Probability Density Functions
|
|
- Jason Wood
- 5 years ago
- Views:
Transcription
1 Statistical Methods in Particle Physics / WS 13 Lecture II Probability Density Functions Niklaus Berger Physics Institute, University of Heidelberg
2 Recap of Lecture I: Kolmogorov Axioms Ingredients: Set S of outcomes of the experiment (sample space) Subsets A, B... of S (technically need to form a Σ-Algebra) Mapping into the real numbers P(A) called the Probability Kolmogorov Axioms: For all A S, P(A) 0 P(S) = 1 If A B = 0, P(A B) = P(A) + P(B) Niklaus Berger SMIPP WS 2013 Slide 2
3 Recap of Lecture I: Bayes Theorem Niklaus Berger SMIPP WS 2013 Slide 3
4 Part II: Catalogue of Probability Density Functions Niklaus Berger SMIPP WS 2013 Slide 4
5 2.1. Binomial Distribution p=0.5 and n=20 p=0.7 and n=20 p=0.5 and n=40 P (x; n, p) = (x) n px (1-p) n-x B Mean: E[x] = np p=0.5 and N=20 p=0.7 and N=20 p=0.5 and N= Variance: V[x] = np(1-p) Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 5
6 Binomial Distribution: Application y z Efficiency measurement: x n particles cross the Device Under Test (DUT) DUT has an efficiency to detect a particle of p Average number of hits seen in DUT x is n p Results of many experiments will follow a binomial distribution - use corresponding error bars Do NOT use counting errors on n and x and perform error propagation Niklaus Berger SMIPP WS 2013 Slide 6
7 Rare events Take limit of binomial distribution for n p 0 with n p fixed to ν Rare events with constant rate Niklaus Berger SMIPP WS 2013 Slide 7
8 2.2. Poisson Distribution ν x P P (x; ν) = x! e -ν Mean: Variance: E[x] = ν V[x] = ν Standard deviation: Σ = ν Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 8
9 Poisson Distribution: Application Any counting experiment with rare events will have results that are Poisson distributed: ν = Σ Ldt expected number of events given luminosity and cross-section Number of entries in a histogram bin Number of cosmic rays passing through you per second (what is ν?) Niklaus Berger SMIPP WS 2013 Slide 9
10 Many rare events Look at Poisson distribution for ν >> 1 Many Rare events with constant rate (and almost every other large N limit...) Niklaus Berger SMIPP WS 2013 Slide 10
11 2.3. Normal (Gaussian) Distribution 1 P (x; μ,σ) = e -(x-μ)2 /2Σ 2 G 2πΣ μ= 0, μ= 0, μ= 0, μ= 2, 2 σ = 0.2, 2 σ = 1.0, 2 σ = 5.0, 2 σ = 0.5, φ μ,σ 2( x) Mean: E[x] = μ μ= 0, μ= 0, μ= 0, μ= 2, 2 σ = 0.2, 2 σ = 1.0, 2 σ = 5.0, 2 σ = 0.5, x Variance: V[x] = Σ 2 Φ μ,σ 2(x) CDF is called error function x Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 11
12 Central limit theorem The arithmetic mean of a sufficiently large number of uncorrelated random variables drawn from a distribution with well defined mean and well defined and finite variance will be approximately normal distributed This is independent of the underlying distribution If your measurement is sufficiently messy (i.e. influenced by enough independent effects) it will be normal distributed Usually leads to the assumption, that everything is normal distributed, which is wrong... Niklaus Berger SMIPP WS 2013 Slide 12
13 Examples for the normal distribution Good example: Size distribution among students of a particular sex in a class Not so good example: Distribution of scattering angles of particles passing through a slab of material (There are large angle scatters (Rutherford!) that produce non-gaussian tails) Bad example: Energy loss of particles passing through a slab of material (Dominated by few large losses - Landau distribution) Niklaus Berger SMIPP WS 2013 Slide 13
14 Quantiles of the Normal Distribution: Standard Deviations % 2.1% 3σ 34.1% 34.1% 13.6% 13.6% 2.1% 0.1% 2σ 1σ µ 1σ 2σ 3σ Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 14
15 2.4. Log Normal Distribution 1 P (x; μ,σ) = e -(ln x-μ)2 /2Σ 2 G x 2πΣ Mean: E[x] = e μ + Σ2 /2 Variance: V[x] = (e Σ2-1) e 2μ+Σ2 Order of magnitude is normally distributed: e.g. growth of droplets Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 15
16 2.5. Exponential Distribution 1 P (x; τ) = e - x/τ for x 0 E Τ Mean: E[x] = Τ Variance: V[x] = Τ 2 Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 16
17 Exponential Distribution: Application Lifetime of a particle Has no memory: f(t -t 0 t > t 0 ) = f(t) Niklaus Berger SMIPP WS 2013 Slide 17
18 2.6. Uniform Distribution 1 P (x; a, b) = for a x b U a-b Mean: E[x] = 1/2 (a+b) Variance: V[x] = 1/12 (a-b) 2 Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 18
Statistics 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 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 informationLectures on Statistical Data Analysis
Lectures on Statistical Data Analysis London Postgraduate Lectures on Particle Physics; University of London MSci course PH4515 Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk
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 informationThe Binomial distribution. Probability theory 2. Example. The Binomial distribution
Probability theory Tron Anders Moger September th 7 The Binomial distribution Bernoulli distribution: One experiment X i with two possible outcomes, probability of success P. If the experiment is repeated
More informationWhy study probability? Set theory. ECE 6010 Lecture 1 Introduction; Review of Random Variables
ECE 6010 Lecture 1 Introduction; Review of Random Variables Readings from G&S: Chapter 1. Section 2.1, Section 2.3, Section 2.4, Section 3.1, Section 3.2, Section 3.5, Section 4.1, Section 4.2, Section
More informationIndependent Events. Two events are independent if knowing that one occurs does not change the probability of the other occurring
Independent Events Two events are independent if knowing that one occurs does not change the probability of the other occurring Conditional probability is denoted P(A B), which is defined to be: P(A and
More informationRWTH Aachen Graduiertenkolleg
RWTH Aachen Graduiertenkolleg 9-13 February, 2009 Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk www.pp.rhul.ac.uk/~cowan Course web page: www.pp.rhul.ac.uk/~cowan/stat_aachen.html
More informationStatistical Data Analysis 2017/18
Statistical Data Analysis 2017/18 London Postgraduate Lectures on Particle Physics; University of London MSci course PH4515 Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk
More informationSTA 111: Probability & Statistical Inference
STA 111: Probability & Statistical Inference Lecture Four Expectation and Continuous Random Variables Instructor: Olanrewaju Michael Akande Department of Statistical Science, Duke University Instructor:
More informationLecture 16. Lectures 1-15 Review
18.440: Lecture 16 Lectures 1-15 Review Scott Sheffield MIT 1 Outline Counting tricks and basic principles of probability Discrete random variables 2 Outline Counting tricks and basic principles of probability
More informationYETI IPPP Durham
YETI 07 @ IPPP Durham Young Experimentalists and Theorists Institute Glen Cowan Physics Department Royal Holloway, University of London g.cowan@rhul.ac.uk www.pp.rhul.ac.uk/~cowan Course web page: www.pp.rhul.ac.uk/~cowan/stat_yeti.html
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 informationAn introduction to Bayesian reasoning in particle physics
An introduction to Bayesian reasoning in particle physics Graduiertenkolleg seminar, May 15th 2013 Overview Overview A concrete example Scientific reasoning Probability and the Bayesian interpretation
More informationNumerical Methods for Data Analysis
Michael O. Distler distler@uni-mainz.de Bosen (Saar), August 29 - September 3, 2010 Fundamentals Probability distributions Expectation values, error propagation Parameter estimation Regression analysis
More informationBMIR Lecture Series on Probability and Statistics Fall, 2015 Uniform Distribution
Lecture #5 BMIR Lecture Series on Probability and Statistics Fall, 2015 Department of Biomedical Engineering and Environmental Sciences National Tsing Hua University s 5.1 Definition ( ) A continuous random
More informationAdvanced Herd Management Probabilities and distributions
Advanced Herd Management Probabilities and distributions Anders Ringgaard Kristensen Slide 1 Outline Probabilities Conditional probabilities Bayes theorem Distributions Discrete Continuous Distribution
More informationModern Methods of Data Analysis - WS 07/08
Modern Methods of Data Analysis Lecture V (12.11.07) Contents: Central Limit Theorem Uncertainties: concepts, propagation and properties Central Limit Theorem Consider the sum X of n independent variables,
More informationIntro to probability concepts
October 31, 2017 Serge Lang lecture This year s Serge Lang Undergraduate Lecture will be given by Keith Devlin of our main athletic rival. The title is When the precision of mathematics meets the messiness
More informationStatistical Methods in Particle Physics. Lecture 2
Statistical Methods in Particle Physics Lecture 2 October 17, 2011 Silvia Masciocchi, GSI Darmstadt s.masciocchi@gsi.de Winter Semester 2011 / 12 Outline Probability Definition and interpretation Kolmogorov's
More informationContinuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.
UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Christopher Barr University of California, Los Angeles,
More informationComputer Science, Informatik 4 Communication and Distributed Systems. Simulation. Discrete-Event System Simulation. Dr.
Simulation Discrete-Event System Simulation Chapter 4 Statistical Models in Simulation Purpose & Overview The world the model-builder sees is probabilistic rather than deterministic. Some statistical model
More informationCME 106: Review Probability theory
: Probability theory Sven Schmit April 3, 2015 1 Overview In the first half of the course, we covered topics from probability theory. The difference between statistics and probability theory is the following:
More informationPhysics 403 Probability Distributions II: More Properties of PDFs and PMFs
Physics 403 Probability Distributions II: More Properties of PDFs and PMFs Segev BenZvi Department of Physics and Astronomy University of Rochester Table of Contents 1 Last Time: Common Probability Distributions
More informationProbability Review. Chao Lan
Probability Review Chao Lan Let s start with a single random variable Random Experiment A random experiment has three elements 1. sample space Ω: set of all possible outcomes e.g.,ω={1,2,3,4,5,6} 2. event
More informationContinuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( )
UCLA STAT 35 Applied Computational and Interactive Probability Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Chris Barr Continuous Random Variables and Probability
More informationIntroduction to Error Analysis
Introduction to Error Analysis Part 1: the Basics Andrei Gritsan based on lectures by Petar Maksimović February 1, 2010 Overview Definitions Reporting results and rounding Accuracy vs precision systematic
More informationProbability and Probability Distributions. Dr. Mohammed Alahmed
Probability and Probability Distributions 1 Probability and Probability Distributions Usually we want to do more with data than just describing them! We might want to test certain specific inferences about
More informationfunctions Poisson distribution Normal distribution Arbitrary functions
Physics 433: Computational Physics Lecture 6 Random number distributions Generation of random numbers of various distribuition functions Normal distribution Poisson distribution Arbitrary functions Random
More informationChapter Learning Objectives. Probability Distributions and Probability Density Functions. Continuous Random Variables
Chapter 4: Continuous Random Variables and Probability s 4-1 Continuous Random Variables 4-2 Probability s and Probability Density Functions 4-3 Cumulative Functions 4-4 Mean and Variance of a Continuous
More informationRandom Variables. Definition: A random variable (r.v.) X on the probability space (Ω, F, P) is a mapping
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
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 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 informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Tobias Neckel, Ionuț-Gabriel Farcaș Lehrstuhl Informatik V Summer Semester 2017 Lecture 2: Repetition of probability theory and statistics Example: coin flip Example
More informationStatistical Methods in Particle Physics
Statistical Methods in Particle Physics. Probability Distributions Prof. Dr. Klaus Reygers (lectures) Dr. Sebastian Neubert (tutorials) Heidelberg University WS 07/8 Gaussian g(x; µ, )= p exp (x µ) https://en.wikipedia.org/wiki/normal_distribution
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 informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
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 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 informationProbability Review. Yutian Li. January 18, Stanford University. Yutian Li (Stanford University) Probability Review January 18, / 27
Probability Review Yutian Li Stanford University January 18, 2018 Yutian Li (Stanford University) Probability Review January 18, 2018 1 / 27 Outline 1 Elements of probability 2 Random variables 3 Multiple
More informationL2: Review of probability and statistics
Probability L2: Review of probability and statistics Definition of probability Axioms and properties Conditional probability Bayes theorem Random variables Definition of a random variable Cumulative distribution
More informationStatistics and data analyses
Statistics and data analyses Designing experiments Measuring time Instrumental quality Precision Standard deviation depends on Number of measurements Detection quality Systematics and methology σ tot =
More informationLearning Objectives for Stat 225
Learning Objectives for Stat 225 08/20/12 Introduction to Probability: Get some general ideas about probability, and learn how to use sample space to compute the probability of a specific event. Set Theory:
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationLecture 2: Repetition of probability theory and statistics
Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:
More informationPHYS 6710: Nuclear and Particle Physics II
Data Analysis Content (~7 Classes) Uncertainties and errors Random variables, expectation value, (co)variance Distributions and samples Binomial, Poisson, and Normal Distribution Student's t-distribution
More information15 Discrete Distributions
Lecture Note 6 Special Distributions (Discrete and Continuous) MIT 4.30 Spring 006 Herman Bennett 5 Discrete Distributions We have already seen the binomial distribution and the uniform distribution. 5.
More informationCopyright c 2006 Jason Underdown Some rights reserved. choose notation. n distinct items divided into r distinct groups.
Copyright & License Copyright c 2006 Jason Underdown Some rights reserved. choose notation binomial theorem n distinct items divided into r distinct groups Axioms Proposition axioms of probability probability
More informationProbability and Estimation. Alan Moses
Probability and Estimation Alan Moses Random variables and probability A random variable is like a variable in algebra (e.g., y=e x ), but where at least part of the variability is taken to be stochastic.
More information3.4. The Binomial Probability Distribution
3.4. The Binomial Probability Distribution Objectives. Binomial experiment. Binomial random variable. Using binomial tables. Mean and variance of binomial distribution. 3.4.1. Four Conditions that determined
More information4. Discrete Probability Distributions. Introduction & Binomial Distribution
4. Discrete Probability Distributions Introduction & Binomial Distribution Aim & Objectives 1 Aims u Introduce discrete probability distributions v Binomial distribution v Poisson distribution 2 Objectives
More informationIntroduction to Probability and Statistics Slides 3 Chapter 3
Introduction to Probability and Statistics Slides 3 Chapter 3 Ammar M. Sarhan, asarhan@mathstat.dal.ca Department of Mathematics and Statistics, Dalhousie University Fall Semester 2008 Dr. Ammar M. Sarhan
More informationSlides 8: Statistical Models in Simulation
Slides 8: Statistical Models in Simulation Purpose and Overview The world the model-builder sees is probabilistic rather than deterministic: Some statistical model might well describe the variations. An
More informationChing-Han Hsu, BMES, National Tsing Hua University c 2015 by Ching-Han Hsu, Ph.D., BMIR Lab. = a + b 2. b a. x a b a = 12
Lecture 5 Continuous Random Variables BMIR Lecture Series in Probability and Statistics Ching-Han Hsu, BMES, National Tsing Hua University c 215 by Ching-Han Hsu, Ph.D., BMIR Lab 5.1 1 Uniform Distribution
More informationData Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2017 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
More informationJoint Probability Distributions, Correlations
Joint Probability Distributions, Correlations What we learned so far Events: Working with events as sets: union, intersection, etc. Some events are simple: Head vs Tails, Cancer vs Healthy Some are more
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 informationWill Landau. Feb 28, 2013
Iowa State University The F Feb 28, 2013 Iowa State University Feb 28, 2013 1 / 46 Outline The F The F Iowa State University Feb 28, 2013 2 / 46 The normal (Gaussian) distribution A random variable X is
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 informationRandom variables. DS GA 1002 Probability and Statistics for Data Science.
Random variables DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Motivation Random variables model numerical quantities
More informationAnalysis of Experimental Designs
Analysis of Experimental Designs p. 1/? Analysis of Experimental Designs Gilles Lamothe Mathematics and Statistics University of Ottawa Analysis of Experimental Designs p. 2/? Review of Probability A short
More informationQuick Tour of Basic Probability Theory and Linear Algebra
Quick Tour of and Linear Algebra Quick Tour of and Linear Algebra CS224w: Social and Information Network Analysis Fall 2011 Quick Tour of and Linear Algebra Quick Tour of and Linear Algebra Outline Definitions
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 informationThis does not cover everything on the final. Look at the posted practice problems for other topics.
Class 7: Review Problems for Final Exam 8.5 Spring 7 This does not cover everything on the final. Look at the posted practice problems for other topics. To save time in class: set up, but do not carry
More informationLecture 3 Continuous Random Variable
Lecture 3 Continuous Random Variable 1 Cumulative Distribution Function Definition Theorem 3.1 For any random variable X, 2 Continuous Random Variable Definition 3 Example Suppose we have a wheel of circumference
More informationReview of Probabilities and Basic Statistics
Alex Smola Barnabas Poczos TA: Ina Fiterau 4 th year PhD student MLD Review of Probabilities and Basic Statistics 10-701 Recitations 1/25/2013 Recitation 1: Statistics Intro 1 Overview Introduction to
More informationBinomial and Poisson Probability Distributions
Binomial and Poisson Probability Distributions Esra Akdeniz March 3, 2016 Bernoulli Random Variable Any random variable whose only possible values are 0 or 1 is called a Bernoulli random variable. What
More informationCS 361: Probability & Statistics
February 26, 2018 CS 361: Probability & Statistics Random variables The discrete uniform distribution If every value of a discrete random variable has the same probability, then its distribution is called
More informationECE 302 Division 2 Exam 2 Solutions, 11/4/2009.
NAME: ECE 32 Division 2 Exam 2 Solutions, /4/29. You will be required to show your student ID during the exam. This is a closed-book exam. A formula sheet is provided. No calculators are allowed. Total
More informationReview for Exam Spring 2018
Review for Exam 1 18.05 Spring 2018 Extra office hours Tuesday: David 3 5 in 2-355 Watch web site for more Friday, Saturday, Sunday March 9 11: no office hours March 2, 2018 2 / 23 Exam 1 Designed to be
More informationContinuous random variables
Continuous random variables Continuous r.v. s take an uncountably infinite number of possible values. Examples: Heights of people Weights of apples Diameters of bolts Life lengths of light-bulbs We cannot
More informationProbability Distributions.
Probability Distributions http://www.pelagicos.net/classes_biometry_fa18.htm Probability Measuring Discrete Outcomes Plotting probabilities for discrete outcomes: 0.6 0.5 0.4 0.3 0.2 0.1 NOTE: Area within
More informationBell-shaped curves, variance
November 7, 2017 Pop-in lunch on Wednesday Pop-in lunch tomorrow, November 8, at high noon. Please join our group at the Faculty Club for lunch. Means If X is a random variable with PDF equal to f (x),
More informationLecture 10: Normal RV. Lisa Yan July 18, 2018
Lecture 10: Normal RV Lisa Yan July 18, 2018 Announcements Midterm next Tuesday Practice midterm, solutions out on website SCPD students: fill out Google form by today Covers up to and including Friday
More informationINF FALL NATURAL LANGUAGE PROCESSING. Jan Tore Lønning
1 INF4080 2018 FALL NATURAL LANGUAGE PROCESSING Jan Tore Lønning 2 Probability distributions Lecture 5, 5 September Today 3 Recap: Bayes theorem Discrete random variable Probability distribution Discrete
More informationMaximum-Likelihood fitting
CMP 0b Lecture F. Sigworth Maximum-Likelihood fitting One of the issues I want to address in this lecture is the fitting of distributions dwell times. We want to find the best curve to draw over a histogram,
More informationCS 5014: Research Methods in Computer Science. Bernoulli Distribution. Binomial Distribution. Poisson Distribution. Clifford A. Shaffer.
Department of Computer Science Virginia Tech Blacksburg, Virginia Copyright c 2015 by Clifford A. Shaffer Computer Science Title page Computer Science Clifford A. Shaffer Fall 2015 Clifford A. Shaffer
More informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationChapter 5. Statistical Models in Simulations 5.1. Prof. Dr. Mesut Güneş Ch. 5 Statistical Models in Simulations
Chapter 5 Statistical Models in Simulations 5.1 Contents Basic Probability Theory Concepts Discrete Distributions Continuous Distributions Poisson Process Empirical Distributions Useful Statistical Models
More informationContinuous-Valued Probability Review
CS 6323 Continuous-Valued Probability Review Prof. Gregory Provan Department of Computer Science University College Cork 2 Overview Review of discrete distributions Continuous distributions 3 Discrete
More informationPart IA Probability. Definitions. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Definitions Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
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 informationContinuous Random Variables. What continuous random variables are and how to use them. I can give a definition of a continuous random variable.
Continuous Random Variables Today we are learning... What continuous random variables are and how to use them. I will know if I have been successful if... I can give a definition of a continuous random
More informationFYST17 Lecture 8 Statistics and hypothesis testing. Thanks to T. Petersen, S. Maschiocci, G. Cowan, L. Lyons
FYST17 Lecture 8 Statistics and hypothesis testing Thanks to T. Petersen, S. Maschiocci, G. Cowan, L. Lyons 1 Plan for today: Introduction to concepts The Gaussian distribution Likelihood functions Hypothesis
More informationPractical Statistics
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 -
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 informationIntroduction to Probability
Introduction to Probability Salvatore Pace September 2, 208 Introduction In a frequentist interpretation of probability, a probability measure P (A) says that if I do something N times, I should see event
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 3: The Exponential Distribution and the Poisson process Section 4.8 The Exponential Distribution 1 / 21 Exponential Distribution
More informationa table or a graph or an equation.
Topic (8) POPULATION DISTRIBUTIONS 8-1 So far: Topic (8) POPULATION DISTRIBUTIONS We ve seen some ways to summarize a set of data, including numerical summaries. We ve heard a little about how to sample
More information4th IIA-Penn State Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur
4th IIA-Penn State Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur Laws of Probability, Bayes theorem, and the Central Limit Theorem Rahul Roy Indian Statistical Institute, Delhi. Adapted
More informationEE/CpE 345. Modeling and Simulation. Fall Class 9
EE/CpE 345 Modeling and Simulation Class 9 208 Input Modeling Inputs(t) Actual System Outputs(t) Parameters? Simulated System Outputs(t) The input data is the driving force for the simulation - the behavior
More informationIntroduction to Scientific Modeling CS 365, Fall 2012 Power Laws and Scaling
Introduction to Scientific Modeling CS 365, Fall 2012 Power Laws and Scaling Stephanie Forrest Dept. of Computer Science Univ. of New Mexico Albuquerque, NM http://cs.unm.edu/~forrest forrest@cs.unm.edu
More informationDecision making and problem solving Lecture 1. Review of basic probability Monte Carlo simulation
Decision making and problem solving Lecture 1 Review of basic probability Monte Carlo simulation Why probabilities? Most decisions involve uncertainties Probability theory provides a rigorous framework
More informationStatistics, Probability Distributions & Error Propagation. James R. Graham
Statistics, Probability Distributions & Error Propagation James R. Graham Sample & Parent Populations Make measurements x x In general do not expect x = x But as you take more and more measurements a pattern
More informationECE 313 Probability with Engineering Applications Fall 2000
Exponential random variables Exponential random variables arise in studies of waiting times, service times, etc X is called an exponential random variable with parameter λ if its pdf is given by f(u) =
More information3. Review of Probability and Statistics
3. Review of Probability and Statistics ECE 830, Spring 2014 Probabilistic models will be used throughout the course to represent noise, errors, and uncertainty in signal processing problems. This lecture
More information+ + ( + ) = Linear recurrent networks. Simpler, much more amenable to analytic treatment E.g. by choosing
Linear recurrent networks Simpler, much more amenable to analytic treatment E.g. by choosing + ( + ) = Firing rates can be negative Approximates dynamics around fixed point Approximation often reasonable
More information1 Random Variable: Topics
Note: Handouts DO NOT replace the book. In most cases, they only provide a guideline on topics and an intuitive feel. 1 Random Variable: Topics Chap 2, 2.1-2.4 and Chap 3, 3.1-3.3 What is a random variable?
More informationJoint Probability Distributions, Correlations
Joint Probability Distributions, Correlations What we learned so far Events: Working with events as sets: union, intersection, etc. Some events are simple: Head vs Tails, Cancer vs Healthy Some are more
More information