Practical Statistics
|
|
- Mary Ford
- 5 years ago
- Views:
Transcription
1 Practical Statistics Lecture 1 (Nov. 9): - Correlation - Hypothesis Testing Lecture 2 (Nov. 16): - Error Estimation - Bayesian Analysis - Rejecting Outliers Lecture 3 (Nov. 18) - Monte Carlo Modeling - Bootstrap + Jack-knife Lecture 4 (Nov. 30): - Detection Effects - Survival Analysis Lecture 5 (Dec. 2): - Fourier Techniques - Filtering - Unevenly Sampled Data Good Reference: Hogg et al
2 Review: Process of Decision Making Ask a Question Take Data Reduce Data Derive Statistics describing data Reflect on what is needed Probability Distribution Error Analysis Does the Statistic answer your question? No Hypothesis Testing Yes Simulation Publish! 2
3 Review: The Binomial distribution You are observing something that has a probability, p, of occurring in a single observation. You observe it N times. Want chance of obtaining n successes. For one, particular sequence of observations the probability is: P 1 (n) =p n (1 p) N n There are many sequences which yield n successes: N! P (n) = n!(n n)! pn (1 p) N n N = p n (1 p) N n n Mean Np Variance Np(1-p) Often said N choose n
4 Review: Mean and Variance of Distributions Distribution Mean Variance Binomial Np Np(1-p) Poisson µ µ Gaussian µ σ 2 Uniform [a,b) (a+b)/2 (b-a)/12
5 Review: Comparing a data set to a distribution Suppose we have N data points and a model we think describes this with M parameters. Our model: y(x) =y(x, a 1...a M ) An intuitive metric is the distance of each data point from the model. Let's use the square of the difference between data and the model. LS = N (yi y(x i,a 1 a M )) 2 Why is this a reasonable metric for Determining the best fit to the data?
6 Review: Chi-squared The statistic chi-squared is defined as: N χ 2 (y i y(x i )) 2 = Chi-squared is not a unique metric, but is commonly used: Mean: µ χ 2 = ν = N M Variance: σχ 2 =2ν 2 Often, reduced chi-squared is quoted: Mean: Variance: i=1 µ reduced χ 2 =1 σ 2 reduced χ 2 = 2 ν σ 2
7 HW 2, Problem 2 10% of G type stars have detectable RV. How many stars should I observe to determine whether M type stars are similar? 7
8 Exam 1: Problem 2 Detector has digital units of measured flux, and 3 DU measured RMS noise at this level. How many photons does this correspond to? At no-light level, we measure 1 DU of RMS noise. How much noise does this add? 8
9 Correlation Often the first approach to analyzing data is to look for correlations in various parameters. - May or may not be physically motivated. - Understand experimental effects first (be skeptical). - Be careful of subclusters of points. - Correlation is not (necessarily) causation (remain skeptical). 9
10 A mass-separation correlation? 10
11 Are people born early in the year better hockey players? See Outliers book by Malcolm Gladwell 11
12 Correlation coefficient The correlation coefficient for two parameters, x and y, is defined as the covariance between parameters over the scatter in the distribution for each parameter: ρ = covariance(x, y) σ x σ y The correlation coefficient can be estimated directly from the data: r = i (X i <X>)(Y i <Y >) i (X i <X>) 2 i (Y i <Y >) 2 12
13 Probability of correlation For a bivariate Gaussian distribution, Bayes theorem can be used to estimate the probability of correlation: prob(ρ data) (1 ρ2 ) (N 1)/2 (1 ρr) ( ρr N 3/2 n 1/ ) 13
14 What if we see a correlation? It s common (but dangerous!) to just fit a line to the data: Anscombe s quartet illustrates the potential pitfalls of line fitting 14
15 Principle Component Analysis If we have N objects, n measured variables (x_n) for each object then: - We want a minimum number of variables that are independent. - These variables will be linear combinations of the observed variables: i = n a ij x j j=1 The goal is to define the new variables to minimize the residual variance in the data 15
16 Geometrical view of PCA Iterative approach of finding the component with maximum variance. 16
17 PCA manipulation 17
18 Statistics for Hypothesis Testing Hypothesis testing uses some metric to determine whether two data sets, or a data set and a model, are distinct. Typically, the problem is set up so that the hypothesis is that the data sets are consistent (the null hypothesis). A probability is calculated that the value found would be obtained again with another sample. Based on the required level of confidence, the hypothesis is rejected or accepted.
19 Parametric Tests Often, the most intuitive way to understand our data is to choose the parameter of interest (say the mean) and compare it to a model. Alternatively, we might be comparing two data sets by asking whether the differences in a statistic are meaningful. These general tests are called Parametric tests They can use frequentist approaches to accept or reject the hypothesis. They can use Bayesian approaches to calculate probabilities of different results. 19
20 Are two data sets drawn from the same distribution? The t statistic quantifies the likelihood that the means are the same. The F statistic quantifies the likelihood that the variances of two data sets are the same. Consider two data sets, x and y, with m and n data points: t = x y s 1/m +1/n F = (xi x) 2 /(n 1) (yi y) 2 /(m 1) s 2 = ns x + ms y n + m S x = (xi x) 2 n
21 Student's t test Calculate the t statistic. A perfect agreement is t=0. Evaluate the probability for t>value. ν = m + n 2 t = x y s 1/m +1/n s 2 = ns x + ms y n + m
22 F test Calculate the F statistic. F = (xi x) 2 /(n 1) (yi y) 2 /(m 1) Calculate the probability that F>value.
23 Non-Parametric Tests If we don t know the underlying distribution, or have small number statistics, there are still tests that can be used to accept or reject a hypothesis. Non-parametric tests still make some assumption about the data: Usually this is something related to the data following counting statistics, or the binomial distribution (randomness assumed, in the appropriate form) 23
24 Chi-squared test The chi-squared statistic can be used to compare any model to a data set: χ 2 = N i=1 (E i O i ) 2 E i Assumes variation in data is due to counting statistics Data must be binned so that E_i is reasonable for the model 24
25 The Kolmogorov-Smirnov Test Calculate the cumulative distribution function for your model (C_model(x)). Calculate the cumulative distribution function for your data(c_data(x). Find maximum of Cmodel(x)-Cdata(x) The variables, x, must be continuous to use K-S test. Don t need to bin the data.
26 K-S test example
27 Assignment: Test your toolbox Download Matlab (or use another tool for this) Download plot data set at: - Familiarize yourself with plotting data, error bars, etc. (This data set will be the basis of HW 7) 27
28 Matlab download go to: - Follow instructions to download and install. - Make sure to use an you@ .arizona.edu for Mathworks registration. 28
Statistical Methods for Astronomy
Statistical Methods for Astronomy Probability (Lecture 1) Statistics (Lecture 2) Why do we need statistics? Useful Statistics Definitions Error Analysis Probability distributions Error Propagation Binomial
More informationStatistical Methods for Astronomy
Statistical Methods for Astronomy If your experiment needs statistics, you ought to have done a better experiment. -Ernest Rutherford Lecture 1 Lecture 2 Why do we need statistics? Definitions Statistical
More informationAST 418/518 Instrumentation and Statistics
AST 418/518 Instrumentation and Statistics Class Website: http://ircamera.as.arizona.edu/astr_518 Class Texts: Practical Statistics for Astronomers, J.V. Wall, and C.R. Jenkins Measuring the Universe,
More informationStatistics notes. A clear statistical framework formulates the logic of what we are doing and why. It allows us to make precise statements.
Statistics notes Introductory comments These notes provide a summary or cheat sheet covering some basic statistical recipes and methods. These will be discussed in more detail in the lectures! What is
More informationStatistical Methods for Astronomy
Statistical Methods for Astronomy Probability (Lecture 1) Statistics (Lecture 2) Why do we need statistics? Useful Statistics Definitions Error Analysis Probability distributions Error Propagation Binomial
More informationSTATS 200: Introduction to Statistical Inference. Lecture 29: Course review
STATS 200: Introduction to Statistical Inference Lecture 29: Course review Course review We started in Lecture 1 with a fundamental assumption: Data is a realization of a random process. The goal throughout
More informationMath Review Sheet, Fall 2008
1 Descriptive Statistics Math 3070-5 Review Sheet, Fall 2008 First we need to know about the relationship among Population Samples Objects The distribution of the population can be given in one of the
More informationSTATISTICS OF OBSERVATIONS & SAMPLING THEORY. Parent Distributions
ASTR 511/O Connell Lec 6 1 STATISTICS OF OBSERVATIONS & SAMPLING THEORY References: Bevington Data Reduction & Error Analysis for the Physical Sciences LLM: Appendix B Warning: the introductory literature
More information* Tuesday 17 January :30-16:30 (2 hours) Recored on ESSE3 General introduction to the course.
Name of the course Statistical methods and data analysis Audience The course is intended for students of the first or second year of the Graduate School in Materials Engineering. The aim of the course
More informationIntroduction to Statistical Methods for High Energy Physics
Introduction to Statistical Methods for High Energy Physics 2011 CERN Summer Student Lectures Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk www.pp.rhul.ac.uk/~cowan
More informationStatistical Data Analysis Stat 3: p-values, parameter estimation
Statistical Data Analysis Stat 3: p-values, parameter estimation London Postgraduate Lectures on Particle Physics; University of London MSci course PH4515 Glen Cowan Physics Department Royal Holloway,
More informationData modelling Parameter estimation
ASTR509-10 Data modelling Parameter estimation Pierre-Simon, Marquis de Laplace 1749-1827 Under Napoleon, Laplace was a member, then chancellor, of the Senate, and received the Legion of Honour in 1805.
More informationStatistics and Data Analysis
Statistics and Data Analysis The Crash Course Physics 226, Fall 2013 "There are three kinds of lies: lies, damned lies, and statistics. Mark Twain, allegedly after Benjamin Disraeli Statistics and Data
More informationUnit 10: Simple Linear Regression and Correlation
Unit 10: Simple Linear Regression and Correlation Statistics 571: Statistical Methods Ramón V. León 6/28/2004 Unit 10 - Stat 571 - Ramón V. León 1 Introductory Remarks Regression analysis is a method for
More informationBrandon C. Kelly (Harvard Smithsonian Center for Astrophysics)
Brandon C. Kelly (Harvard Smithsonian Center for Astrophysics) Probability quantifies randomness and uncertainty How do I estimate the normalization and logarithmic slope of a X ray continuum, assuming
More informationMy data doesn t look like that..
Testing assumptions My data doesn t look like that.. We have made a big deal about testing model assumptions each week. Bill Pine Testing assumptions Testing assumptions We have made a big deal about testing
More informationIf we want to analyze experimental or simulated data we might encounter the following tasks:
Chapter 1 Introduction If we want to analyze experimental or simulated data we might encounter the following tasks: Characterization of the source of the signal and diagnosis Studying dependencies Prediction
More informationPhysics 6720 Introduction to Statistics April 4, 2017
Physics 6720 Introduction to Statistics April 4, 2017 1 Statistics of Counting Often an experiment yields a result that can be classified according to a set of discrete events, giving rise to an integer
More informationBERTINORO 2 (JVW) Yet more probability Bayes' Theorem* Monte Carlo! *The Reverend Thomas Bayes
BERTINORO 2 (JVW) Yet more probability Bayes' Theorem* Monte Carlo! *The Reverend Thomas Bayes 1702-61 1 The Power-law (Scale-free) Distribution N(>L) = K L (γ+1) (integral form) dn = (γ+1) K L γ dl (differential
More informationSubject CS1 Actuarial Statistics 1 Core Principles
Institute of Actuaries of India Subject CS1 Actuarial Statistics 1 Core Principles For 2019 Examinations Aim The aim of the Actuarial Statistics 1 subject is to provide a grounding in mathematical and
More informationStatistical Methods in Particle Physics
Statistical Methods in Particle Physics Lecture 3 October 29, 2012 Silvia Masciocchi, GSI Darmstadt s.masciocchi@gsi.de Winter Semester 2012 / 13 Outline Reminder: Probability density function Cumulative
More informationUnit 14: Nonparametric Statistical Methods
Unit 14: Nonparametric Statistical Methods Statistics 571: Statistical Methods Ramón V. León 8/8/2003 Unit 14 - Stat 571 - Ramón V. León 1 Introductory Remarks Most methods studied so far have been based
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2018 Examinations Subject CT3 Probability and Mathematical Statistics Core Technical Syllabus 1 June 2017 Aim The
More informationDeciding, Estimating, Computing, Checking
Deciding, Estimating, Computing, Checking How are Bayesian posteriors used, computed and validated? Fundamentalist Bayes: The posterior is ALL knowledge you have about the state Use in decision making:
More informationDeciding, Estimating, Computing, Checking. How are Bayesian posteriors used, computed and validated?
Deciding, Estimating, Computing, Checking How are Bayesian posteriors used, computed and validated? Fundamentalist Bayes: The posterior is ALL knowledge you have about the state Use in decision making:
More informationPhysics 509: Bootstrap and Robust Parameter Estimation
Physics 509: Bootstrap and Robust Parameter Estimation Scott Oser Lecture #20 Physics 509 1 Nonparametric parameter estimation Question: what error estimate should you assign to the slope and intercept
More informationApplied Regression. Applied Regression. Chapter 2 Simple Linear Regression. Hongcheng Li. April, 6, 2013
Applied Regression Chapter 2 Simple Linear Regression Hongcheng Li April, 6, 2013 Outline 1 Introduction of simple linear regression 2 Scatter plot 3 Simple linear regression model 4 Test of Hypothesis
More informationIntroduction to Statistics and Error Analysis II
Introduction to Statistics and Error Analysis II Physics116C, 4/14/06 D. Pellett References: Data Reduction and Error Analysis for the Physical Sciences by Bevington and Robinson Particle Data Group notes
More informationREVIEW 8/2/2017 陈芳华东师大英语系
REVIEW Hypothesis testing starts with a null hypothesis and a null distribution. We compare what we have to the null distribution, if the result is too extreme to belong to the null distribution (p
More informationTable of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z).
Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z). For example P(X.04) =.8508. For z < 0 subtract the value from,
More informationPractice Problems Section Problems
Practice Problems Section 4-4-3 4-4 4-5 4-6 4-7 4-8 4-10 Supplemental Problems 4-1 to 4-9 4-13, 14, 15, 17, 19, 0 4-3, 34, 36, 38 4-47, 49, 5, 54, 55 4-59, 60, 63 4-66, 68, 69, 70, 74 4-79, 81, 84 4-85,
More informationLecture 2: Linear Models. Bruce Walsh lecture notes Seattle SISG -Mixed Model Course version 23 June 2011
Lecture 2: Linear Models Bruce Walsh lecture notes Seattle SISG -Mixed Model Course version 23 June 2011 1 Quick Review of the Major Points The general linear model can be written as y = X! + e y = vector
More informationMATH 10 INTRODUCTORY STATISTICS
MATH 10 INTRODUCTORY STATISTICS Tommy Khoo Your friendly neighbourhood graduate student. It is Time for Homework! ( ω `) First homework + data will be posted on the website, under the homework tab. And
More informationFourier and Stats / Astro Stats and Measurement : Stats Notes
Fourier and Stats / Astro Stats and Measurement : Stats Notes Andy Lawrence, University of Edinburgh Autumn 2013 1 Probabilities, distributions, and errors Laplace once said Probability theory is nothing
More informationContents. Preface to Second Edition Preface to First Edition Abbreviations PART I PRINCIPLES OF STATISTICAL THINKING AND ANALYSIS 1
Contents Preface to Second Edition Preface to First Edition Abbreviations xv xvii xix PART I PRINCIPLES OF STATISTICAL THINKING AND ANALYSIS 1 1 The Role of Statistical Methods in Modern Industry and Services
More informationCE 3710: Uncertainty Analysis in Engineering
FINAL EXAM Monday, December 14, 10:15 am 12:15 pm, Chem Sci 101 Open book and open notes. Exam will be cumulative, but emphasis will be on material covered since Exam II Learning Expectations for Final
More informationMATH 10 INTRODUCTORY STATISTICS
MATH 10 INTRODUCTORY STATISTICS Ramesh Yapalparvi It is Time for Homework! ( ω `) First homework + data will be posted on the website, under the homework tab. And also sent out via email. 30% weekly homework.
More informationMock Exam - 2 hours - use of basic (non-programmable) calculator is allowed - all exercises carry the same marks - exam is strictly individual
Mock Exam - 2 hours - use of basic (non-programmable) calculator is allowed - all exercises carry the same marks - exam is strictly individual Question 1. Suppose you want to estimate the percentage of
More informationSome Statistics. V. Lindberg. May 16, 2007
Some Statistics V. Lindberg May 16, 2007 1 Go here for full details An excellent reference written by physicists with sample programs available is Data Reduction and Error Analysis for the Physical Sciences,
More informationHypothesis testing:power, test statistic CMS:
Hypothesis testing:power, test statistic The more sensitive the test, the better it can discriminate between the null and the alternative hypothesis, quantitatively, maximal power In order to achieve this
More informationCh. 1: Data and Distributions
Ch. 1: Data and Distributions Populations vs. Samples How to graphically display data Histograms, dot plots, stem plots, etc Helps to show how samples are distributed Distributions of both continuous and
More informationFundamental Probability and Statistics
Fundamental Probability and Statistics "There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are
More informationLecture 3: Linear Models. Bruce Walsh lecture notes Uppsala EQG course version 28 Jan 2012
Lecture 3: Linear Models Bruce Walsh lecture notes Uppsala EQG course version 28 Jan 2012 1 Quick Review of the Major Points The general linear model can be written as y = X! + e y = vector of observed
More informationStatistical Methods in Particle Physics Lecture 1: Bayesian methods
Statistical Methods in Particle Physics Lecture 1: Bayesian methods SUSSP65 St Andrews 16 29 August 2009 Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk www.pp.rhul.ac.uk/~cowan
More informationLecture 1: Probability Fundamentals
Lecture 1: Probability Fundamentals IB Paper 7: Probability and Statistics Carl Edward Rasmussen Department of Engineering, University of Cambridge January 22nd, 2008 Rasmussen (CUED) Lecture 1: Probability
More informationConfidence Intervals, Testing and ANOVA Summary
Confidence Intervals, Testing and ANOVA Summary 1 One Sample Tests 1.1 One Sample z test: Mean (σ known) Let X 1,, X n a r.s. from N(µ, σ) or n > 30. Let The test statistic is H 0 : µ = µ 0. z = x µ 0
More informationSTA 2201/442 Assignment 2
STA 2201/442 Assignment 2 1. This is about how to simulate from a continuous univariate distribution. Let the random variable X have a continuous distribution with density f X (x) and cumulative distribution
More informationTesting Statistical Hypotheses
E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition 4y Springer Preface vii I Small-Sample Theory 1 1 The General Decision Problem 3 1.1 Statistical Inference and Statistical Decisions
More informationStatistical Methods in Particle Physics
Statistical Methods in Particle Physics Lecture 11 January 7, 2013 Silvia Masciocchi, GSI Darmstadt s.masciocchi@gsi.de Winter Semester 2012 / 13 Outline How to communicate the statistical uncertainty
More informationParameter estimation! and! forecasting! Cristiano Porciani! AIfA, Uni-Bonn!
Parameter estimation! and! forecasting! Cristiano Porciani! AIfA, Uni-Bonn! Questions?! C. Porciani! Estimation & forecasting! 2! Cosmological parameters! A branch of modern cosmological research focuses
More informationLecture 10: Generalized likelihood ratio test
Stat 200: Introduction to Statistical Inference Autumn 2018/19 Lecture 10: Generalized likelihood ratio test Lecturer: Art B. Owen October 25 Disclaimer: These notes have not been subjected to the usual
More informationLECTURE NOTES FYS 4550/FYS EXPERIMENTAL HIGH ENERGY PHYSICS AUTUMN 2013 PART I A. STRANDLIE GJØVIK UNIVERSITY COLLEGE AND UNIVERSITY OF OSLO
LECTURE NOTES FYS 4550/FYS9550 - EXPERIMENTAL HIGH ENERGY PHYSICS AUTUMN 2013 PART I PROBABILITY AND STATISTICS A. STRANDLIE GJØVIK UNIVERSITY COLLEGE AND UNIVERSITY OF OSLO Before embarking on the concept
More informationCSE 312 Final Review: Section AA
CSE 312 TAs December 8, 2011 General Information General Information Comprehensive Midterm General Information Comprehensive Midterm Heavily weighted toward material after the midterm Pre-Midterm Material
More informationLecture 2. G. Cowan Lectures on Statistical Data Analysis Lecture 2 page 1
Lecture 2 1 Probability (90 min.) Definition, Bayes theorem, probability densities and their properties, catalogue of pdfs, Monte Carlo 2 Statistical tests (90 min.) general concepts, test statistics,
More information1 Statistics Aneta Siemiginowska a chapter for X-ray Astronomy Handbook October 2008
1 Statistics Aneta Siemiginowska a chapter for X-ray Astronomy Handbook October 2008 1.1 Introduction Why do we need statistic? Wall and Jenkins (2003) give a good description of the scientific analysis
More information18.05 Practice Final Exam
No calculators. 18.05 Practice Final Exam Number of problems 16 concept questions, 16 problems. Simplifying expressions Unless asked to explicitly, you don t need to simplify complicated expressions. For
More informationStatistics Introductory Correlation
Statistics Introductory Correlation Session 10 oscardavid.barrerarodriguez@sciencespo.fr April 9, 2018 Outline 1 Statistics are not used only to describe central tendency and variability for a single variable.
More informationOverview of Spatial analysis in ecology
Spatial Point Patterns & Complete Spatial Randomness - II Geog 0C Introduction to Spatial Data Analysis Chris Funk Lecture 8 Overview of Spatial analysis in ecology st step in understanding ecological
More informationPart III: Unstructured Data
Inf1-DA 2010 2011 III: 51 / 89 Part III Unstructured Data Data Retrieval: III.1 Unstructured data and data retrieval Statistical Analysis of Data: III.2 Data scales and summary statistics III.3 Hypothesis
More informationStatistics. Lent Term 2015 Prof. Mark Thomson. 2: The Gaussian Limit
Statistics Lent Term 2015 Prof. Mark Thomson Lecture 2 : The Gaussian Limit Prof. M.A. Thomson Lent Term 2015 29 Lecture Lecture Lecture Lecture 1: Back to basics Introduction, Probability distribution
More informationLecture 3. G. Cowan. Lecture 3 page 1. Lectures on Statistical Data Analysis
Lecture 3 1 Probability (90 min.) Definition, Bayes theorem, probability densities and their properties, catalogue of pdfs, Monte Carlo 2 Statistical tests (90 min.) general concepts, test statistics,
More informationPrimer on statistics:
Primer on statistics: MLE, Confidence Intervals, and Hypothesis Testing ryan.reece@gmail.com http://rreece.github.io/ Insight Data Science - AI Fellows Workshop Feb 16, 018 Outline 1. Maximum likelihood
More informationReview. DS GA 1002 Statistical and Mathematical Models. Carlos Fernandez-Granda
Review DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos Fernandez-Granda Probability and statistics Probability: Framework for dealing with
More informationRobustness and Distribution Assumptions
Chapter 1 Robustness and Distribution Assumptions 1.1 Introduction In statistics, one often works with model assumptions, i.e., one assumes that data follow a certain model. Then one makes use of methodology
More informationSimulation. Where real stuff starts
1 Simulation Where real stuff starts ToC 1. What is a simulation? 2. Accuracy of output 3. Random Number Generators 4. How to sample 5. Monte Carlo 6. Bootstrap 2 1. What is a simulation? 3 What is a simulation?
More informationSTAT 461/561- Assignments, Year 2015
STAT 461/561- Assignments, Year 2015 This is the second set of assignment problems. When you hand in any problem, include the problem itself and its number. pdf are welcome. If so, use large fonts and
More informationStatistical Inference: Estimation and Confidence Intervals Hypothesis Testing
Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing 1 In most statistics problems, we assume that the data have been generated from some unknown probability distribution. We desire
More informationStatistical techniques for data analysis in Cosmology
Statistical techniques for data analysis in Cosmology arxiv:0712.3028; arxiv:0911.3105 Numerical recipes (the bible ) Licia Verde ICREA & ICC UB-IEEC http://icc.ub.edu/~liciaverde outline Lecture 1: Introduction
More informationStatistical Models with Uncertain Error Parameters (G. Cowan, arxiv: )
Statistical Models with Uncertain Error Parameters (G. Cowan, arxiv:1809.05778) Workshop on Advanced Statistics for Physics Discovery aspd.stat.unipd.it Department of Statistical Sciences, University of
More informationStat 535 C - Statistical Computing & Monte Carlo Methods. Arnaud Doucet.
Stat 535 C - Statistical Computing & Monte Carlo Methods Arnaud Doucet Email: arnaud@cs.ubc.ca 1 CS students: don t forget to re-register in CS-535D. Even if you just audit this course, please do register.
More informationHypothesis testing. 1 Principle of hypothesis testing 2
Hypothesis testing Contents 1 Principle of hypothesis testing One sample tests 3.1 Tests on Mean of a Normal distribution..................... 3. Tests on Variance of a Normal distribution....................
More informationMath 562 Homework 1 August 29, 2006 Dr. Ron Sahoo
Math 56 Homework August 9, 006 Dr. Ron Sahoo He who labors diligently need never despair; for all things are accomplished by diligence and labor. Menander of Athens Direction: This homework worths 60 points
More informationStat 5101 Lecture Notes
Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random
More informationChapter 27 Summary Inferences for Regression
Chapter 7 Summary Inferences for Regression What have we learned? We have now applied inference to regression models. Like in all inference situations, there are conditions that we must check. We can test
More informationPhysics 403. Segev BenZvi. Parameter Estimation, Correlations, and Error Bars. Department of Physics and Astronomy University of Rochester
Physics 403 Parameter Estimation, Correlations, and Error Bars Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Review of Last Class Best Estimates and Reliability
More informationSYSM 6303: Quantitative Introduction to Risk and Uncertainty in Business Lecture 4: Fitting Data to Distributions
SYSM 6303: Quantitative Introduction to Risk and Uncertainty in Business Lecture 4: Fitting Data to Distributions M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu
More informationy n 1 ( x i x )( y y i n 1 i y 2
STP3 Brief Class Notes Instructor: Ela Jackiewicz Chapter Regression and Correlation In this chapter we will explore the relationship between two quantitative variables, X an Y. We will consider n ordered
More informationLecture 5. G. Cowan Lectures on Statistical Data Analysis Lecture 5 page 1
Lecture 5 1 Probability (90 min.) Definition, Bayes theorem, probability densities and their properties, catalogue of pdfs, Monte Carlo 2 Statistical tests (90 min.) general concepts, test statistics,
More informationM & M Project. Think! Crunch those numbers! Answer!
M & M Project Think! Crunch those numbers! Answer! Chapters 1-2 Exploring Data and Describing Location in a Distribution Univariate Data: Length Stemplot and Frequency Table Stem (Units Digit) 0 1 1 Leaf
More informationRecall the Basics of Hypothesis Testing
Recall the Basics of Hypothesis Testing The level of significance α, (size of test) is defined as the probability of X falling in w (rejecting H 0 ) when H 0 is true: P(X w H 0 ) = α. H 0 TRUE H 1 TRUE
More informationProbability Density Functions
Statistical Methods in Particle Physics / WS 13 Lecture II Probability Density Functions Niklaus Berger Physics Institute, University of Heidelberg Recap of Lecture I: Kolmogorov Axioms Ingredients: Set
More informationData Analysis I. Dr Martin Hendry, Dept of Physics and Astronomy University of Glasgow, UK. 10 lectures, beginning October 2006
Astronomical p( y x, I) p( x, I) p ( x y, I) = p( y, I) Data Analysis I Dr Martin Hendry, Dept of Physics and Astronomy University of Glasgow, UK 10 lectures, beginning October 2006 4. Monte Carlo Methods
More informationQuantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing
Quantitative Introduction ro Risk and Uncertainty in Business Module 5: Hypothesis Testing M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More informationBiostatistics for physicists fall Correlation Linear regression Analysis of variance
Biostatistics for physicists fall 2015 Correlation Linear regression Analysis of variance Correlation Example: Antibody level on 38 newborns and their mothers There is a positive correlation in antibody
More informationTesting Statistical Hypotheses
E.L. Lehmann Joseph P. Romano, 02LEu1 ttd ~Lt~S Testing Statistical Hypotheses Third Edition With 6 Illustrations ~Springer 2 The Probability Background 28 2.1 Probability and Measure 28 2.2 Integration.........
More informationSTAT 518 Intro Student Presentation
STAT 518 Intro Student Presentation Wen Wei Loh April 11, 2013 Title of paper Radford M. Neal [1999] Bayesian Statistics, 6: 475-501, 1999 What the paper is about Regression and Classification Flexible
More informationChapte The McGraw-Hill Companies, Inc. All rights reserved.
er15 Chapte Chi-Square Tests d Chi-Square Tests for -Fit Uniform Goodness- Poisson Goodness- Goodness- ECDF Tests (Optional) Contingency Tables A contingency table is a cross-tabulation of n paired observations
More informationMath 50: Final. 1. [13 points] It was found that 35 out of 300 famous people have the star sign Sagittarius.
Math 50: Final 180 minutes, 140 points. No algebra-capable calculators. Try to use your calculator only at the end of your calculation, and show working/reasoning. Please do look up z, t, χ 2 values for
More informationStatistics Boot Camp. Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018
Statistics Boot Camp Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018 March 21, 2018 Outline of boot camp Summarizing and simplifying data Point and interval estimation Foundations of statistical
More informationModeling and Performance Analysis with Discrete-Event Simulation
Simulation Modeling and Performance Analysis with Discrete-Event Simulation Chapter 9 Input Modeling Contents Data Collection Identifying the Distribution with Data Parameter Estimation Goodness-of-Fit
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 4.0 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationAPPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 651 APPENDIX B. BIBLIOGRAPHY 677 APPENDIX C. ANSWERS TO SELECTED EXERCISES 679
APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 1 Table I Summary of Common Probability Distributions 2 Table II Cumulative Standard Normal Distribution Table III Percentage Points, 2 of the Chi-Squared
More informationSTATISTICS ANCILLARY SYLLABUS. (W.E.F. the session ) Semester Paper Code Marks Credits Topic
STATISTICS ANCILLARY SYLLABUS (W.E.F. the session 2014-15) Semester Paper Code Marks Credits Topic 1 ST21012T 70 4 Descriptive Statistics 1 & Probability Theory 1 ST21012P 30 1 Practical- Using Minitab
More informationStatistics for Data Analysis. Niklaus Berger. PSI Practical Course Physics Institute, University of Heidelberg
Statistics for Data Analysis PSI Practical Course 2014 Niklaus Berger Physics Institute, University of Heidelberg Overview You are going to perform a data analysis: Compare measured distributions to theoretical
More informationProbability and Statistics. Joyeeta Dutta-Moscato June 29, 2015
Probability and Statistics Joyeeta Dutta-Moscato June 29, 2015 Terms and concepts Sample vs population Central tendency: Mean, median, mode Variance, standard deviation Normal distribution Cumulative distribution
More informationQualifying Exam CS 661: System Simulation Summer 2013 Prof. Marvin K. Nakayama
Qualifying Exam CS 661: System Simulation Summer 2013 Prof. Marvin K. Nakayama Instructions This exam has 7 pages in total, numbered 1 to 7. Make sure your exam has all the pages. This exam will be 2 hours
More informationOverview. Confidence Intervals Sampling and Opinion Polls Error Correcting Codes Number of Pet Unicorns in Ireland
Overview Confidence Intervals Sampling and Opinion Polls Error Correcting Codes Number of Pet Unicorns in Ireland Confidence Intervals When a random variable lies in an interval a X b with a specified
More informationTHE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 004 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER II STATISTICAL METHODS The Society provides these solutions to assist candidates preparing for the examinations in future
More informationAST 418/518 Instrumentation and Statistics
AST 418/518 Instrumentation and Statistics Cass Website: http://ircamera.as.arizona.edu/astr_518 Cass Texts: Practica Statistics for Astronomers, J.V. Wa, and C.R. Jenkins, Second Edition. Measuring the
More informationPhysics 509: Non-Parametric Statistics and Correlation Testing
Physics 509: Non-Parametric Statistics and Correlation Testing Scott Oser Lecture #19 Physics 509 1 What is non-parametric statistics? Non-parametric statistics is the application of statistical tests
More information