Probability Distributions
|
|
- Zoe Warren
- 6 years ago
- Views:
Transcription
1 02/07/07 PHY310: Statistical Data Analysis 1 PHY310: Lecture 05 Probability Distributions Road Map The Gausssian Describing Distributions Expectation Value Variance Basic Distributions Generating Random Numbers
2 02/07/07 PHY310: Statistical Data Analysis 2 The Gaussian or Normal Distribution Also known as a Bell Curve. It's important since when you add enough continuous random variables (i.e. almost any measurement) the sum becomes a Gaussian. g x ;, 2 dx = x e 2 2 dx The central value is given by μ. Called the mean The width is given by σ The variance is σ² The standard deviation is σ
3 02/07/07 PHY310: Statistical Data Analysis 3 The Expectation Value The Expectation Value of a P.D.F. is the value of a random variable that you expect to measure Also called the Population Mean, or just the Mean The expectation value is written E[x] This is an abbreviation that means The expectation value of x Depends on the P.D.F. and isn't a function of x. If a random variable x has a P.D.F. f(x), then the expectation value is: E[ x] = dx x f x If you want to find the expectation value of q(x) (q is a function of x): E[q x ] = dx q x f x The average, <x>, and the expectation value, E[x], are different The average is an estimator of the expectation.
4 02/07/07 PHY310: Statistical Data Analysis 4 Example: Gaussian Expectation Value E[ x] = dx x g x ;, 2 E[ x] = E[ x] = dx x 2 x 2 2 e 2 2 mcgrew@boxer:macros$ maxima Maxima Using Lisp GNU Common Lisp (GCL) GCL (aka GCL) Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. This is a development version of Maxima. The function bug_report() provides bug reporting information. (%i1) integrate((x/sqrt(2*%pi*sigma^2))*exp(-(x-mu)^2/(2*sigma^2)),x,minf,inf); Is sigma positive or negative? positive; mu sigma (%o1) abs(sigma) (%i2) quit(); mcgrew@boxer:macros$
5 02/07/07 PHY310: Statistical Data Analysis 5 The Variance The Variance describes the width of a distribution Also called the Population Variance The variance of x is written as V[x] V[ x]=e [ x E [ x] 2 ] V[ x]=e[ x 2 2 x E[ x ] E[ x ] 2 ] V[ x]=e[ x 2 ] 2E[ x]e[ x ] E[ x] 2 V[ x]=e[ x 2 ] E[ x] 2 Multiply out the argument If k is a constant, then E[k] = k Simplify
6 02/07/07 PHY310: Statistical Data Analysis 6 Example: Gaussian Variance E[ x 2 ] = dx x 2 g x ;, 2 E[ x 2 ] = dx E[ x 2 ] = 2 2 x e x V [ x]=e[ x 2 ] E[ x] 2 V [ x]= V[ x]= 2 mcgrew@boxer:macros$ maxima Maxima Using Lisp GNU Common Lisp (GCL) GCL (aka GCL) Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. This is a development version of Maxima. The function bug_report() provides bug reporting information. (%i1) integrate((x^2/sqrt(2*%pi*%sigma^2))*exp(-(x-%mu)^2/(2*%sigma^2)),x,minf,inf); Is %sigma positive or negative? positive; sqrt(2) sqrt(%pi) %sigma + 2 sqrt(2) sqrt(%pi) %mu %sigma (%o1) sqrt(2) sqrt(%pi) abs(%sigma) (%i2) quit(); mcgrew@boxer:macros$
7 (How I Really Do Integrals) 02/07/07 PHY310: Statistical Data Analysis 7
8 02/07/07 PHY310: Statistical Data Analysis 8 Multi-Dimensional Correlations Multi-dimensional p.d.f.s can have internal correlations between variables Not all correlations are linear Variables with a Linear Correlation Variables with a Non-Linear Correlation
9 The Multi-Dimensional Variance: The Covariance The Covariance describes the width of a multi-dimensional distribution like f(x 1, x 2,..., x n ) Cov[ x i, x j ] = E[ x i i x j j ] = E[ x i x j ] i j The Covariance is usually written as an n x n matrix V ij = Cov[x i, x j ] The Correlation Coefficient is a dimensionless measure of the correlation between two random variables ij = Cov[ x i, x j ] V [ x i ] V [ x j ] = V ij x y 02/07/07 PHY310: Statistical Data Analysis 9
10 02/07/07 PHY310: Statistical Data Analysis 10 Binomial Distribution (Discrete) If the probability of a single positive outcome is p, what is the probability of n positive outcomes in a total of N measurements? P n ; N, p = N! n! N n! pn 1 p N n A is too hard to get in math mode, so I'm going to use ; from now on! Equivalently: If the probability of a single positive measurement is p, and a negative measurement is q=1-p, what is the probability of n positive and m=n-n negative outcomes in of a total of N measurements? P n, m ; N, p = N! n!m! pn q m The Expectation Value is E[n] = Np N! E[n]= n n=0 n! N n! pn 1 p N n =N p The Variance is V[n] = Np(1-p) = Npq The Standard Deviation is n = V[n]= N p 1 p
11 Binomial Example If there were 8 children born at the Hospital today: What do we know (assume)? It's 50/50 a given child will be a girl There were exactly 8 children born. The point is that the number of measurements is fixed. What is the expected distribution of boys and girls? P n ; N, p = N! n! N n! pn 1 p N n 8! P n ; 8,0.5 = n! 8 n! 0.5 n n What is the expected number of girls? E[n]=N p=8 0.5=4 What is the expected variation in the number of girls? V[n]=N p 1 p = =2 ; n = V[n]= 2 02/07/07 PHY310: Statistical Data Analysis 11
12 02/07/07 PHY310: Statistical Data Analysis 12 Multinomial Distribution (Discrete) When there are several possible outcomes for a single measurement; What is the probability of n positive outcomes in a total of N measurements? e.g. What is the probability of drawing a spade and a diamond from a deck of cards P n 1,...,n m ; N, p 1,..., p m = N! n 1!...n m! p n 1... n m 1 pm The Trinomial can be used for any multinomial distribution where you are only interested in pairs of variables (e.g. n, m, everything else) P n, m ; N, p,q = N! n!m! N n m! pn q m 1 p q N n m Trinomial expectation values are E[n] = Np, E[m] = Nq Trinomial variances are V[n] = Np(1-p), V[m] = Nq(1-q) Trinomial covariance is Cov[n,m] = -Npq Notice the negative correlation!
13 02/07/07 PHY310: Statistical Data Analysis 13 Poisson Distribution (Discrete) The Poisson Distribution is the limit of the Binomial for μ = Np constant as N and p 0. It describes processes like the number of radioactive decays per unit time. P n ; = n e n! The expectation value is E[n] = μ The variance is V[n] = μ The standard deviation is
14 02/07/07 PHY310: Statistical Data Analysis 14 Poisson Example How many cosmic rays enter a detector in 10 seconds if the mean rate is 20 decays/second? At 20 decays/second for 10 seconds, we expect 200 cosmic rays. This is a Poisson process so the uncertainty on the expectation is CR = 200 How can we tell this is a Poisson process? Poisson statistics apply when you are counting events, but there isn't an upper limit on the number you might measure (e.g. not binomial) The number of events could be zero The number of events could be hung The expected number might be less than one. Since we live in a quantum world, many (most) processes are fundamentally Poissonian. CHEAP TRICK: When the Poisson mean is large (>15) a Gaussian (at integer values) is a really good approximation for a Poissonian. And, it's much easier to calculate.
15 02/07/07 PHY310: Statistical Data Analysis 15 Continuous Distributions The Uniform Distribution This is what most computer pseudo-random number generators approximate 1 f x ;, ={, x 0, otherwise E[x] = (α+β)/2 V[x] = (β-α)²/12 σ = (β-α)/ 12 The Exponential Distribution Usually seen when you measure the time until a random process happens The inverse of Poisson: you measure seconds/count instead of counts/second f t ; = 1 e t / E[x] = λ V[x] = λ² σ = λ
16 02/07/07 PHY310: Statistical Data Analysis 16 Gaussian Distribution (Continuous) How do you get a Gaussian? It's the limiting distribution if you take a sum of continuous random variables (add figure showing the sum of uniform variables). If you added enough discrete variables together they become continuous If you add enough random variables together, you get a Gaussian g x ;, 2 dx = x e 2 2 dx The expectation value is E[x] = μ. The variance is V[x] = σ² The standard deviation is σ
17 Binomial Approaches Gaussian As the number of measurements gets large, the binomial distributions gets more and more like a Gaussian 02/07/07 PHY310: Statistical Data Analysis 17
18 Chi-Squared (χ²) Distribution Chi-squared is the distribution of the sum of squares of Gaussian distributed random variables. When you compare measurements to the average, the deviation is described by chi-squared. The expectation value is the number of degrees of freedom The variance is two times the d.o.f. The standard deviation is sqrt(2 d.o.f.) We will usually use the cumulative distribution for Chi-Squared Can be calculated with the ROOT TMATH::Prob(z,dof) function... 02/07/07 PHY310: Statistical Data Analysis 18
19 02/07/07 PHY310: Statistical Data Analysis 19 Deciding Which Distribution to Use You are counting things (boys born today) If you have a fixed number of trials use the binomial distribution e.g. There were eight children born in the hospital, how many were boys? If you don't have a fixed number use the Poisson distribution e.g. How many children were born in NY City? You have a continuous distribution If you are measuring the distance between discrete events (time or space), use the exponential distribution. e.g. How long do you have to wait until a C.R. muon arrives? e.g. The time between radioactive decays? Almost everything else is Gaussian Anything that depends on a sum of multiple random processes will have a Gaussian distribution.
20 Pseudo-Random Numbers MC's depend on a source of random numbers, but computers are deterministic Pseudo-random number generators are algorithms that sequences of apparently uncorrelated numbers Be very careful, the numbers are correlated, and are repeatable There are about a half dozen good generators Most computer generators return a uniform distribution between 0 and 1 Example of a simple generator: The Multiplicative Linear Congruential Generator (The MLC Generator) Start with an initial integer, n 1, called the seed Generate a sequence of new integers, n i+1 = (A n i + 1) mod M The integers will lie between 0 and M-1 MLC Generators are common and dangerous Very Fast Consecutive pairs of integers fall on a finite number of hyperplanes I'll use the term random (not pseudo-random ) unless I'm making a point 02/07/07 PHY310: Statistical Data Analysis 20
21 02/07/07 PHY310: Statistical Data Analysis 21 Generating Non-Uniform Distributions Start with a sequence of uniform random numbers and get a sequence of non-uniform random numbers The Transform Method You need to solve the equation Usually can't be done x r dx 'f x ' = r Functions where solution exists are usually tabulated in libraries More details available in Statistical Data Analysis The Accept/Reject Method Always works! But, can be inefficient
22 02/07/07 PHY310: Statistical Data Analysis 22 Basic Acceptance Works for functions over a finite interval To generate a random value from the p.d.f. f(x) over an interval [α,β] Generate a uniform random value, x, between α and β. Generate a second random value, y, between 0 and 1 If y < f(x) use x, otherwise reject x and start over.
23 02/07/07 PHY310: Statistical Data Analysis 23 General Acceptance The basic acceptance method only works over a fixed interval. A more general method uses the ratio of two functions. f(x) The P.D.F. you want to generate g(x) A P.D.F. that you have a transform generator for Must satisfy εf(x) < g(x) for all x, and ε>0 The algorithm: Generate x according to g(x) Generate a uniform y between 0 and g(x) if y < εf(x), accept x, otherwise reject x and start over
24 General Acceptance Example Generate random numbers for f x = Ae x 2 2 sin 2 2x double x, g, y, f; x = grandom->gaus(); g = exp(-x*x/2); y = grandom->uniform(0.0,g); f = exp(-x*x/2)*sin(2*x)*sin(2*x); if (y<f) h2->fill(x); Generate a Gaussian distributed number, x Generate a number, y, between 0 and g(x) Check if y is less than f(x) 02/07/07 PHY310: Statistical Data Analysis 24
Statistical 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 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 informationStatistics, Data Analysis, and Simulation SS 2013
Statistics, Data Analysis, and Simulation SS 213 8.128.73 Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, 23. April 213 What we ve learned so far Fundamental
More informationSUMMARY OF PROBABILITY CONCEPTS SO FAR (SUPPLEMENT FOR MA416)
SUMMARY OF PROBABILITY CONCEPTS SO FAR (SUPPLEMENT FOR MA416) D. ARAPURA This is a summary of the essential material covered so far. The final will be cumulative. I ve also included some review problems
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 informationRandom processes and probability distributions. Phys 420/580 Lecture 20
Random processes and probability distributions Phys 420/580 Lecture 20 Random processes Many physical processes are random in character: e.g., nuclear decay (Poisson distributed event count) P (k, τ) =
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 informationChapter 5 continued. Chapter 5 sections
Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationStatistics, Data Analysis, and Simulation SS 2015
Statistics, Data Analysis, and Simulation SS 2015 08.128.730 Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, 27. April 2015 Dr. Michael O. Distler
More informationProbability Distributions Columns (a) through (d)
Discrete Probability Distributions Columns (a) through (d) Probability Mass Distribution Description Notes Notation or Density Function --------------------(PMF or PDF)-------------------- (a) (b) (c)
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 informationLIST OF FORMULAS FOR STK1100 AND STK1110
LIST OF FORMULAS FOR STK1100 AND STK1110 (Version of 11. November 2015) 1. Probability Let A, B, A 1, A 2,..., B 1, B 2,... be events, that is, subsets of a sample space Ω. a) Axioms: A probability function
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 informationChapter 5. Chapter 5 sections
1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More information1 Review of Probability and Distributions
Random variables. A numerically valued function X of an outcome ω from a sample space Ω X : Ω R : ω X(ω) is called a random variable (r.v.), and usually determined by an experiment. We conventionally denote
More informationStat 5101 Notes: Brand Name Distributions
Stat 5101 Notes: Brand Name Distributions Charles J. Geyer February 14, 2003 1 Discrete Uniform Distribution DiscreteUniform(n). Discrete. Rationale Equally likely outcomes. The interval 1, 2,..., n of
More informationLectures on Elementary Probability. William G. Faris
Lectures on Elementary Probability William G. Faris February 22, 2002 2 Contents 1 Combinatorics 5 1.1 Factorials and binomial coefficients................. 5 1.2 Sampling with replacement.....................
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 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 informationStatistics, Data Analysis, and Simulation SS 2017
Statistics, Data Analysis, and Simulation SS 2017 08.128.730 Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, 27. April 2017 Dr. Michael O. Distler
More informationUniform random numbers generators
Uniform random numbers generators Lecturer: Dmitri A. Moltchanov E-mail: moltchan@cs.tut.fi http://www.cs.tut.fi/kurssit/tlt-2707/ OUTLINE: The need for random numbers; Basic steps in generation; Uniformly
More informationSlides 3: Random Numbers
Slides 3: Random Numbers We previously considered a few examples of simulating real processes. In order to mimic real randomness of events such as arrival times we considered the use of random numbers
More informationPractice Questions for Final
Math 39 Practice Questions for Final June. 8th 4 Name : 8. Continuous Probability Models You should know Continuous Random Variables Discrete Probability Distributions Expected Value of Discrete Random
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 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 informationBivariate distributions
Bivariate distributions 3 th October 017 lecture based on Hogg Tanis Zimmerman: Probability and Statistical Inference (9th ed.) Bivariate Distributions of the Discrete Type The Correlation Coefficient
More informationPHYSICS 2150 LABORATORY
PHYSICS 2150 LABORATORY Professor John Cumalat TAs: Adam Green John Houlton Lab Coordinator: Scott Pinegar Lecture 6 Feb. 17, 2015 ANNOUNCEMENT The problem set will be posted on the course website or you
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 informationEXAM. Exam #1. Math 3342 Summer II, July 21, 2000 ANSWERS
EXAM Exam # Math 3342 Summer II, 2 July 2, 2 ANSWERS i pts. Problem. Consider the following data: 7, 8, 9, 2,, 7, 2, 3. Find the first quartile, the median, and the third quartile. Make a box and whisker
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationLecture 2 Binomial and Poisson Probability Distributions
Binomial Probability Distribution Lecture 2 Binomial and Poisson Probability Distributions Consider a situation where there are only two possible outcomes (a Bernoulli trial) Example: flipping a coin James
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 informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01: Probability and Statistics for Engineers Spring 2013 Contents 1 Joint Probability Distributions 2 1.1 Two Discrete
More informationStat 5101 Notes: Brand Name Distributions
Stat 5101 Notes: Brand Name Distributions Charles J. Geyer September 5, 2012 Contents 1 Discrete Uniform Distribution 2 2 General Discrete Uniform Distribution 2 3 Uniform Distribution 3 4 General Uniform
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 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 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 informationWeek 4: Chap. 3 Statistics of Radioactivity
Week 4: Chap. 3 Statistics of Radioactivity Vacuum Technology General use of Statistical Distributions in Radiation Measurements -- Fluctuations in Number --- distribution function models -- Fluctuations
More informationA Few Special Distributions and Their Properties
A Few Special Distributions and Their Properties Econ 690 Purdue University Justin L. Tobias (Purdue) Distributional Catalog 1 / 20 Special Distributions and Their Associated Properties 1 Uniform Distribution
More informationSources of randomness
Random Number Generator Chapter 7 In simulations, we generate random values for variables with a specified distribution Ex., model service times using the exponential distribution Generation of random
More informationWeek 1 Quantitative Analysis of Financial Markets Distributions A
Week 1 Quantitative Analysis of Financial Markets Distributions A Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October
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 informationMonte Carlo Techniques
Physics 75.502 Part III: Monte Carlo Methods 40 Monte Carlo Techniques Monte Carlo refers to any procedure that makes use of random numbers. Monte Carlo methods are used in: Simulation of natural phenomena
More informationError propagation. Alexander Khanov. October 4, PHYS6260: Experimental Methods is HEP Oklahoma State University
Error propagation Alexander Khanov PHYS660: Experimental Methods is HEP Oklahoma State University October 4, 017 Why error propagation? In many cases we measure one thing and want to know something else
More informationStatistical Tests: Discriminants
PHY310: Lecture 13 Statistical Tests: Discriminants Road Map Introduction to Statistical Tests (Take 2) Use in a Real Experiment Likelihood Ratio 1 Some Terminology for Statistical Tests Hypothesis This
More informationTopic 2: Probability & Distributions. Road Map Probability & Distributions. ECO220Y5Y: Quantitative Methods in Economics. Dr.
Topic 2: Probability & Distributions ECO220Y5Y: Quantitative Methods in Economics Dr. Nick Zammit University of Toronto Department of Economics Room KN3272 n.zammit utoronto.ca November 21, 2017 Dr. Nick
More information15-388/688 - Practical Data Science: Basic probability. J. Zico Kolter Carnegie Mellon University Spring 2018
15-388/688 - Practical Data Science: Basic probability J. Zico Kolter Carnegie Mellon University Spring 2018 1 Announcements Logistics of next few lectures Final project released, proposals/groups due
More informationCommon ontinuous random variables
Common ontinuous random variables CE 311S Earlier, we saw a number of distribution families Binomial Negative binomial Hypergeometric Poisson These were useful because they represented common situations:
More informationRandom Number Generation. CS1538: Introduction to simulations
Random Number Generation CS1538: Introduction to simulations Random Numbers Stochastic simulations require random data True random data cannot come from an algorithm We must obtain it from some process
More informationIntroduction to Statistics and Error Analysis
Introduction to Statistics and Error Analysis Physics116C, 4/3/06 D. Pellett References: Data Reduction and Error Analysis for the Physical Sciences by Bevington and Robinson Particle Data Group notes
More informationContinuous Random Variables and Continuous Distributions
Continuous Random Variables and Continuous Distributions Continuous Random Variables and Continuous Distributions Expectation & Variance of Continuous Random Variables ( 5.2) The Uniform Random Variable
More informationSpecial Discrete RV s. Then X = the number of successes is a binomial RV. X ~ Bin(n,p).
Sect 3.4: Binomial RV Special Discrete RV s 1. Assumptions and definition i. Experiment consists of n repeated trials ii. iii. iv. There are only two possible outcomes on each trial: success (S) or failure
More informationReview of Probability Theory
Review of Probability Theory Arian Maleki and Tom Do Stanford University Probability theory is the study of uncertainty Through this class, we will be relying on concepts from probability theory for deriving
More informationProbability Distributions - Lecture 5
Probability Distributions - Lecture 5 1 Introduction There are a number of mathematical models of probability density functions that represent the behavior of physical systems. In this lecture we explore
More informationProbability and Statistics
Probability and Statistics 1 Contents some stochastic processes Stationary Stochastic Processes 2 4. Some Stochastic Processes 4.1 Bernoulli process 4.2 Binomial process 4.3 Sine wave process 4.4 Random-telegraph
More informationProbabilities and distributions
Appendix B Probabilities and distributions B.1 Expectation value and variance Definition B.1. Suppose a (not necessarily quantum) experiment to measure a quantity Q can yield any one of N possible outcomes
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 informationHYPERGEOMETRIC and NEGATIVE HYPERGEOMETIC DISTRIBUTIONS
HYPERGEOMETRIC and NEGATIVE HYPERGEOMETIC DISTRIBUTIONS A The Hypergeometric Situation: Sampling without Replacement In the section on Bernoulli trials [top of page 3 of those notes], it was indicated
More informationthe amount of the data corresponding to the subinterval the width of the subinterval e x2 to the left by 5 units results in another PDF g(x) = 1 π
Math 10A with Professor Stankova Worksheet, Discussion #42; Friday, 12/8/2017 GSI name: Roy Zhao Problems 1. For each of the following distributions, derive/find all of the following: PMF/PDF, CDF, median,
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 informationSTAT2201. Analysis of Engineering & Scientific Data. Unit 3
STAT2201 Analysis of Engineering & Scientific Data Unit 3 Slava Vaisman The University of Queensland School of Mathematics and Physics What we learned in Unit 2 (1) We defined a sample space of a random
More informationNorthwestern University Department of Electrical Engineering and Computer Science
Northwestern University Department of Electrical Engineering and Computer Science EECS 454: Modeling and Analysis of Communication Networks Spring 2008 Probability Review As discussed in Lecture 1, probability
More informationFINAL EXAM: Monday 8-10am
ECE 30: Probabilistic Methods in Electrical and Computer Engineering Fall 016 Instructor: Prof. A. R. Reibman FINAL EXAM: Monday 8-10am Fall 016, TTh 3-4:15pm (December 1, 016) This is a closed book exam.
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 informationDr. Junchao Xia Center of Biophysics and Computational Biology. Fall /13/2016 1/33
BIO5312 Biostatistics Lecture 03: Discrete and Continuous Probability Distributions Dr. Junchao Xia Center of Biophysics and Computational Biology Fall 2016 9/13/2016 1/33 Introduction In this lecture,
More information2008 Winton. Review of Statistical Terminology
1 Review of Statistical Terminology 2 Formal Terminology An experiment is a process whose outcome is not known with certainty The experiment s sample space S is the set of all possible outcomes. A random
More informationSTAT Chapter 5 Continuous Distributions
STAT 270 - Chapter 5 Continuous Distributions June 27, 2012 Shirin Golchi () STAT270 June 27, 2012 1 / 59 Continuous rv s Definition: X is a continuous rv if it takes values in an interval, i.e., range
More informationMath 3215 Intro. Probability & Statistics Summer 14. Homework 5: Due 7/3/14
Math 325 Intro. Probability & Statistics Summer Homework 5: Due 7/3/. Let X and Y be continuous random variables with joint/marginal p.d.f. s f(x, y) 2, x y, f (x) 2( x), x, f 2 (y) 2y, y. Find the conditional
More informationProbability and Distributions
Probability and Distributions What is a statistical model? A statistical model is a set of assumptions by which the hypothetical population distribution of data is inferred. It is typically postulated
More informationLecture 8 Hypothesis Testing
Lecture 8 Hypothesis Testing Taylor Ch. 6 and 10.6 Introduction l The goal of hypothesis testing is to set up a procedure(s) to allow us to decide if a mathematical model ("theory") is acceptable in light
More information( x) ( ) F ( ) ( ) ( ) Prob( ) ( ) ( ) X x F x f s ds
Applied Numerical Analysis Pseudo Random Number Generator Lecturer: Emad Fatemizadeh What is random number: A sequence in which each term is unpredictable 29, 95, 11, 60, 22 Application: Monte Carlo Simulations
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 information04. Random Variables: Concepts
University of Rhode Island DigitalCommons@URI Nonequilibrium Statistical Physics Physics Course Materials 215 4. Random Variables: Concepts Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative
More informationPhysics 509: Error Propagation, and the Meaning of Error Bars. Scott Oser Lecture #10
Physics 509: Error Propagation, and the Meaning of Error Bars Scott Oser Lecture #10 1 What is an error bar? Someone hands you a plot like this. What do the error bars indicate? Answer: you can never be
More informationB.N.Bandodkar College of Science, Thane. Random-Number Generation. Mrs M.J.Gholba
B.N.Bandodkar College of Science, Thane Random-Number Generation Mrs M.J.Gholba Properties of Random Numbers A sequence of random numbers, R, R,., must have two important statistical properties, uniformity
More informationPart 3: Parametric Models
Part 3: Parametric Models Matthew Sperrin and Juhyun Park April 3, 2009 1 Introduction Is the coin fair or not? In part one of the course we introduced the idea of separating sampling variation from a
More informationSection 8.1. Vector Notation
Section 8.1 Vector Notation Definition 8.1 Random Vector A random vector is a column vector X = [ X 1 ]. X n Each Xi is a random variable. Definition 8.2 Vector Sample Value A sample value of a random
More informationChapter 5. Means and Variances
1 Chapter 5 Means and Variances Our discussion of probability has taken us from a simple classical view of counting successes relative to total outcomes and has brought us to the idea of a probability
More informationProbability. Machine Learning and Pattern Recognition. Chris Williams. School of Informatics, University of Edinburgh. August 2014
Probability Machine Learning and Pattern Recognition Chris Williams School of Informatics, University of Edinburgh August 2014 (All of the slides in this course have been adapted from previous versions
More informationMath 416 Lecture 2 DEFINITION. Here are the multivariate versions: X, Y, Z iff P(X = x, Y = y, Z =z) = p(x, y, z) of X, Y, Z iff for all sets A, B, C,
Math 416 Lecture 2 DEFINITION. Here are the multivariate versions: PMF case: p(x, y, z) is the joint Probability Mass Function of X, Y, Z iff P(X = x, Y = y, Z =z) = p(x, y, z) PDF case: f(x, y, z) is
More informationACM 116: Lecture 2. Agenda. Independence. Bayes rule. Discrete random variables Bernoulli distribution Binomial distribution
1 ACM 116: Lecture 2 Agenda Independence Bayes rule Discrete random variables Bernoulli distribution Binomial distribution Continuous Random variables The Normal distribution Expected value of a random
More informationStatistics 224 Solution key to EXAM 2 FALL 2007 Friday 11/2/07 Professor Michael Iltis (Lecture 2)
NOTE : For the purpose of review, I have added some additional parts not found on the original exam. These parts are indicated with a ** beside them Statistics 224 Solution key to EXAM 2 FALL 2007 Friday
More informationData Analysis and Monte Carlo Methods
Lecturer: Allen Caldwell, Max Planck Institute for Physics & TUM Recitation Instructor: Oleksander (Alex) Volynets, MPP & TUM General Information: - Lectures will be held in English, Mondays 16-18:00 -
More information2905 Queueing Theory and Simulation PART IV: SIMULATION
2905 Queueing Theory and Simulation PART IV: SIMULATION 22 Random Numbers A fundamental step in a simulation study is the generation of random numbers, where a random number represents the value of a random
More informationChapter 5 Joint Probability Distributions
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 5 Joint Probability Distributions 5 Joint Probability Distributions CHAPTER OUTLINE 5-1 Two
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 informationALL TEXTS BELONG TO OWNERS. Candidate code: glt090 TAKEN FROM
How are Generating Functions used in finding the closed form of sequences involving recurrence relations and in the analysis of probability distributions? Mathematics Extended Essay Word count: 3865 Abstract
More informationReview of Statistical Terminology
Review of Statistical Terminology An experiment is a process whose outcome is not known with certainty. The experiment s sample space S is the set of all possible outcomes. A random variable is a function
More informationISyE 6644 Fall 2014 Test #2 Solutions (revised 11/7/16)
1 NAME ISyE 6644 Fall 2014 Test #2 Solutions (revised 11/7/16) This test is 85 minutes. You are allowed two cheat sheets. Good luck! 1. Some short answer questions to get things going. (a) Consider the
More information2 Random Variable Generation
2 Random Variable Generation Most Monte Carlo computations require, as a starting point, a sequence of i.i.d. random variables with given marginal distribution. We describe here some of the basic methods
More information1 Exercises for lecture 1
1 Exercises for lecture 1 Exercise 1 a) Show that if F is symmetric with respect to µ, and E( X )
More informationStat 100a, Introduction to Probability.
Stat 100a, Introduction to Probability. Outline for the day: 1. Geometric random variables. 2. Negative binomial random variables. 3. Moment generating functions. 4. Poisson random variables. 5. Continuous
More informationGuidelines for Solving Probability Problems
Guidelines for Solving Probability Problems CS 1538: Introduction to Simulation 1 Steps for Problem Solving Suggested steps for approaching a problem: 1. Identify the distribution What distribution does
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 informationS.A. Teukolsky, Computers in Physics, Vol. 6, No. 5,
Physics 75.502 Part III: Monte Carlo Methods 139 Part III: Monte Carlo Methods Topics: Introduction Random Number generators Special distributions General Techniques Multidimensional simulation References:
More information18.440: Lecture 28 Lectures Review
18.440: Lecture 28 Lectures 18-27 Review Scott Sheffield MIT Outline Outline It s the coins, stupid Much of what we have done in this course can be motivated by the i.i.d. sequence X i where each X i is
More information1 Presessional Probability
1 Presessional Probability Probability theory is essential for the development of mathematical models in finance, because of the randomness nature of price fluctuations in the markets. This presessional
More informationSTAT/MA 416 Answers Homework 6 November 15, 2007 Solutions by Mark Daniel Ward PROBLEMS
STAT/MA 4 Answers Homework November 5, 27 Solutions by Mark Daniel Ward PROBLEMS Chapter Problems 2a. The mass p, corresponds to neither of the first two balls being white, so p, 8 7 4/39. The mass p,
More information(y 1, y 2 ) = 12 y3 1e y 1 y 2 /2, y 1 > 0, y 2 > 0 0, otherwise.
54 We are given the marginal pdfs of Y and Y You should note that Y gamma(4, Y exponential( E(Y = 4, V (Y = 4, E(Y =, and V (Y = 4 (a With U = Y Y, we have E(U = E(Y Y = E(Y E(Y = 4 = (b Because Y and
More informationPROBABILITY DISTRIBUTION
PROBABILITY DISTRIBUTION DEFINITION: If S is a sample space with a probability measure and x is a real valued function defined over the elements of S, then x is called a random variable. Types of Random
More information