ΔP(x) Δx. f "Discrete Variable x" (x) dp(x) dx. (x) f "Continuous Variable x" Module 3 Statistics. I. Basic Statistics. A. Statistics and Physics
|
|
- Bernice Tate
- 5 years ago
- Views:
Transcription
1 Module 3 Statistics I. Basic Statistics A. Statistics and Physics 1. Why Statistics Up to this point, your courses in physics and engineering have considered systems from a macroscopic point of view. For instance, we have described baseballs, blocks, airplanes, etc. as rigid bodies. In our discussion of the kinetic theory of gases in University Physics, we demonstrated that the macroscopic properties of a gas including temperature, pressure, and volume are derived from the microscopic motion of the molecules that compose the gas. Since all matter is composed of atoms, we should epect that this approach is universal and would provide a greater understanding than just studying macroscopic properties (the average of microscopic properties)! This particular field of physics is called Statistical Mechanics since it combines mechanics (classical or quantum) and statistics. In statistical mechanics and quantum mechanics, we talk about calculating the probability that a particle has some physical attribute. The attribute might be energy, linear momentum, position, etc. In the discussion below, we will consider the attribute to be position, but any attribute could be inserted. The reason for choosing position is to simplify the tet and because position is easier to visualize. Humans are equipped with position detectors called eyes.. Probability Distribution Function - f () The probability distribution function is a function that determines the probability that an object is located between and + d. It is defined as the change in probability function over the change in and is most useful in dealing with problems continuous physical quantities. () ΔP() Δ f "Discrete Variable " () dp() d f "Continuous Variable "
2 3. Calculating Probabilities Using the Probability Distribution Function Given the probability distribution function, we can calculate the probability that a particle is located in the region between i and f by f P(1 ) f ( )Δ ΔP( ) "Discrete Variable " k i k P(1 ) f i f k i k f ()d "Continuous Variable " 4. Normalization If we consider all possibilities, we will have a 100% probability of finding the particle. Therefore, the sum/integral of the probability distribution function over all possible values of must equal one!! P ( ) P( ) 1 "Discrete Variable " P( ) k k f () d "Continuous Variable " 1 If the function f() doesn't have this property then it is said to be unnormalized and can NOT be a probability distribution function! In order to create a probability distribution function, we divided the function by the result of the previous equation. This process is called normalization! 5. Calculating the Average (Epectation) Value - or The average location of the object can be calculated using ΔP() "Discrete" f ()d "Continuous"
3 In physics, the average value of a physical quantity is usually called its "epectation value." This is because it is the value that is epected on average from multiple measurements of the quantity even though no single measurement may give this value. Consider a class in which half the students score 100 and the other half score 0 on a test. The class average (epectation value) is 50% even though no individual student had this result! Mathematicians call this type of average value the "mean." 6. Standard Deviation () and Variance ( ) Although the epectation value is important, it doesn't completely specify how a system is behaving. For eample, the average voltage out of a wall plug is zero (no DC voltage). However, the standard deviation is 110 volts!! Obviously, the standard deviation is important since this is what makes your TV, radio, and other appliances work! The variance and standard deviation tells us how much the location of the particle will vary (i.e. the spread) on average as we make several measurements. Consider the following graph. Both the red system and blue system have the same average. Average Measurement From the definition of the average, we know that if we sum the distance between each red data point and the average line it will add up to zero. Obviously, the same is true for the blue data points! This is epressed mathematically by the equation 0 "Definition of an Average" We can obtain a measure of the spread of the data by summing the square of the distance between each data point and the average line. Since taking additional data points will increase the sum even though the data points might be closer to the average (less spread), we must divide by the number of data points. Thus, we are finding the average of the square of the distance between the data points and the average line. This is called the variance!
4 σ ( ) Since we have to calculate the epectation value to compute the variance, we find the following formula more useful for computations: σ The equation says that the variance is the average of the squares of a quantity minus the square of the average of the quantity. From dimensional analysis, you should realize that the variance doesn't have the same units as. Thus, we need to take the square root of the variance in order to obtain a quantity that can represent the spread of. This quantity is called the standard deviation! σ σ In physics, we usually refer to the standard deviation as the uncertainty as it represents the uncertainty in a measurement. We will find that uncertainty has a very important place in quantum mechanics. II. Binomial Distribution A. We often find events where there are only two possible results. For eample, a nucleus has either decayed or has not decayed; a particle is detected or not detected; a flipped coin either comes up heads or tails. If a system contains a number of identical events each of which are independent, have only two outcomes, and the probability for each outcome is constant, then the statistics of the system is described by the binomial distribution. B. Formula If the probability that a single event can occur is given by p than the probability f that events will occur out of n independent trials is given by f() = n! p (1 p) n! (n )!
5 The mean number of events that occur is given by = np The standard deviation is given by σ = np(1 p) Although many physical systems are described by the binomial distribution, it is often difficult if not impossible to use the distribution for actual calculations especially when the number of trials is large. For instance, a gram of Uranium contains more than 10 5 nuclei (n = 10 5 ) that can decay so the number of calculations to be performed would be enormous. For this reason, we usually approimate the binomial distribution with a continuous function. III. Gaussian Distribution A. The Gaussian distribution is one of the more important probability distributions. It finds application in a wide range of fields. It is sometimes called the normal or standard distribution or the Bell curve. It is also sometimes referred to as the "drunken walk." The Gaussian distribution is a special case of the discrete binomial distribution for large numbers of trials, n, where the probability of success, p, is not too small (see Appendi D of Rohlf). A common eample of this condition occurs when the physical quantity that is being measured depends on the "sum" of a set of large random numbers. Start The displacement of a drunk undergoing a random walk is the sum of several random steps. The drunk should have a much greater probability for small
6 Probability displacements where his/her individual steps cancel each other than for large displacements where more of the steps must be in the same direction! B. Formula The probability that events occur for a system with an average of a and a standard deviation of "σ" is given by: f () 1 πσ e ( a) /(σ ) C. Graph The graph for a Gaussian with a mean of 5 and a standard deviation of 6 is shown below. Gaussian Distribution The solid green line shows that: 1) the "most probable" value (peak) is at = 5 ) the median (50% level) is at = 5 3) the mean is at = 5 The dashed red lines (inner pair) show the region where - < < +. The probability that a particle is located in this region is (68.3%).
7 The dashed purple lines (outer pair) show the region where - < < +. The probability that a particle is located in this region is 0.95 (95%). IV. Poisson Distribution A. From the graph in the previous section, we see that the Gaussian distribution is symmetric about the mean. This symmetrical shape cannot describe a system in which there is a very small probability of an event occurring as in the case where one is trying to detect a rare particle such as the Higgs boson. In these cases, the peak of the distribution has to be near zero and be asymmetrical. A system described by a binomial distribution with a large number of trials, n, where the probability, p, for an individual event is so small that np is small is instead approimated by the Poisson distribution. This has great application in nuclear counting statistics as well as engineering applications like measuring small concentrations of a pollutant. B. For a system of n trials where the probability of a single event is p, the probability for having events is given by: f() = (np) e np! For the Poisson distribution, the mean and standard deviations are given by = np σ = np In sampling problems, where the probability of an event may not be known apriori, the probability for later samples is often assumed to be the same as from a previous sample. For instance, if a batch of 50,000 parts had three defective parts then the average for the net batch of 50,000 parts is assumed to be three and the probability is for calculations as to the number of defects epected in later batches.
8 C. Graph The graph of a system described by a Poisson distribution with p = and n = 10000, shows the asymmetrical nature of the distribution Poisson Distribution One can see that for even 10,000 trials there is almost no probability for more than four events to occur. The average of the distribution is one.
Chapter 2: The Random Variable
Chapter : The Random Variable The outcome of a random eperiment need not be a number, for eample tossing a coin or selecting a color ball from a bo. However we are usually interested not in the outcome
More informationBasic Concepts and Tools in Statistical Physics
Chapter 1 Basic Concepts and Tools in Statistical Physics 1.1 Introduction Statistical mechanics provides general methods to study properties of systems composed of a large number of particles. It establishes
More informationII. Probability. II.A General Definitions
II. Probability II.A General Definitions The laws of thermodynamics are based on observations of macroscopic bodies, and encapsulate their thermal properties. On the other hand, matter is composed of atoms
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 informationDiscrete Probability Distribution
Shapes of binomial distributions Discrete Probability Distribution Week 11 For this activity you will use a web applet. Go to http://socr.stat.ucla.edu/htmls/socr_eperiments.html and choose Binomial coin
More informationChapter 2. Random Variable. Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance.
Chapter 2 Random Variable CLO2 Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance. 1 1. Introduction In Chapter 1, we introduced the concept
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 informationX = X X n, + X 2
CS 70 Discrete Mathematics for CS Fall 2003 Wagner Lecture 22 Variance Question: At each time step, I flip a fair coin. If it comes up Heads, I walk one step to the right; if it comes up Tails, I walk
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 informationChapter 3 Single Random Variables and Probability Distributions (Part 1)
Chapter 3 Single Random Variables and Probability Distributions (Part 1) Contents What is a Random Variable? Probability Distribution Functions Cumulative Distribution Function Probability Density Function
More informationthat relative errors are dimensionless. When reporting relative errors it is usual to multiply the fractional error by 100 and report it as a percenta
Error Analysis and Significant Figures Errors using inadequate data are much less than those using no data at all. C. Babbage No measurement of a physical quantity can be entirely accurate. It is important
More informationScientific Measurement
Scientific Measurement SPA-4103 Dr Alston J Misquitta Lecture 5 - The Binomial Probability Distribution Binomial Distribution Probability of k successes in n trials. Gaussian/Normal Distribution Poisson
More informationPhysics 115A: Statistical Physics
Physics 115A: Statistical Physics Prof. Clare Yu email: cyu@uci.edu phone: 949-824-6216 Office: 210E RH Fall 2013 LECTURE 1 Introduction So far your physics courses have concentrated on what happens to
More informationCHAPTER 5. Department of Medical Physics, University of the Free State, Bloemfontein, South Africa
CHAPTE 5 STATISTICS FO ADIATIO MEASUEMET M.G. LÖTTE Department of Medical Physics, University of the Free State, Bloemfontein, South Africa 5.1. SOUCES OF EO I UCLEA MEDICIE MEASUEMET Measurement errors
More informationRandom Variable. Discrete Random Variable. Continuous Random Variable. Discrete Random Variable. Discrete Probability Distribution
Random Variable Theoretical Probability Distribution Random Variable Discrete Probability Distributions A variable that assumes a numerical description for the outcome of a random eperiment (by chance).
More informationMATH 3670 First Midterm February 17, No books or notes. No cellphone or wireless devices. Write clearly and show your work for every answer.
No books or notes. No cellphone or wireless devices. Write clearly and show your work for every answer. Name: Question: 1 2 3 4 Total Points: 30 20 20 40 110 Score: 1. The following numbers x i, i = 1,...,
More informationCOUNTING ERRORS AND STATISTICS RCT STUDY GUIDE Identify the five general types of radiation measurement errors.
LEARNING OBJECTIVES: 2.03.01 Identify the five general types of radiation measurement errors. 2.03.02 Describe the effect of each source of error on radiation measurements. 2.03.03 State the two purposes
More informationUnit 4 Probability. Dr Mahmoud Alhussami
Unit 4 Probability Dr Mahmoud Alhussami Probability Probability theory developed from the study of games of chance like dice and cards. A process like flipping a coin, rolling a die or drawing a card from
More informationChapter 8. Some Approximations to Probability Distributions: Limit Theorems
Chapter 8. Some Approximations to Probability Distributions: Limit Theorems Sections 8.2 -- 8.3: Convergence in Probability and in Distribution Jiaping Wang Department of Mathematical Science 04/22/2013,
More informationCentral Limit Theorem and the Law of Large Numbers Class 6, Jeremy Orloff and Jonathan Bloom
Central Limit Theorem and the Law of Large Numbers Class 6, 8.5 Jeremy Orloff and Jonathan Bloom Learning Goals. Understand the statement of the law of large numbers. 2. Understand the statement of the
More information213 Midterm coming up
213 Midterm coming up Monday April 8 @ 7 pm (conflict exam @ 5:15pm) Covers: Lectures 1-12 (not including thermal radiation) HW 1-4 Discussion 1-4 Labs 1-2 Review Session Sunday April 7, 3-5 PM, 141 Loomis
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 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 informationStatistics Random Variables
1 Statistics Statistics are used in a variety of ways in neuroscience. Perhaps the most familiar example is trying to decide whether some experimental results are reliable, using tests such as the t-test.
More informationL06. Chapter 6: Continuous Probability Distributions
L06 Chapter 6: Continuous Probability Distributions Probability Chapter 6 Continuous Probability Distributions Recall Discrete Probability Distributions Could only take on particular values Continuous
More informationExamples of common quantum mechanical procedures and calculations carried out in Mathcad.
Eample_QM_calculations.mcd page Eamples of common quantum mechanical procedures and calculations carried out in Mathcad. Erica Harvey Fairmont State College Department of Chemistry Fairmont State University
More informationLecture 6 Examples and Problems
Lecture 6 Examples and Problems Heat capacity of solids & liquids Thermal diffusion Thermal conductivity Irreversibility Hot Cold Random Walk and Particle Diffusion Counting and Probability Microstates
More informationLecture 6. Probability events. Definition 1. The sample space, S, of a. probability experiment is the collection of all
Lecture 6 1 Lecture 6 Probability events Definition 1. The sample space, S, of a probability experiment is the collection of all possible outcomes of an experiment. One such outcome is called a simple
More informationRadioactivity. PC1144 Physics IV. 1 Objectives. 2 Equipment List. 3 Theory
PC1144 Physics IV Radioactivity 1 Objectives Investigate the analogy between the decay of dice nuclei and radioactive nuclei. Determine experimental and theoretical values of the decay constant λ and the
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 informationExpected Value - Revisited
Expected Value - Revisited An experiment is a Bernoulli Trial if: there are two outcomes (success and failure), the probability of success, p, is always the same, the trials are independent. Expected Value
More informationDiscrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 20
CS 70 Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 20 Today we shall discuss a measure of how close a random variable tends to be to its expectation. But first we need to see how to compute
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 informationWhere would you rather live? (And why?)
Where would you rather live? (And why?) CityA CityB Where would you rather live? (And why?) CityA CityB City A is San Diego, CA, and City B is Evansville, IN Measures of Dispersion Suppose you need a new
More informationnoise = function whose amplitude is is derived from a random or a stochastic process (i.e., not deterministic)
SIMG-716 Linear Imaging Mathematics I, Handout 05 1 1-D STOCHASTIC FUCTIOS OISE noise = function whose amplitude is is derived from a random or a stochastic process (i.e., not deterministic) Deterministic:
More informationn px p x (1 p) n x. p x n(n 1)... (n x + 1) x!
Lectures 3-4 jacques@ucsd.edu 7. Classical discrete distributions D. The Poisson Distribution. If a coin with heads probability p is flipped independently n times, then the number of heads is Bin(n, p)
More informationCSE 103 Homework 8: Solutions November 30, var(x) = np(1 p) = P r( X ) 0.95 P r( X ) 0.
() () a. X is a binomial distribution with n = 000, p = /6 b. The expected value, variance, and standard deviation of X is: E(X) = np = 000 = 000 6 var(x) = np( p) = 000 5 6 666 stdev(x) = np( p) = 000
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 informationPage Max. Possible Points Total 100
Math 3215 Exam 2 Summer 2014 Instructor: Sal Barone Name: GT username: 1. No books or notes are allowed. 2. You may use ONLY NON-GRAPHING and NON-PROGRAMABLE scientific calculators. All other electronic
More informationChapter 8: An Introduction to Probability and Statistics
Course S3, 200 07 Chapter 8: An Introduction to Probability and Statistics This material is covered in the book: Erwin Kreyszig, Advanced Engineering Mathematics (9th edition) Chapter 24 (not including
More information3/30/2009. Probability Distributions. Binomial distribution. TI-83 Binomial Probability
Random variable The outcome of each procedure is determined by chance. Probability Distributions Normal Probability Distribution N Chapter 6 Discrete Random variables takes on a countable number of values
More informationLecture 5 (Sep. 20, 2017)
Lecture 5 8.321 Quantum Theory I, Fall 2017 22 Lecture 5 (Sep. 20, 2017) 5.1 The Position Operator In the last class, we talked about operators with a continuous spectrum. A prime eample is the position
More informationExplaining Periodic Trends. Saturday, January 20, 18
Explaining Periodic Trends Many observable trends in the chemical and physical properties of elements are observable in the periodic table. Let s review a trend that you should already be familiar with,
More informationIntroduction to Thermodynamic States Gases
Chapter 1 Introduction to Thermodynamic States Gases We begin our study in thermodynamics with a survey of the properties of gases. Gases are one of the first things students study in general chemistry.
More informationKinetic Theory 1 / Probabilities
Kinetic Theory 1 / Probabilities 1. Motivations: statistical mechanics and fluctuations 2. Probabilities 3. Central limit theorem 1 The need for statistical mechanics 2 How to describe large systems In
More informationRandom Variables Example:
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 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 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 informationLecture 3. The Population Variance. The population variance, denoted σ 2, is the sum. of the squared deviations about the population
Lecture 5 1 Lecture 3 The Population Variance The population variance, denoted σ 2, is the sum of the squared deviations about the population mean divided by the number of observations in the population,
More informationChemistry 121: Atomic and Molecular Chemistry Topic 3: Atomic Structure and Periodicity
Text Chapter 2, 8 & 9 3.1 Nature of light, elementary spectroscopy. 3.2 The quantum theory and the Bohr atom. 3.3 Quantum mechanics; the orbital concept. 3.4 Electron configurations of atoms 3.5 The periodic
More informationDiscrete probability distributions
Discrete probability s BSAD 30 Dave Novak Fall 08 Source: Anderson et al., 05 Quantitative Methods for Business th edition some slides are directly from J. Loucks 03 Cengage Learning Covered so far Chapter
More informationLecture 8 Sampling Theory
Lecture 8 Sampling Theory Thais Paiva STA 111 - Summer 2013 Term II July 11, 2013 1 / 25 Thais Paiva STA 111 - Summer 2013 Term II Lecture 8, 07/11/2013 Lecture Plan 1 Sampling Distributions 2 Law of Large
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 informationUniversity of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra
University of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra Table of Contents Chapter The Algebra of Polynomials Chapter Factoring 7 Chapter 3 Fractions Chapter 4 Eponents and Radicals
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 informationIntroduction to Probability Theory for Graduate Economics Fall 2008
Introduction to Probability Theory for Graduate Economics Fall 008 Yiğit Sağlam October 10, 008 CHAPTER - RANDOM VARIABLES AND EXPECTATION 1 1 Random Variables A random variable (RV) is a real-valued function
More information3 Multiple Discrete Random Variables
3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f
More informationUNIT 4 MATHEMATICAL METHODS SAMPLE REFERENCE MATERIALS
UNIT 4 MATHEMATICAL METHODS SAMPLE REFERENCE MATERIALS EXTRACTS FROM THE ESSENTIALS EXAM REVISION LECTURES NOTES THAT ARE ISSUED TO STUDENTS Students attending our mathematics Essentials Year & Eam Revision
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 informationSystem Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models
System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models Fatih Cavdur fatihcavdur@uludag.edu.tr March 29, 2014 Introduction Introduction The world of the model-builder
More informationCard #1/28. Card #2/28. Science Revision P2. Science Revision P2. Science Revision P2. Card #4/28. Topic: F = ma. Topic: Resultant Forces
Card #1/28 Card #2/28 Topic: Resultant Forces Topic: F = ma Topic: Distance-TIme Graphs Card #3/28 Card #4/28 Topic: Velocity-Time Graphs Card #2/28 Card #1/28 Card #4/28 Card #3/28 Card #5/28 Card #6/28
More informationElectron Probability Accelerated Chemistry I
Introduction: Electron Probability Accelerated Chemistry I Could you determine the exact position and momentum of a baseball as it soared through the air? Of course you could by taking a timed series of
More informationRandom Variables and Probability Distributions Chapter 4
Random Variables and Probability Distributions Chapter 4 4.2 a. The closing price of a particular stock on the New York Stock Echange is discrete. It can take on only a countable number of values. b. The
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 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 information1 Some Statistical Basics.
Q Some Statistical Basics. Statistics treats random errors. (There are also systematic errors e.g., if your watch is 5 minutes fast, you will always get the wrong time, but it won t be random.) The two
More informationISP209 Spring Exam #3. Name: Student #:
ISP209 Spring 2014 Exam #3 Name: Student #: Please write down your name and student # on both the exam and the scoring sheet. After you are finished with the exam, please place the scoring sheet inside
More informationIntroduction. Chapter The Purpose of Statistical Mechanics
Chapter 1 Introduction 1.1 The Purpose of Statistical Mechanics Statistical Mechanics is the mechanics developed to treat a collection of a large number of atoms or particles. Such a collection is, for
More informationCondensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras
Condensed Matter Physics Prof. G. Rangarajan Department of Physics Indian Institute of Technology, Madras Lecture - 10 The Free Electron Theory of Metals - Electrical Conductivity (Refer Slide Time: 00:20)
More informationAnswers to All Exercises
CAPER 10 CAPER 10 CAPER10 CAPER REFRESING YOUR SKILLS FOR CAPER 10 1a. 5 1 0.5 10 1b. 6 3 0.6 10 5 1c. 0. 10 5 a. 10 36 5 1 0. 7 b. 7 is most likely; probability of 7 is 6 36 1 6 0.1 6. c. 1 1 0.5 36 3a.
More informationLecture 6. Statistical Processes. Irreversibility. Counting and Probability. Microstates and Macrostates. The Meaning of Equilibrium Ω(m) 9 spins
Lecture 6 Statistical Processes Irreversibility Counting and Probability Microstates and Macrostates The Meaning of Equilibrium Ω(m) 9 spins -9-7 -5-3 -1 1 3 5 7 m 9 Lecture 6, p. 1 Irreversibility Have
More informationError analysis in biology
Error analysis in biology Marek Gierliński Division of Computational Biology Hand-outs available at http://is.gd/statlec Errors, like straws, upon the surface flow; He who would search for pearls must
More informationELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Random Variable Discrete Random
More informationChapter 4. Probability-The Study of Randomness
Chapter 4. Probability-The Study of Randomness 4.1.Randomness Random: A phenomenon- individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.
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 informationApplied Statistics I
Applied Statistics I (IMT224β/AMT224β) Department of Mathematics University of Ruhuna A.W.L. Pubudu Thilan Department of Mathematics University of Ruhuna Applied Statistics I(IMT224β/AMT224β) 1/158 Chapter
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions
Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions 1999 Prentice-Hall, Inc. Chap. 4-1 Chapter Topics Basic Probability Concepts: Sample
More informationSTA 4321/5325 Solution to Extra Homework 1 February 8, 2017
STA 431/535 Solution to Etra Homework 1 February 8, 017 1. Show that for any RV X, V (X 0. (You can assume X to be discrete, but this result holds in general. Hence or otherwise show that E(X E (X. Solution.
More information/60 (multiple choice) II /20 III /30 IV /10 V /60 (essay)
1 PHYSICS 6 HOUR EXAM 2 SPRING 2003 NAME This is a closed book, closed notes exam, except for a copy of Copenhagen. You may use calculators. Make sure you show all your work! You will get partial credit
More information1: PROBABILITY REVIEW
1: PROBABILITY REVIEW Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) Slides 1: Probability Review 1 / 56 Outline We will review the following
More informationProbability Distribution. Stat Camp for the MBA Program. Debbon Air Seat Release
Stat Camp for the MBA Program Daniel Solow Lecture 3 Random Variables and Distributions 136 Probability Distribution Recall that a random variable is a quantity of interest whose value is uncertain and
More informationStatistics for Managers Using Microsoft Excel (3 rd Edition)
Statistics for Managers Using Microsoft Excel (3 rd Edition) Chapter 4 Basic Probability and Discrete Probability Distributions 2002 Prentice-Hall, Inc. Chap 4-1 Chapter Topics Basic probability concepts
More informationNuclear Physics Lab I: Geiger-Müller Counter and Nuclear Counting Statistics
Nuclear Physics Lab I: Geiger-Müller Counter and Nuclear Counting Statistics PART I Geiger Tube: Optimal Operating Voltage and Resolving Time Objective: To become acquainted with the operation and characteristics
More informationThe Exciting Guide To Probability Distributions Part 2. Jamie Frost v1.1
The Exciting Guide To Probability Distributions Part 2 Jamie Frost v. Contents Part 2 A revisit of the multinomial distribution The Dirichlet Distribution The Beta Distribution Conjugate Priors The Gamma
More informationAP Online Quiz KEY Chapter 7: Sampling Distributions
AP Online Quiz KEY Chapter 7: Sampling Distributions 1. A news website claims that 30% of all Major League Baseball players use performanceenhancing drugs ( PEDs ) Indignant at this claim, league officials
More informationLecture 10: The Schrödinger Equation Lecture 10, p 1
Lecture 10: The Schrödinger Equation Lecture 10, p 1 Overview Probability distributions Schrödinger s Equation Particle in a Bo Matter waves in an infinite square well Quantized energy levels y() U= n=1
More informationCENTRAL LIMIT THEOREM (CLT)
CENTRAL LIMIT THEOREM (CLT) A sampling distribution is the probability distribution of the sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic
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 informationPrentice Hall. Physics: Principles with Applications, Updated 6th Edition (Giancoli) High School
Prentice Hall Physics: Principles with Applications, Updated 6th Edition (Giancoli) 2009 High School C O R R E L A T E D T O Physics I Students should understand that scientific knowledge is gained from
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 informationMath/Stat 352 Lecture 9. Section 4.5 Normal distribution
Math/Stat 352 Lecture 9 Section 4.5 Normal distribution 1 Abraham de Moivre, 1667-1754 Pierre-Simon Laplace (1749 1827) A French mathematician, who introduced the Normal distribution in his book The doctrine
More informationLecture 10: Random Walks in One Dimension
Physical Principles in Biology Biology 3550 Fall 2018 Lecture 10: Random Walks in One Dimension Wednesday, 12 September 2018 c David P. Goldenberg University of Utah goldenberg@biology.utah.edu Announcements
More informationGCSE PHYSICS REVISION LIST
GCSE PHYSICS REVISION LIST OCR Gateway Physics (J249) from 2016 Topic P1: Matter P1.1 Describe how and why the atomic model has changed over time Describe the structure of the atom and discuss the charges
More informationStatistical Intervals (One sample) (Chs )
7 Statistical Intervals (One sample) (Chs 8.1-8.3) Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to normally distributed with expected value µ and
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationLecture 6 Examples and Problems
Lecture 6 Examples and Problems Heat capacity of solids & liquids Thermal diffusion Thermal conductivity Irreversibility Hot Cold Random Walk and Particle Diffusion Counting and Probability Microstates
More information1. Discrete Distributions
Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 1. Discrete Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space Ω.
More informationBrief Review of Probability
Maura Department of Economics and Finance Università Tor Vergata Outline 1 Distribution Functions Quantiles and Modes of a Distribution 2 Example 3 Example 4 Distributions Outline Distribution Functions
More informationChapter # classifications of unlikely, likely, or very likely to describe possible buying of a product?
A. Attribute data B. Numerical data C. Quantitative data D. Sample data E. Qualitative data F. Statistic G. Parameter Chapter #1 Match the following descriptions with the best term or classification given
More informationExponential Growth and Decay - M&M's Activity
Eponential Growth and Decay - M&M's Activity Activity 1 - Growth 1. The results from the eperiment are as follows: Group 1 0 4 1 5 2 6 3 4 15 5 22 6 31 2. The scatterplot of the result is as follows: 3.
More information