Lecture 1 Probability and Statistics
|
|
- Byron Lewis
- 5 years ago
- Views:
Transcription
1 Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure ( ): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark Twai the statemet shows the persuasive power of umbers use of statistics to bolster weak argumets tedecy of people to disparage statistics that do ot support their positios The purpose of P3700: how to uderstad the statistical ucertaity of observatio/measuremet how to use statistics to argue agaist a weak argumet (or bolster a weak argumet?) how to argue agaist people disparagig statistics that do ot support their positios how to lie with statistics? K.K. Ga L1: Probability ad Statistics 1
2 Why there is statistical ucertaity? You sell 7 cryogeic equipmet last moth You kow how to cout ad 7 is the exact umber of equipmet sold there is o ucertaity o 7! However if you used the statistics of 7 to predict the future sale or compare with past sale there is a ucertaity o 7 the umber of equipmet sold could be 5, 8, or 10! must iclude the ucertaity i the calculatio What is the ucertaity o 7? Lecture 2: 7 = 2.6 there is a 68% chace that the expected umber of equipmet sold per moth is However the umber of equipmet sold per moth is a discrete umber there is a ~68% chace that the expected umber of equipmet sold per moth is 4-10 should use Poisso statistics as i Lecture 2 for more precise predictio K.K. Ga L1: Probability ad Statistics 2
3 Itroductio: Uderstadig of may physical pheomea deped o statistical ad probabilistic cocepts: Statistical Mechaics (physics of systems composed of may parts: gases, liquids, solids.) 1 mole of aythig cotais 6x10 23 particles (Avogadro's umber) impossible to keep track of all 6x10 23 particles eve with the fastest computer imagiable resort to learig about the group properties of all the particles partitio fuctio: calculate eergy, etropy, pressure... of a system Quatum Mechaics (physics at the atomic or smaller scale) wavefuctio = probability amplitude probability of a electro beig located at (x,y,z) at a certai time. Uderstadig/iterpretatio of experimetal data deped o statistical ad probabilistic cocepts: how do we extract the best value of a quatity from a set of measuremets? how do we decide if our experimet is cosistet/icosistet with a give theory? how do we decide if our experimet is iterally cosistet? how do we decide if our experimet is cosistet with other experimets? I this course we will cocetrate o the above experimetal issues! K.K. Ga L1: Probability ad Statistics 3
4 Defiitio of probability: Suppose we have N trials ad a specified evet occurs r times. example: rollig a dice ad the evet could be rollig a 6. defie probability (P) of a evet (E) occurrig as: P(E) = r/n whe N examples: six sided dice: P(6) = 1/6 coi toss: P(heads) = 0.5 P(heads) should approach 0.5 the more times you toss the coi. for a sigle coi toss we ca ever get P(heads) = 0.5! by defiitio probability is a o-egative real umber bouded by 0 P 1 if P = 0 the the evet ever occurs if P = 1 the the evet always occurs sum (or itegral) of all probabilities if they are mutually exclusive must = 1. evets are idepedet if: P(AÇB) = P(A)P(B) Ǻitersectio, Ⱥ uio coi tosses are idepedet evets, the result of ext toss does ot deped o previous toss. evets are mutually exclusive (disjoit) if: P(AÇB) = 0 or P(AÈB) = P(A) + P(B) i coi tossig, we either get a head or a tail. K.K. Ga L1: Probability ad Statistics 4
5 Probability ca be a discrete or a cotiuous variable. Discrete probability: P ca have certai values oly. examples: tossig a six-sided dice: P(x i ) = P i here x i = 1, 2, 3, 4, 5, 6 ad P i = 1/6 for all x i. tossig a coi: oly 2 choices, heads or tails. for both of the above discrete examples (ad i geeral) whe we sum over all mutually exclusive possibilities: P( x i ) =1 Cotiuous probability: P ca be ay umber betwee 0 ad 1. defie a probability desity fuctio, pdf, f ( x) f ( x)dx = dp( x α x + dx) with a cotiuous variable probability for x to be i the rage a x b is: just like the discrete case the sum of all probabilities must equal 1. + f x dx =1 i P(a x b) = ( ) b a ( ) f x dx f(x) is ormalized to oe. probability for x to be exactly some umber is zero sice: x=a ( ) f x dx = 0 x=a Notatio: x i is called a radom variable K.K. Ga L1: Probability ad Statistics 5
6 Examples of some commo P(x) s ad f(x) s: Discrete = P(x) biomial Poisso Cotiuous = f(x) uiform, i.e. costat Gaussia expoetial chi square How do we describe a probability distributio? mea, mode, media, ad variace for a cotiuous distributio, these quatities are defied by: Mea Mode Media Variace average most probable 50% poit width of distributio + a + f x µ = xf (x)dx = = f (x)dx σ 2 = f (x) x µ ( ) x x = a ( ) 2 dx for a discrete distributio, the mea ad variace are defied by: µ = 1 x i σ 2 = 1 (x i µ) 2 K.K. Ga L1: Probability ad Statistics 6
7 Some cotiuous pdf: Probability is the area uder the curves! mode media mea σ symmetric distributio (gaussia) For a Gaussia pdf, the mea, mode, ad media are all at the same x. Asymmetric distributio showig the mea, media ad mode For most pdfs, the mea, mode, ad media are at differet locatios. K.K. Ga L1: Probability ad Statistics 7
8 Calculatio of mea ad variace: example: a discrete data set cosistig of three umbers: {1, 2, 3} average (µ) is just: x µ = i = = 2 3 complicatio: suppose some measuremet are more precise tha others. if each measuremet x i have a weight w i associated with it: µ = x i w i / w i variace (s 2 ) or average squared deviatio from the mea is just: σ 2 = 1 (x i µ) 2 variace describes the width of the pdf! s is called the stadard deviatio rewrite the above expressio by expadig the summatios: σ 2 = 1 % x 2 i + µ 2 ( ' 2µ x i * & ) = 1 = 1 x 2 i + µ 2 2µ 2 x 2 i µ 2 = x 2 x 2 < > º average weighted average i the deomiator would be -1 if we determied the average (µ) from the data itself. K.K. Ga L1: Probability ad Statistics 8
9 usig the defiitio of µ from above we have for our example of {1,2,3}: σ 2 = 1 x 2 i µ 2 = = 0.67 the case where the measuremets have differet weights is more complicated: σ 2 = µ is the weighted mea w i (x i µ) 2 2 / w i = w i x i / w i µ 2 if we calculated µ from the data, s 2 gets multiplied by a factor /(-1). example: a cotiuous probability distributio, f (x) = si 2 x for 0 x 2π has two modes! has same mea ad media, but differ from the mode(s). f(x) is ot properly ormalized: ormalized pdf: f (x) = si 2 x / 2π 0 2π 0 si 2 xdx = π 1 si 2 xdx = 1 π si2 x K.K. Ga L1: Probability ad Statistics 9
10 for cotiuous probability distributios, the mea, mode, ad media are calculated usig either itegrals or derivatives: µ = 1 π 2π 0 x si 2 xdx = π mode : x si2 x = 0 π 2, 3π 2 α media : 1 π si2 xdx = α = π example: Gaussia distributio fuctio, a cotiuous probability distributio p(x) = (x µ ) 1 2 σ 2π e 2σ 2 gaussia K.K. Ga L1: Probability ad Statistics 10
11 Accuracy ad Precisio: Accuracy: The accuracy of a experimet refers to how close the experimetal measuremet is to the true value of the quatity beig measured. Precisio: This refers to how well the experimetal result has bee determied, without regard to the true value of the quatity beig measured. just because a experimet is precise it does ot mea it is accurate!! accurate but ot precise precise but ot accurate K.K. Ga L1: Probability ad Statistics 11
12 Measuremet Errors (Ucertaities) Use results from probability ad statistics as a way of idicatig how good a measuremet is. most commo quality idicator: relative precisio = [ucertaity of measuremet]/measuremet example: we measure a table to be 10 iches with ucertaity of 1 ich. relative precisio = 1/10 = 0.1 or 10% (% relative precisio) ucertaity i measuremet is usually square root of variace: s = stadard deviatio usually calculated usig the techique of propagatio of errors (Lecture 4). Statistics ad Systematic Errors Results from experimets are ofte preseted as: N ± XX ± YY N: value of quatity measured (or determied) by experimet. XX: statistical error, usually assumed to be from a Gaussia distributio. with the assumptio of Gaussia statistics we ca say (calculate) somethig about how well our experimet agrees with other experimets ad/or theories. Expect a 68% chace that the true value is betwee N - XX ad N + XX. YY: systematic error. Hard to estimate, distributio of errors usually ot kow. examples: mass of proto = ± GeV (oly statistical error give) mass of W boso = 80.8 ± 1.5 ± 2.4 GeV K.K. Ga L1: Probability ad Statistics 12
13 What s the differece betwee statistical ad systematic errors? N ± XX ± YY statistical errors are radom i the sese that if we repeat the measuremet eough times: XX -> 0 systematic errors do ot -> 0 with repetitio. examples of sources of systematic errors: voltmeter ot calibrated properly a ruler ot the legth we thik is (meter stick might really be < meter!) because of systematic errors, a experimetal result ca be precise, but ot accurate! How do we combie systematic ad statistical errors to get oe estimate of precisio? big problem! two choices: s tot = XX + YY add them liearly s tot = (XX 2 + YY 2 ) 1/2 add them i quadrature widely accepted practice if XX ad YY are ot correlated errors ot of same origi, e.g. from the same voltmeter smaller s tot! Some other ways of quotig experimetal results lower limit: the mass of particle X is > 100 GeV upper limit: the mass of particle X is < 100 GeV +4 asymmetric errors: mass of particle X = GeV K.K. Ga L1: Probability ad Statistics 13
14 How to preset your measured values: Do t quote ay measuremet to more tha three sigificat digits three sigificat digits meas you measure a quatity to 1 part i a thousad or 0.1% precisio difficult to achieve 0.1% precisio acceptable to quote more tha three sigificat digits if you have a large data sample (e.g. large simulatios) Do t quote ay ucertaity to more tha two sigificat digits Measuremet ad ucertaity should have the same umber of digits 991± ± (5.98± 0.43)x10-5 follow this rule i the lab report! K.K. Ga L1: Probability ad Statistics 14
Lecture 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 informationPhysics 3700 Probability, Statistics, & Data Analysis
R.Kass/Sp15 P3700 Lecture 1 1 Physics 3700 Probability, Statistics, & Data Aalysis Itroductio: I) The uderstadig of may physical pheomea relies o statistical ad probabilistic cocepts: Statistical Mechaics
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 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 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 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 informationRandom Variables, Sampling and Estimation
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
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 informationExpectation 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 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 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 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 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 informationUnderstanding Samples
1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We
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 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 informationLecture 4. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za
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 informationLecture 5. Random variable and distribution of probability
Itroductio to theory of probability ad statistics Lecture 5. Radom variable ad distributio of probability prof. dr hab.iż. Katarzya Zarzewsa Katedra Eletroii, AGH e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za
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 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 informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationVariance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio
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 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 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 informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationNO! This is not evidence in favor of ESP. We are rejecting the (null) hypothesis that the results are
Hypothesis Testig Suppose you are ivestigatig extra sesory perceptio (ESP) You give someoe a test where they guess the color of card 100 times They are correct 90 times For guessig at radom you would expect
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 probability 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 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 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 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 informationPH 425 Quantum Measurement and Spin Winter SPINS Lab 1
PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured
More informationAnalysis of Experimental Measurements
Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example,
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 informationMath 152. Rumbos Fall Solutions to Review Problems for Exam #2. Number of Heads Frequency
Math 152. Rumbos Fall 2009 1 Solutios to Review Problems for Exam #2 1. I the book Experimetatio ad Measuremet, by W. J. Youde ad published by the by the Natioal Sciece Teachers Associatio i 1962, the
More informationDiscrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21. Some Important Distributions
CS 70 Discrete Mathematics for CS Sprig 2005 Clacy/Wager Notes 21 Some Importat Distributios Questio: A biased coi with Heads probability p is tossed repeatedly util the first Head appears. What is the
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationPoisson approximations
The Bi, p) ca be thought of as the distributio of a sum of idepedet idicator radom variables X +...+ X, with {X i = } deotig a head o the ith toss of a coi. The ormal approximatio to the Biomial works
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 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 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 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 informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
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 informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
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 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 informationIntroduction to probability Stochastic Process Queuing systems. TELE4642: Week2
Itroductio to probability Stochastic Process Queuig systems TELE4642: Week2 Overview Refresher: Probability theory Termiology, defiitio Coditioal probability, idepedece Radom variables ad distributios
More information4. Basic probability theory
Cotets Basic cocepts Discrete radom variables Discrete distributios (br distributios) Cotiuous radom variables Cotiuous distributios (time distributios) Other radom variables Lect04.ppt S-38.45 - Itroductio
More informationNote: we can take the Real and Imaginary part of the Schrödinger equation and write it in a way similar to the electromagnetic field. p = n!
Quatum Mechaics //8 Staford class with Leoard Susskid. Lecture 5. The ext class i this series has the title Special Relativity ad icludes classical field theory, least actio, Lagragia techiques, tesor
More informationANALYSIS OF EXPERIMENTAL ERRORS
ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder
More informationAs stated by Laplace, Probability is common sense reduced to calculation.
Note: Hadouts DO NOT replace the book. I most cases, they oly provide a guidelie o topics ad a ituitive feel. The math details will be covered i class, so it is importat to atted class ad also you MUST
More informationData Analysis and Statistical Methods Statistics 651
Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio
More informationMassachusetts Institute of Technology
Solutios to Quiz : Sprig 006 Problem : Each of the followig statemets is either True or False. There will be o partial credit give for the True False questios, thus ay explaatios will ot be graded. Please
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 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 informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationLecture 6 Simple alternatives and the Neyman-Pearson lemma
STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull
More informationClosed book and notes. No calculators. 60 minutes, but essentially unlimited time.
IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of
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 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 informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 15
CS 70 Discrete Mathematics ad Probability Theory Summer 2014 James Cook Note 15 Some Importat Distributios I this ote we will itroduce three importat probability distributios that are widely used to model
More informationLecture Chapter 6: Convergence of Random Sequences
ECE5: Aalysis of Radom Sigals Fall 6 Lecture Chapter 6: Covergece of Radom Sequeces Dr Salim El Rouayheb Scribe: Abhay Ashutosh Doel, Qibo Zhag, Peiwe Tia, Pegzhe Wag, Lu Liu Radom sequece Defiitio A ifiite
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 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 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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationLimit Theorems. Convergence in Probability. Let X be the number of heads observed in n tosses. Then, E[X] = np and Var[X] = np(1-p).
Limit Theorems Covergece i Probability Let X be the umber of heads observed i tosses. The, E[X] = p ad Var[X] = p(-p). L O This P x p NM QP P x p should be close to uity for large if our ituitio is correct.
More informationSTAT Homework 1 - Solutions
STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationIntroduction to Probability I: Expectations, Bayes Theorem, Gaussians, and the Poisson Distribution. 1
Itroductio to Probability I: Expectatios, Bayes Theorem, Gaussias, ad the Poisso Distributio. 1 Pakaj Mehta February 25, 2019 1 Read: This will itroduce some elemetary ideas i probability theory that we
More informationSample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for
Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they
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 informationEmpirical Distributions
Empirical Distributios A empirical distributio is oe for which each possible evet is assiged a probability derived from experimetal observatio. It is assumed that the evets are idepedet ad the sum of the
More informationSample Size Estimation in the Proportional Hazards Model for K-sample or Regression Settings Scott S. Emerson, M.D., Ph.D.
ample ie Estimatio i the Proportioal Haards Model for K-sample or Regressio ettigs cott. Emerso, M.D., Ph.D. ample ie Formula for a Normally Distributed tatistic uppose a statistic is kow to be ormally
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 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 information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
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 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 informationDiscrete Mathematics and Probability Theory Spring 2012 Alistair Sinclair Note 15
CS 70 Discrete Mathematics ad Probability Theory Sprig 2012 Alistair Siclair Note 15 Some Importat Distributios The first importat distributio we leared about i the last Lecture Note is the biomial distributio
More informationActivity 3: Length Measurements with the Four-Sided Meter Stick
Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter
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 informationChapter 18 Summary Sampling Distribution Models
Uit 5 Itroductio to Iferece Chapter 18 Summary Samplig Distributio Models What have we leared? Sample proportios ad meas will vary from sample to sample that s samplig error (samplig variability). Samplig
More informationCH.25 Discrete Random Variables
CH.25 Discrete Radom Variables 25B PG.784-785 #1, 3, 4, 6 25C PG.788-789 #1, 3, 5, 8, 10, 11 25D PG.791-792 #1, 3, 6 25E PG.794-795 #1, 2, 3, 7, 10 25F.1 PG.798-799 #2, 3, 5 25F.2 PG. 800-802 #2, 4, 7,
More informationSTATS 200: Introduction to Statistical Inference. Lecture 1: Course introduction and polling
STATS 200: Itroductio to Statistical Iferece Lecture 1: Course itroductio ad pollig U.S. presidetial electio projectios by state (Source: fivethirtyeight.com, 25 September 2016) Pollig Let s try to uderstad
More informationIntroduction to Probability and Statistics Twelfth Edition
Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver Itroductio to Probability ad Statistics Twelfth
More informationLecture 2: Poisson Sta*s*cs Probability Density Func*ons Expecta*on and Variance Es*mators
Lecture 2: Poisso Sta*s*cs Probability Desity Fuc*os Expecta*o ad Variace Es*mators Biomial Distribu*o: P (k successes i attempts) =! k!( k)! p k s( p s ) k prob of each success Poisso Distributio Note
More informationExponential Families and Bayesian Inference
Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where
More informationDISTRIBUTION LAW Okunev I.V.
1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated
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 informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
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 informationNOTES ON DISTRIBUTIONS
NOTES ON DISTRIBUTIONS MICHAEL N KATEHAKIS Radom Variables Radom variables represet outcomes from radom pheomea They are specified by two objects The rage R of possible values ad the frequecy fx with which
More information