STAT 241/251 - Chapter 7: Central Limit Theorem
|
|
- Matilda Freeman
- 5 years ago
- Views:
Transcription
1 STAT 241/251 - Chapter 7: Central Limit Theorem In this chapter we will introduce the most important theorem in statistics; the central limit theorem. What have we seen so far? First, we saw that for an i.i.d random sample X 1, X 2,..., X n, where E(X i ) = µ and Var(X i ) = σ 2 E( X) = µ X = µ, Var( X) = σ 2 X = σ2 n E(S) = µ S = nµ, Var(S) = σ 2 S = nσ2 Further, in chapter 5 we saw that if the X i N(µ, σ 2 ), then... X N(µ, σ2 n ) S N(nµ, nσ 2 ) (ie) Linear functions of a normal RV are also normally distributed. These are nice results, although often we have a random sample where each of the X i s do not follow a normal distribution. (The distribution we are sampling from is not Normal) (eg) We may take a random sample of new fluorescent lightbulbs and measure the life-time of each bulb. In this case, we might model the life-time of each bulb as an exponential RV. For this random sample, we will still probably look at the mean life-time of all the bulbs or maybe the sum of the life-times. So, what distribution do X and S follow then? 1
2 The Central Limit Theorem (CLT) If our random sample X 1, X 2,..., X n is i.i.d and comes from any particular distribution (may not be normal) with mean µ and variance σ 2, then when n is large enough (and the samples are independent of one another and are made randomly), the sample mean X and the sample sum S approximately follow a normal distribution. (ie) If X i Almost any shape distribution with mean µ and variance σ 2, and large n then... X N(µ, σ2 n ) S N(nµ, nσ 2 ) This result is very important and is used extensively throughout statistics, as it tells us that no matter what distribution our random sample comes from, that the sample mean and sample sum follow a Normal distribution as long as certain conditions are met. The CLT is an asymptotic or limit result, meaning that when n =, X and S are Normally distributed, but when n <, X and S are only approximately Normally distributed. This raises the question, when is n large enough to say that X and S are Normally distributed? Answer: The more symmetric and light tailed the distribution of the X i s, the quicker that X and S will converge to Normality. Provided that the distribution of the X i s is not too skewed or asymetric, a sample size of n 20 is usually adequate for the CLT to kick in. If the distribution we are sampling from is very skewed, then we need a larger sample size. For the sake of this class, we will use n=20 as the magic number. In reality it depends more on the shape of the distribution that we are sampling from. 2
3 First we will look at a simulation to explain the central limit theorem. Then I will draw a picture to illustrate the concept. The CLT can also be proven mathematically, although that is beyond the scope of this course. Simulating The Sampling Distribution of a Mean: Below is a picture of histograms of some simulated rolls of dice. I will talk about these in class. Note: I used dice as the example, as these cover many areas. (ie) A dice can lead to proportions, it can be binomial, and it is an ordinal variable, which is similar to a quantitative variable Rolls of 1 Die The Mean of Rolls of 2 Dice The Mean of Rolls of 3 Dice die die die3 The Mean of Rolls of 5 Dice The Mean of Rolls of 25 Dice die die25 We can see that as the sample size increases (1, 2, 3, 5, 25), the sampling distribution of the mean begins to look like a Normal distribution. Here is another picture interpretation of the CLT... 3
4 Examples: 1. You have designed a new sattelite that is planned to orbit in space for the next 150 years. It is set up in the following way. It has 25 battery packs. One powers the sattelite, and when it burns out the next battery takes over, and when that one burns out the next takes over, and so on. The lifetime of each battery follows an exponential distribution with a mean of 8 years and a standard deviation of 8 years. What is the probability that the sattelite runs out of batteries before the 150 years is up? 2. A standard bottle of beer advertises that it contains 341mL of beer. In fact, the machine that pours the beer into the bottle pours a mean amount of 343mL with a standard deviation of 2mL. The amount of beer poured follows a normal distribution. (a) What is the probability that a randomly selected bottle of beer is underfilled? (b) If you buy a two-four, what is the probability that no more than 4 bottles are underfilled? (c) If you buy a 6-pack, what is the probability that the average amount of liquid is less than 341mL? 3. An elevator has a limit of 10 people or 2000lbs. If 10 people get on the elevator what is the probability that they surpass the limit? Suppose that the weights of people follow a normal distribution with a mean of 170lbs and a standard deviation of 30lbs. 4. You have developed a new type of concrete that is reinforced with steel fibers. Suppose you know that a concrete block of this type has a true mean breaking strength of 500lbs with a standard deviation of 12lbs. If you take a sample of 25 blocks and test their breaking strengths, what is the probability that the 25 blocks have a mean breaking strength greater than 505lbs? 5. Suppose you will go to the casino and make 81 bets, each of $1, on the colour of the number coming up. What is the mean and variance of the gain/loss you will make from these 81 bets? What is the probability that you leave the casino having made money? 4
5 So far, we have only dealt with the CLT and continuous distributions. But what happens when we have a random sample where each X i comes from some discrete distribution? We present a few results here that follow from the CLT. Shortly, we will see that under certain conditions, we can use a normal distribution to approximate the binomial and the poisson distributions. Normal Approximation to the Binomial Recall: That if X BIN(n, p), then µ x = np and σ 2 x = np(1 p) When n is large and p is not too close to 0 or 1, then... BIN(n, p) N(np, np(1-p)) The rule-of-thumb for this approximation to work is that min{np, n(1 p)} 10 Continuity Correction: Because we are approximating a discrete distribution with a continuous distribution, we must make a continuity correction. (ie) In the discrete case, P(X x) P(X > x) P (X = k) = P (k 0.5 X k + 0.5) P (a X b) = P (a 0.5 X b + 0.5) P (a < X < b) = P (a X b 0.5) P (X < a) = P (X a 0.5) P (X a) = P (X a + 0.5) P (X > a) = P (X a + 0.5) P (X a) = P (X a 0.5) Note: The continuity correction makes little difference when n is large. I will explain the idea of the continuity correction more in depth during lecture, and the above should make more sense then. 5
6 Examples: 1. Consider rolling a die 150 times. What is the probability that you get... (a) Exactly 23 6 s? (b) Between 15 and 35 6 s? (c) More than 3 6 s? 2. You go to the casino and make 81 bets, each of $1, on the colour of the number coming up. What is the probability that you leave the casino having made money? 3. It is believed that 4% of children have a gene that may be linked to juvenille diabetes. Researchers are hoping to track 20 or more of these children (with the defect) for several years. they will test 732 newborn babies for the presence of this gene, and if the gene is present, they will track the child for several years. What is the probability that they find 20 or more subjects to be in the study? 4. Suppose you will toss a coin 100 times. Create an interval that you are 95% sure that the number of heads tossed will be in. Center this interval around the expected number of heads. 6
7 Normal Approximation to the Poisson Recall: That if X POISSON(λ ), then µ x = λ and σ 2 x = λ When λ is large, then... POISSON(λ ) N(λ, λ ) The rule-of-thumb for this approximation to work is that λ 20 Like in the Binomial case, here we are approximating a discrete distribution using a continuous distribution, so we must use the same continuity correction as in the Binomial case. Example: 1. Recall example (1) from chapter 6, in the section on Poisson Processes. We were monitoring the number of earthquakes in California over 6.7, and there were an average of 1.5 per year. (a) What is the probability of having more than 14 large earthquakes in the next 15 years? (b) What is the probability of having exactly 1 large earthquake in the coming year? 2. A factory produces sheet metal. Impurities occur at a rate of 0.1 per meter squared, and are equally likely to occur anywhere on a given sheet. The occurrence of impurities are independent of one another. If you buy 500 square meters of this sheet metal, what is the probability that there is more than 40 impurities on the sheet? 7
STAT Chapter 7: Central Limit Theorem
STAT 251 - Chapter 7: Central Limit Theorem In this chapter we will introduce the most important theorem in statistics; the central limit theorem. What have we seen so far? First, we saw that for an i.i.d
More informationBIOL The Normal Distribution and the Central Limit Theorem
BIOL 300 - The Normal Distribution and the Central Limit Theorem In the first week of the course, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are
More informationChapter 7: Point Estimation and Sampling Distributions
Chapter 7: Point Estimation and Sampling Distributions Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 20 Motivation In chapter 3, we learned
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationElementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
More informationRandom Variable: Definition
Random Variables Random Variable: Definition A Random Variable is a numerical description of the outcome of an experiment Experiment Roll a die 10 times Inspect a shipment of 100 parts Open a gas station
More informationCommonly Used Distributions
Chapter 4: Commonly Used Distributions 1 Introduction Statistical inference involves drawing a sample from a population and analyzing the sample data to learn about the population. We often have some knowledge
More informationSection The Sampling Distribution of a Sample Mean
Section 5.2 - The Sampling Distribution of a Sample Mean Statistics 104 Autumn 2004 Copyright c 2004 by Mark E. Irwin The Sampling Distribution of a Sample Mean Example: Quality control check of light
More informationChapter 3 - Lecture 5 The Binomial Probability Distribution
Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009 Experiment Examples Moments and moment generating function of a Binomial Random Variable Outline Experiment Examples A binomial experiment
More informationSTAT Chapter 4/6: Random Variables and Probability Distributions
STAT 251 - Chapter 4/6: Random Variables and Probability Distributions We use random variables (RV) to represent the numerical features of a random experiment. In chapter 3, we defined a random experiment
More informationPart V - Chance Variability
Part V - Chance Variability Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Part V - Chance Variability 1 / 78 Law of Averages In Chapter 13 we discussed the Kerrich coin-tossing experiment.
More informationEngineering Statistics ECIV 2305
Engineering Statistics ECIV 2305 Section 5.3 Approximating Distributions with the Normal Distribution Introduction A very useful property of the normal distribution is that it provides good approximations
More informationStatistics and Probability
Statistics and Probability Continuous RVs (Normal); Confidence Intervals Outline Continuous random variables Normal distribution CLT Point estimation Confidence intervals http://www.isrec.isb-sib.ch/~darlene/geneve/
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More informationAMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Summer 2014 1 / 26 Sampling Distributions!!!!!!
More informationChapter 4: Commonly Used Distributions. Statistics for Engineers and Scientists Fourth Edition William Navidi
Chapter 4: Commonly Used Distributions Statistics for Engineers and Scientists Fourth Edition William Navidi 2014 by Education. This is proprietary material solely for authorized instructor use. Not authorized
More informationStatistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More informationPoint Estimation. Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
6 Point Estimation Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Point Estimation Statistical inference: directed toward conclusions about one or more parameters. We will use the generic
More informationChapter 5 Student Lecture Notes 5-1. Department of Quantitative Methods & Information Systems. Business Statistics
Chapter 5 Student Lecture Notes 5-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals
More information5.3 Statistics and Their Distributions
Chapter 5 Joint Probability Distributions and Random Samples Instructor: Lingsong Zhang 1 Statistics and Their Distributions 5.3 Statistics and Their Distributions Statistics and Their Distributions Consider
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationTOPIC: PROBABILITY DISTRIBUTIONS
TOPIC: PROBABILITY DISTRIBUTIONS There are two types of random variables: A Discrete random variable can take on only specified, distinct values. A Continuous random variable can take on any value within
More informationChapter 5: Statistical Inference (in General)
Chapter 5: Statistical Inference (in General) Shiwen Shen University of South Carolina 2016 Fall Section 003 1 / 17 Motivation In chapter 3, we learn the discrete probability distributions, including Bernoulli,
More informationCentral Limit Thm, Normal Approximations
Central Limit Thm, Normal Approximations Engineering Statistics Section 5.4 Josh Engwer TTU 23 March 2016 Josh Engwer (TTU) Central Limit Thm, Normal Approximations 23 March 2016 1 / 26 PART I PART I:
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationProbability is the tool used for anticipating what the distribution of data should look like under a given model.
AP Statistics NAME: Exam Review: Strand 3: Anticipating Patterns Date: Block: III. Anticipating Patterns: Exploring random phenomena using probability and simulation (20%-30%) Probability is the tool used
More informationMLLunsford 1. Activity: Central Limit Theorem Theory and Computations
MLLunsford 1 Activity: Central Limit Theorem Theory and Computations Concepts: The Central Limit Theorem; computations using the Central Limit Theorem. Prerequisites: The student should be familiar with
More informationBusiness Statistics. Chapter 5 Discrete Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationSimple Random Sample
Simple Random Sample A simple random sample (SRS) of size n consists of n elements from the population chosen in such a way that every set of n elements has an equal chance to be the sample actually selected.
More informationIEOR 3106: Introduction to OR: Stochastic Models. Fall 2013, Professor Whitt. Class Lecture Notes: Tuesday, September 10.
IEOR 3106: Introduction to OR: Stochastic Models Fall 2013, Professor Whitt Class Lecture Notes: Tuesday, September 10. The Central Limit Theorem and Stock Prices 1. The Central Limit Theorem (CLT See
More informationLecture 8 - Sampling Distributions and the CLT
Lecture 8 - Sampling Distributions and the CLT Statistics 102 Kenneth K. Lopiano September 18, 2013 1 Basics Improvements 2 Variability of Estimates Activity Sampling distributions - via simulation Sampling
More informationIntroduction to Business Statistics QM 120 Chapter 6
DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 When a RV x is discrete, we can
More informationChapter 5. Statistical inference for Parametric Models
Chapter 5. Statistical inference for Parametric Models Outline Overview Parameter estimation Method of moments How good are method of moments estimates? Interval estimation Statistical Inference for Parametric
More information4-1. Chapter 4. Commonly Used Distributions by The McGraw-Hill Companies, Inc. All rights reserved.
4-1 Chapter 4 Commonly Used Distributions 2014 by The Companies, Inc. All rights reserved. Section 4.1: The Bernoulli Distribution 4-2 We use the Bernoulli distribution when we have an experiment which
More informationChapter 7: Random Variables
Chapter 7: Random Variables 7.1 Discrete and Continuous Random Variables 7.2 Means and Variances of Random Variables 1 Introduction A random variable is a function that associates a unique numerical value
More informationProbability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
More informationStatistics for Managers Using Microsoft Excel 7 th Edition
Statistics for Managers Using Microsoft Excel 7 th Edition Chapter 5 Discrete Probability Distributions Statistics for Managers Using Microsoft Excel 7e Copyright 014 Pearson Education, Inc. Chap 5-1 Learning
More informationMidterm Exam III Review
Midterm Exam III Review Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Midterm Exam III Review 1 / 25 Permutations and Combinations ORDER In order to count the number of possible ways
More informationTutorial 11: Limit Theorems. Baoxiang Wang & Yihan Zhang bxwang, April 10, 2017
Tutorial 11: Limit Theorems Baoxiang Wang & Yihan Zhang bxwang, yhzhang@cse.cuhk.edu.hk April 10, 2017 1 Outline The Central Limit Theorem (CLT) Normal Approximation Based on CLT De Moivre-Laplace Approximation
More informationLecture 3. Sampling distributions. Counts, Proportions, and sample mean.
Lecture 3 Sampling distributions. Counts, Proportions, and sample mean. Statistical Inference: Uses data and summary statistics (mean, variances, proportions, slopes) to draw conclusions about a population
More informationMATH 3200 Exam 3 Dr. Syring
. Suppose n eligible voters are polled (randomly sampled) from a population of size N. The poll asks voters whether they support or do not support increasing local taxes to fund public parks. Let M be
More informationCHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS A random variable is the description of the outcome of an experiment in words. The verbal description of a random variable tells you how to find or calculate
More informationChapter 5. Sampling Distributions
Lecture notes, Lang Wu, UBC 1 Chapter 5. Sampling Distributions 5.1. Introduction In statistical inference, we attempt to estimate an unknown population characteristic, such as the population mean, µ,
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More information8.1 Binomial Distributions
8.1 Binomial Distributions The Binomial Setting The 4 Conditions of a Binomial Setting: 1.Each observation falls into 1 of 2 categories ( success or fail ) 2 2.There is a fixed # n of observations. 3.All
More informationChapter 6: Random Variables
Chapter 6: Random Variables Section 6.1 Discrete and Continuous Random Variables The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous
More informationECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10 Fall 2011 Lecture 8 Part 2 (Fall 2011) Probability Distributions Lecture 8 Part 2 1 / 23 Normal Density Function f
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic Probability Distributions: Binomial and Poisson Distributions Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College
More informationLecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
More informationRandom Variables. 6.1 Discrete and Continuous Random Variables. Probability Distribution. Discrete Random Variables. Chapter 6, Section 1
6.1 Discrete and Continuous Random Variables Random Variables A random variable, usually written as X, is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types
More informationECE 340 Probabilistic Methods in Engineering M/W 3-4:15. Lecture 10: Continuous RV Families. Prof. Vince Calhoun
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Lecture 10: Continuous RV Families Prof. Vince Calhoun 1 Reading This class: Section 4.4-4.5 Next class: Section 4.6-4.7 2 Homework 3.9, 3.49, 4.5,
More informationMA 1125 Lecture 14 - Expected Values. Wednesday, October 4, Objectives: Introduce expected values.
MA 5 Lecture 4 - Expected Values Wednesday, October 4, 27 Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More information4.2 Bernoulli Trials and Binomial Distributions
Arkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan 4.2 Bernoulli Trials and Binomial Distributions A Bernoulli trial 1 is an experiment with exactly two outcomes: Success and
More informationPoint Estimation. Some General Concepts of Point Estimation. Example. Estimator quality
Point Estimation Some General Concepts of Point Estimation Statistical inference = conclusions about parameters Parameters == population characteristics A point estimate of a parameter is a value (based
More informationSTAT 201 Chapter 6. Distribution
STAT 201 Chapter 6 Distribution 1 Random Variable We know variable Random Variable: a numerical measurement of the outcome of a random phenomena Capital letter refer to the random variable Lower case letters
More informationLecture 9 - Sampling Distributions and the CLT
Lecture 9 - Sampling Distributions and the CLT Sta102/BME102 Colin Rundel September 23, 2015 1 Variability of Estimates Activity Sampling distributions - via simulation Sampling distributions - via CLT
More informationCentral Limit Theorem, Joint Distributions Spring 2018
Central Limit Theorem, Joint Distributions 18.5 Spring 218.5.4.3.2.1-4 -3-2 -1 1 2 3 4 Exam next Wednesday Exam 1 on Wednesday March 7, regular room and time. Designed for 1 hour. You will have the full
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
More information1 Sampling Distributions
1 Sampling Distributions 1.1 Statistics and Sampling Distributions When a random sample is selected the numerical descriptive measures calculated from such a sample are called statistics. These statistics
More information5. In fact, any function of a random variable is also a random variable
Random Variables - Class 11 October 14, 2012 Debdeep Pati 1 Random variables 1.1 Expectation of a function of a random variable 1. Expectation of a function of a random variable 2. We know E(X) = x xp(x)
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 Fall 216 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / Fall 216 1
More informationModule 4: Probability
Module 4: Probability 1 / 22 Probability concepts in statistical inference Probability is a way of quantifying uncertainty associated with random events and is the basis for statistical inference. Inference
More information4.3 Normal distribution
43 Normal distribution Prof Tesler Math 186 Winter 216 Prof Tesler 43 Normal distribution Math 186 / Winter 216 1 / 4 Normal distribution aka Bell curve and Gaussian distribution The normal distribution
More informationChapter 5. Discrete Probability Distributions. McGraw-Hill, Bluman, 7 th ed, Chapter 5 1
Chapter 5 Discrete Probability Distributions McGraw-Hill, Bluman, 7 th ed, Chapter 5 1 Chapter 5 Overview Introduction 5-1 Probability Distributions 5-2 Mean, Variance, Standard Deviation, and Expectation
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationSTAT Chapter 6: Sampling Distributions
STAT 515 -- Chapter 6: Sampling Distributions Definition: Parameter = a number that characterizes a population (example: population mean ) it s typically unknown. Statistic = a number that characterizes
More informationSTATS 200: Introduction to Statistical Inference. Lecture 4: Asymptotics and simulation
STATS 200: Introduction to Statistical Inference Lecture 4: Asymptotics and simulation Recap We ve discussed a few examples of how to determine the distribution of a statistic computed from data, assuming
More informationFigure 1: 2πσ is said to have a normal distribution with mean µ and standard deviation σ. This is also denoted
Figure 1: Math 223 Lecture Notes 4/1/04 Section 4.10 The normal distribution Recall that a continuous random variable X with probability distribution function f(x) = 1 µ)2 (x e 2σ 2πσ is said to have a
More informationModule 3: Sampling Distributions and the CLT Statistics (OA3102)
Module 3: Sampling Distributions and the CLT Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chpt 7.1-7.3, 7.5 Revision: 1-12 1 Goals for
More informationStatistics, Their Distributions, and the Central Limit Theorem
Statistics, Their Distributions, and the Central Limit Theorem MATH 3342 Sections 5.3 and 5.4 Sample Means Suppose you sample from a popula0on 10 0mes. You record the following sample means: 10.1 9.5 9.6
More informationSampling Distributions For Counts and Proportions
Sampling Distributions For Counts and Proportions IPS Chapter 5.1 2009 W. H. Freeman and Company Objectives (IPS Chapter 5.1) Sampling distributions for counts and proportions Binomial distributions for
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationRandom Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES
Random Variables CHAPTER 6.3 BINOMIAL AND GEOMETRIC RANDOM VARIABLES Essential Question How can I determine whether the conditions for using binomial random variables are met? Binomial Settings When the
More informationMathematics of Randomness
Ch 5 Probability: The Mathematics of Randomness 5.1.1 Random Variables and Their Distributions A random variable is a quantity that (prior to observation) can be thought of as dependent on chance phenomena.
More informationWeek 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
More informationExamples: Random Variables. Discrete and Continuous Random Variables. Probability Distributions
Random Variables Examples: Random variable a variable (typically represented by x) that takes a numerical value by chance. Number of boys in a randomly selected family with three children. Possible values:
More informationSTAT 111 Recitation 3
STAT 111 Recitation 3 Linjun Zhang stat.wharton.upenn.edu/~linjunz/ September 23, 2017 Misc. The unpicked-up homeworks will be put in the STAT 111 box in the Stats Department lobby (It s on the 4th floor
More informationStatistical Methods in Practice STAT/MATH 3379
Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete
More informationBernoulli and Binomial Distributions
Bernoulli and Binomial Distributions Bernoulli Distribution a flipped coin turns up either heads or tails an item on an assembly line is either defective or not defective a piece of fruit is either damaged
More informationChapter 7 Sampling Distributions and Point Estimation of Parameters
Chapter 7 Sampling Distributions and Point Estimation of Parameters Part 1: Sampling Distributions, the Central Limit Theorem, Point Estimation & Estimators Sections 7-1 to 7-2 1 / 25 Statistical Inferences
More informationCentral Limit Theorem (cont d) 7/28/2006
Central Limit Theorem (cont d) 7/28/2006 Central Limit Theorem for Binomial Distributions Theorem. For the binomial distribution b(n, p, j) we have lim npq b(n, p, np + x npq ) = φ(x), n where φ(x) is
More information4.2 Probability Distributions
4.2 Probability Distributions Definition. A random variable is a variable whose value is a numerical outcome of a random phenomenon. The probability distribution of a random variable tells us what the
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationValue (x) probability Example A-2: Construct a histogram for population Ψ.
Calculus 111, section 08.x The Central Limit Theorem notes by Tim Pilachowski If you haven t done it yet, go to the Math 111 page and download the handout: Central Limit Theorem supplement. Today s lecture
More informationPart 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going?
1 The Law of Averages The Expected Value & The Standard Error Where Are We Going? Sums of random numbers The law of averages Box models for generating random numbers Sums of draws: the Expected Value Standard
More informationData Analysis and Statistical Methods Statistics 651
Review of previous lecture: Why confidence intervals? Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Suppose you want to know the
More information***SECTION 8.1*** The Binomial Distributions
***SECTION 8.1*** The Binomial Distributions CHAPTER 8 ~ The Binomial and Geometric Distributions In practice, we frequently encounter random phenomenon where there are two outcomes of interest. For example,
More informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationSTA258H5. Al Nosedal and Alison Weir. Winter Al Nosedal and Alison Weir STA258H5 Winter / 41
STA258H5 Al Nosedal and Alison Weir Winter 2017 Al Nosedal and Alison Weir STA258H5 Winter 2017 1 / 41 NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION. Al Nosedal and Alison Weir STA258H5 Winter 2017
More informationPROBABILITY DISTRIBUTIONS
CHAPTER 3 PROBABILITY DISTRIBUTIONS Page Contents 3.1 Introduction to Probability Distributions 51 3.2 The Normal Distribution 56 3.3 The Binomial Distribution 60 3.4 The Poisson Distribution 64 Exercise
More informationMidTerm 1) Find the following (round off to one decimal place):
MidTerm 1) 68 49 21 55 57 61 70 42 59 50 66 99 Find the following (round off to one decimal place): Mean = 58:083, round off to 58.1 Median = 58 Range = max min = 99 21 = 78 St. Deviation = s = 8:535,
More informationEcon 6900: Statistical Problems. Instructor: Yogesh Uppal
Econ 6900: Statistical Problems Instructor: Yogesh Uppal Email: yuppal@ysu.edu Lecture Slides 4 Random Variables Probability Distributions Discrete Distributions Discrete Uniform Probability Distribution
More informationNormal Distribution. Definition A continuous rv X is said to have a normal distribution with. the pdf of X is
Normal Distribution Normal Distribution Definition A continuous rv X is said to have a normal distribution with parameter µ and σ (µ and σ 2 ), where < µ < and σ > 0, if the pdf of X is f (x; µ, σ) = 1
More informationChapter 4 Discrete Random variables
Chapter 4 Discrete Random variables A is a variable that assumes numerical values associated with the random outcomes of an experiment, where only one numerical value is assigned to each sample point.
More informationThe Binomial Distribution
MATH 382 The Binomial Distribution Dr. Neal, WKU Suppose there is a fixed probability p of having an occurrence (or success ) on any single attempt, and a sequence of n independent attempts is made. Then
More information