terms, the goal of estimatio theory is to aswer the questio: What is that umber? What is the legth, the reactio rate, the fractio
More informationn outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationWhat is Probability?
Quatificatio of ucertaity. What is Probability? Mathematical model for thigs that occur radomly. Radom ot haphazard, do t kow what will happe o ay oe experimet, but has a log ru order. The cocept of probability
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationDistribution of Random Samples & Limit theorems
STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to
More informationIt is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.
MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied
More informationProbability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationLecture 5: Parametric Hypothesis Testing: Comparing Means. GENOME 560, Spring 2016 Doug Fowler, GS
Lecture 5: Parametric Hypothesis Testig: Comparig Meas GENOME 560, Sprig 2016 Doug Fowler, GS (dfowler@uw.edu) 1 Review from last week What is a cofidece iterval? 2 Review from last week What is a cofidece
More informationBig Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.
5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Math 36Q Fall 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2 to
More informationStatistical Properties of OLS estimators
1 Statistical Properties of OLS estimators Liear Model: Y i = β 0 + β 1 X i + u i OLS estimators: β 0 = Y β 1X β 1 = Best Liear Ubiased Estimator (BLUE) Liear Estimator: β 0 ad β 1 are liear fuctio of
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationEconomics Spring 2015
1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures
More informationEconomics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls
Ecoomics 250 Assigmet 1 Suggested Aswers 1. We have the followig data set o the legths (i miutes) of a sample of log-distace phoe calls 1 20 10 20 13 23 3 7 18 7 4 5 15 7 29 10 18 10 10 23 4 12 8 6 (1)
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationEECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1
EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More informationStat410 Probability and Statistics II (F16)
Some Basic Cocepts of Statistical Iferece (Sec 5.) Suppose we have a rv X that has a pdf/pmf deoted by f(x; θ) or p(x; θ), where θ is called the parameter. I previous lectures, we focus o probability problems
More informationAdvanced Stochastic Processes.
Advaced Stochastic Processes. David Gamarik LECTURE 2 Radom variables ad measurable fuctios. Strog Law of Large Numbers (SLLN). Scary stuff cotiued... Outlie of Lecture Radom variables ad measurable fuctios.
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More information