Chapter 17. The. Value Example. The Standard Error. Example The Short Cut. Classifying and Counting. Chapter 17. The.
|
|
- Miles Morris
- 5 years ago
- Views:
Transcription
1 Context Short Part V Chance Variability and Short Last time, we learned that it can be helpful to take real-life chance processes and turn them into a box model. outcome of the chance process then corresponds to drawing tickets from the box and summing up the numbers on the tickets. Today, we will focus on the sum of from the box. Suppose we draw at random with replacement from a box. sum of is a random output which varies around the expected value, the amounts o being similar is size to the standard error (SE) for the sum. 1 1 Short We make 100 draws at random with replacement from the box What do we expect the sum to be? 9 should come up about every forth time, or 25 times; 1 should come up about three out of four times, or 75 times. sum should thus be around = 300. Short We make 100 draws at random with replacement from the box What do we expect the sum to be? re is another way to compute the expected value of the sum. average of the box is = 3. 4 This is the expected value. On average, each draws will thus add around 3 to the sum. With 100 draws, the sum should be around = 300.
2 expected value of the sum of standard error for the sum of Short expected value for the sum of draws made at random with replacement from a box equals (number of draws) (average of box) Short actual sum will likely be dierent from the expected value. It will be o by the chance error sum = expected value + chance error Does the formula make sense? What happens if the number of draws is doubled? n the expected value of the sum of doubles. What happens if the average of the box is doubled? n the expected value of the sum of doubles. chance error is the amount above (+) or below (-) the expected value. standard error (SE) for the sum tells us how big the chance error is likely to be. standard error for the sum of standard error for the sum of Short A sum is likely to be around its expected value, but to be o by an amount similar in size to the standard error. Short orem ( square root law) When drawing at random with replacement from a box of numbered tickets, the standard error for the sum of is number of draws (SD of the box). To compute the SE for a sum, we use the following law: orem ( square root law) When drawing at random with replacement from a box of numbered tickets, the standard error for the sum of is number of draws (SD of the box). Does the formula make sense? What happens if the number of draws is doubled? n the SE of the sum of is multiplied by a factor 2. chance error grows as we make more draws, but only slowly. What happens if we double the SD of the box? n the SE of the sum of doubles.
3 standard error for the sum of 1 (continued) Short For our chance process, we have the formula observed value = expected value + chance error chance error is likely to be similar in size to the standard error (SE) for the sum of from the corresponding box Large SE Large chance errors Observed values are widely spread around the expected value Short We make 100 draws at random with replacement from the box average of the box is 3. expected value of the sum is = 300. SD of the box is (1 3)2 + (1 3) 2 + (1 3) 2 + (9 3) 2 = Small SE Small chance errors Observed values are tightly clustered around the expected value Observed values are rarely more than 2 or 3 SEs away from the expected value SE for the sum is = 35. Thus, the sum of is likely to be around 300, give or take 35 or so. Short cut for calculating the SE 2 Short When the tickets in the box show only two dierent numbers ('big' and 'small'), the SD of the box is ( big number - small number ) Short We make 25 draws from the box (fraction with big number) (fraction with small number) Fill in the blanks: 1 (continued): SD is 1 (9 1) = sum of is around..., give or take... or so.
4 Another way to look at this... Another way to look at this... Short 3: Say we use the box model to count the number of heads in 100 coin tosses. We repeat the process 1000 times, and get a variety of results: Density Number of heads in 100 coin tosses, repeated times Short number of heads is a random variable, with a distribution center of the distribution is the expected value spread of the distribution is the standard error nr of heads normal approximation 2 (continued) Consider the sum of 25 draws from the box Short If the number of draws is large, we can use the normal approximation to estimate chances. We should use a new average and new SD: New average = expected value for sum of New SD = SE for the sum of new standard units tell us how many SEs a number is away from the expected value Short Density Histogram of sum of, when repeated 1000 times sum of
5 2 (continued) and counting Short We make 25 draws from the box About what percentage of observed values should be between 50 and 100? Short Some chance processes involce counting. How can we set up a box model? 4: A die is tossed 60 times. number of sixes should be around..., give or take... or so. Replace tickets by 0s and 1s Short If we want to count the number of a certain ticket (or tickets), then we put 0 on the tickets that we don't want to count put 1 on the ticket (tickets) that we want to count Using this new box, we have that Short the count is like the sum of from the new box we can compute the expected value and SE as before we can also use the normal curve to approximate probabilities as before
6 Short Summary expected value for the sum of draws made at random with replacement from a box equals (number of draws) (average of box) When drawing at random with replacement from a box of numbered tickets, the standard error for the sum of the draws is number of draws (SD of the box). If the number of draws is large, we can use the normal approximation to estimate chances. If we have to classify and count, put 0's and 1's on the tickets. Mark 1 on the tickets that should count, and 0 on the others.
Chapter 23: accuracy of averages
Chapter 23: accuracy of averages Context: previous chapters...................................................... 2 Context: previous chapters...................................................... 3 Context:
More informationChance Error in Sampling
1 Chance Error in Sampling How different is the sample percentage from the population percentage? The purpose of this chapter is to show how box models can be used to understand the error in simple random
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 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 informationAP Statistics Chapter 6 - Random Variables
AP Statistics Chapter 6 - Random 6.1 Discrete and Continuous Random Objective: Recognize and define discrete random variables, and construct a probability distribution table and a probability histogram
More informationThe Accuracy of Percentages. Confidence Intervals
The Accuracy of Percentages Confidence Intervals 1 Review: a 0-1 Box Box average = fraction of tickets which equal 1 Box SD = (fraction of 0 s) x (fraction of 1 s) 2 With a simple random sample, the expected
More informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.1 Discrete and Continuous Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Discrete and Continuous Random
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 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 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 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 informationExpected Value of a Random Variable
Knowledge Article: Probability and Statistics Expected Value of a Random Variable Expected Value of a Discrete Random Variable You're familiar with a simple mean, or average, of a set. The mean value of
More informationLAB 2 Random Variables, Sampling Distributions of Counts, and Normal Distributions
LAB 2 Random Variables, Sampling Distributions of Counts, and Normal Distributions The ECA 225 has open lab hours if you need to finish LAB 2. The lab is open Monday-Thursday 6:30-10:00pm and Saturday-Sunday
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 informationKey Objectives. Module 2: The Logic of Statistical Inference. Z-scores. SGSB Workshop: Using Statistical Data to Make Decisions
SGSB Workshop: Using Statistical Data to Make Decisions Module 2: The Logic of Statistical Inference Dr. Tom Ilvento January 2006 Dr. Mugdim Pašić Key Objectives Understand the logic of statistical inference
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 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 informationShifting our focus. We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why?
Probability Introduction Shifting our focus We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why? What is Probability? Probability is used
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 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 informationMATH1215: Mathematical Thinking Sec. 08 Spring Worksheet 9: Solution. x P(x)
N. Name: MATH: Mathematical Thinking Sec. 08 Spring 0 Worksheet 9: Solution Problem Compute the expected value of this probability distribution: x 3 8 0 3 P(x) 0. 0.0 0.3 0. Clearly, a value is missing
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 informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 6 Learning Objectives Define terms random variable and probability distribution. Distinguish between discrete and continuous probability distributions. Calculate
More information5.7 Probability Distributions and Variance
160 CHAPTER 5. PROBABILITY 5.7 Probability Distributions and Variance 5.7.1 Distributions of random variables We have given meaning to the phrase expected value. For example, if we flip a coin 100 times,
More informationMeasuring Risk. Expected value and expected return 9/4/2018. Possibilities, Probabilities and Expected Value
Chapter Five Understanding Risk Introduction Risk cannot be avoided. Everyday decisions involve financial and economic risk. How much car insurance should I buy? Should I refinance my mortgage now or later?
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 informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 6 McGraw-Hill/Irwin Copyright 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Learning Objectives LO1 Identify the characteristics of a probability
More informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 13: Binomial Formula Tessa L. Childers-Day UC Berkeley 14 July 2014 By the end of this lecture... You will be able to: Calculate the ways an event can
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 information4.1 Probability Distributions
Probability and Statistics Mrs. Leahy Chapter 4: Discrete Probability Distribution ALWAYS KEEP IN MIND: The Probability of an event is ALWAYS between: and!!!! 4.1 Probability Distributions Random Variables
More informationChapter 8: Binomial and Geometric Distributions
Chapter 8: Binomial and Geometric Distributions Section 8.1 Binomial Distributions The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Section 8.1 Binomial Distribution Learning Objectives
More informationSampling Distributions and the Central Limit Theorem
Sampling Distributions and the Central Limit Theorem February 18 Data distributions and sampling distributions So far, we have discussed the distribution of data (i.e. of random variables in our sample,
More informationWe use probability distributions to represent the distribution of a discrete random variable.
Now we focus on discrete random variables. We will look at these in general, including calculating the mean and standard deviation. Then we will look more in depth at binomial random variables which are
More information1. Three draws are made at random from the box [ 3, 4, 4, 5, 5, 5 ].
Stat 1040 Review 2 1. Three draws are made at random from the box [ 3, 4, 4, 5, 5, 5 ]. a) If the draws are made with replacement, find the probability that a "4" is drawn each time. b) If the draws are
More informationMA 1125 Lecture 12 - Mean and Standard Deviation for the Binomial Distribution. Objectives: Mean and standard deviation for the binomial distribution.
MA 5 Lecture - Mean and Standard Deviation for the Binomial Distribution Friday, September 9, 07 Objectives: Mean and standard deviation for the binomial distribution.. Mean and Standard Deviation of the
More informationA random variable is a quantitative variable that represents a certain
Section 6.1 Discrete Random Variables Example: Probability Distribution, Spin the Spinners Sum of Numbers on Spinners Theoretical Probability 2 0.04 3 0.08 4 0.12 5 0.16 6 0.20 7 0.16 8 0.12 9 0.08 10
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics Let s make our own sampling! If we use a random sample (a survey) or if we randomly assign treatments to subjects (an experiment) we can come up with proper, unbiased conclusions
More informationSec$on 6.1: Discrete and Con.nuous Random Variables. Tuesday, November 14 th, 2017
Sec$on 6.1: Discrete and Con.nuous Random Variables Tuesday, November 14 th, 2017 Discrete and Continuous Random Variables Learning Objectives After this section, you should be able to: ü COMPUTE probabilities
More informationCHAPTER 5 SAMPLING DISTRIBUTIONS
CHAPTER 5 SAMPLING DISTRIBUTIONS Sampling Variability. We will visualize our data as a random sample from the population with unknown parameter μ. Our sample mean Ȳ is intended to estimate population mean
More informationThe Binomial and Geometric Distributions. Chapter 8
The Binomial and Geometric Distributions Chapter 8 8.1 The Binomial Distribution A binomial experiment is statistical experiment that has the following properties: The experiment consists of n repeated
More informationProbability & Statistics Chapter 5: Binomial Distribution
Probability & Statistics Chapter 5: Binomial Distribution Notes and Examples Binomial Distribution When a variable can be viewed as having only two outcomes, call them success and failure, it may be considered
More informationChapter 5 Basic Probability
Chapter 5 Basic Probability Probability is determining the probability that a particular event will occur. Probability of occurrence = / T where = the number of ways in which a particular event occurs
More informationSTA 6166 Fall 2007 Web-based Course. Notes 10: Probability Models
STA 6166 Fall 2007 Web-based Course 1 Notes 10: Probability Models We first saw the normal model as a useful model for the distribution of some quantitative variables. We ve also seen that if we make a
More informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.3 Binomial and Geometric Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Binomial and Geometric Random
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 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 informationProbability and Statistics. Copyright Cengage Learning. All rights reserved.
Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.3 Binomial Probability Copyright Cengage Learning. All rights reserved. Objectives Binomial Probability The Binomial Distribution
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 informationChapter 6: Random Variables
Chapter 6: Random Variables Section 6.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous Random Variables 6.2 Transforming and
More informationA probability distribution shows the possible outcomes of an experiment and the probability of each of these outcomes.
Introduction In the previous chapter we discussed the basic concepts of probability and described how the rules of addition and multiplication were used to compute probabilities. In this chapter we expand
More information5.1 Personal Probability
5. Probability Value Page 1 5.1 Personal Probability Although we think probability is something that is confined to math class, in the form of personal probability it is something we use to make decisions
More informationGOALS. Discrete Probability Distributions. A Distribution. What is a Probability Distribution? Probability for Dice Toss. A Probability Distribution
GOALS Discrete Probability Distributions Chapter 6 Dr. Richard Jerz Define the terms probability distribution and random variable. Distinguish between discrete and continuous probability distributions.
More informationDiscrete Probability Distributions Chapter 6 Dr. Richard Jerz
Discrete Probability Distributions Chapter 6 Dr. Richard Jerz 1 GOALS Define the terms probability distribution and random variable. Distinguish between discrete and continuous probability distributions.
More informationData Science Essentials
Data Science Essentials Probability and Random Variables As data scientists, we re often concerned with understanding the qualities and relationships of a set of data points. For example, you may need
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 informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number
More informationChapter 5: Discrete Probability Distributions
Chapter 5: Discrete Probability Distributions Section 5.1: Basics of Probability Distributions As a reminder, a variable or what will be called the random variable from now on, is represented by the letter
More informationDiscrete Probability Distributions
90 Discrete Probability Distributions Discrete Probability Distributions C H A P T E R 6 Section 6.2 4Example 2 (pg. 00) Constructing a Binomial Probability Distribution In this example, 6% of the human
More informationchapter 13: Binomial Distribution Exercises (binomial)13.6, 13.12, 13.22, 13.43
chapter 13: Binomial Distribution ch13-links binom-tossing-4-coins binom-coin-example ch13 image Exercises (binomial)13.6, 13.12, 13.22, 13.43 CHAPTER 13: Binomial Distributions The Basic Practice of Statistics
More informationThe Central Limit Theorem
The Central Limit Theorem Patrick Breheny March 1 Patrick Breheny University of Iowa Introduction to Biostatistics (BIOS 4120) 1 / 29 Kerrich s experiment Introduction The law of averages Mean and SD of
More informationProbability distributions
Probability distributions Introduction What is a probability? If I perform n eperiments and a particular event occurs on r occasions, the relative frequency of this event is simply r n. his is an eperimental
More informationASSIGNMENT 14 section 10 in the probability and statistics module
McMaster University Math 1LT3 ASSIGNMENT 14 section 10 in the probability and statistics module 1. (a) A shipment of 2,000 containers has arrived at the port of Vancouver. As part of the customs inspection,
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 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 informationReview: Population, sample, and sampling distributions
Review: Population, sample, and sampling distributions A population with mean µ and standard deviation σ For instance, µ = 0, σ = 1 0 1 Sample 1, N=30 Sample 2, N=30 Sample 100000000000 InterquartileRange
More informationMean, Variance, and Expectation. Mean
3 Mean, Variance, and Expectation The mean, variance, and standard deviation for a probability distribution are computed differently from the mean, variance, and standard deviation for samples. This section
More informationNumerical Descriptive Measures. Measures of Center: Mean and Median
Steve Sawin Statistics Numerical Descriptive Measures Having seen the shape of a distribution by looking at the histogram, the two most obvious questions to ask about the specific distribution is where
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 informationThe topics in this section are related and necessary topics for both course objectives.
2.5 Probability Distributions The topics in this section are related and necessary topics for both course objectives. A probability distribution indicates how the probabilities are distributed for outcomes
More information6.1 Discrete and Continuous Random Variables. 6.1A Discrete random Variables, Mean (Expected Value) of a Discrete Random Variable
6.1 Discrete and Continuous Random Variables 6.1A Discrete random Variables, Mean (Expected Value) of a Discrete Random Variable Random variable Takes numerical values that describe the outcomes of some
More informationProbability Distributions. Chapter 6
Probability Distributions Chapter 6 McGraw-Hill/Irwin The McGraw-Hill Companies, Inc. 2008 GOALS Define the terms probability distribution and random variable. Distinguish between discrete and continuous
More informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 6 Copyright 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Learning
More information4.1 Introduction Estimating a population mean The problem with estimating a population mean with a sample mean: an example...
Chapter 4 Point estimation Contents 4.1 Introduction................................... 2 4.2 Estimating a population mean......................... 2 4.2.1 The problem with estimating a population mean
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 useful modeling tricks.
.7 Joint models for more than two outcomes We saw that we could write joint models for a pair of variables by specifying the joint probabilities over all pairs of outcomes. In principal, we could do this
More informationExample. Chapter 8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables
Chapter 8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables You are dealt a hand of 5 cards. Find the probability distribution table for the number of hearts. Graph
More informationMarket Volatility and Risk Proxies
Market Volatility and Risk Proxies... an introduction to the concepts 019 Gary R. Evans. This slide set by Gary R. Evans is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International
More informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
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 information5.2 Random Variables, Probability Histograms and Probability Distributions
Chapter 5 5.2 Random Variables, Probability Histograms and Probability Distributions A random variable (r.v.) can be either continuous or discrete. It takes on the possible values of an experiment. It
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter7 Probability Distributions and Statistics Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number of boys in
More informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 6 McGraw-Hill/Irwin Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved. GOALS 6-2 1. Define the terms probability distribution and random variable.
More informationMAKING SENSE OF DATA Essentials series
MAKING SENSE OF DATA Essentials series THE NORMAL DISTRIBUTION Copyright by City of Bradford MDC Prerequisites Descriptive statistics Charts and graphs The normal distribution Surveys and sampling Correlation
More informationWeek 1 Variables: Exploration, Familiarisation and Description. Descriptive Statistics.
Week 1 Variables: Exploration, Familiarisation and Description. Descriptive Statistics. Convergent validity: the degree to which results/evidence from different tests/sources, converge on the same conclusion.
More informationSECTION 4.4: Expected Value
15 SECTION 4.4: Expected Value This section tells you why most all gambling is a bad idea. And also why carnival or amusement park games are a bad idea. Random Variables Definition: Random Variable A random
More informationA continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x)
Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density
More informationSampling variability. Data Science Team
Sampling variability Data Science Team What we have learned so far Often the data is a sample from a population and we want to use it to learn something about this bigger population A summary of the data
More information6.1 Discrete & Continuous Random Variables. Nov 4 6:53 PM. Objectives
6.1 Discrete & Continuous Random Variables examples vocab Objectives Today we will... - Compute probabilities using the probability distribution of a discrete random variable. - Calculate and interpret
More informationChapter 7. Random Variables
Chapter 7 Random Variables Making quantifiable meaning out of categorical data Toss three coins. What does the sample space consist of? HHH, HHT, HTH, HTT, TTT, TTH, THT, THH In statistics, we are most
More informationMA131 Lecture 8.2. The normal distribution curve can be considered as a probability distribution curve for normally distributed variables.
Normal distribution curve as probability distribution curve The normal distribution curve can be considered as a probability distribution curve for normally distributed variables. The area under the normal
More informationThe following content is provided under a Creative Commons license. Your support
MITOCW Recitation 6 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make
More informationRandom variables The binomial distribution The normal distribution Other distributions. Distributions. Patrick Breheny.
Distributions February 11 Random variables Anything that can be measured or categorized is called a variable If the value that a variable takes on is subject to variability, then it the variable is a random
More informationChapter 7. Introduction to Risk, Return, and the Opportunity Cost of Capital. Principles of Corporate Finance. Slides by Matthew Will
Principles of Corporate Finance Seventh Edition Richard A. Brealey Stewart C. Myers Chapter 7 Introduction to Risk, Return, and the Opportunity Cost of Capital Slides by Matthew Will - Topics Covered 75
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 informationWeb Extension: Continuous Distributions and Estimating Beta with a Calculator
19878_02W_p001-008.qxd 3/10/06 9:51 AM Page 1 C H A P T E R 2 Web Extension: Continuous Distributions and Estimating Beta with a Calculator This extension explains continuous probability distributions
More informationSection 0: Introduction and Review of Basic Concepts
Section 0: Introduction and Review of Basic Concepts Carlos M. Carvalho The University of Texas McCombs School of Business mccombs.utexas.edu/faculty/carlos.carvalho/teaching 1 Getting Started Syllabus
More informationRandom variables The binomial distribution The normal distribution Sampling distributions. Distributions. Patrick Breheny.
Distributions September 17 Random variables Anything that can be measured or categorized is called a variable If the value that a variable takes on is subject to variability, then it the variable is a
More informationConsider the following examples: ex: let X = tossing a coin three times and counting the number of heads
Overview Both chapters and 6 deal with a similar concept probability distributions. The difference is that chapter concerns itself with discrete probability distribution while chapter 6 covers continuous
More informationChapter 5 Normal Probability Distributions
Chapter 5 Normal Probability Distributions Section 5-1 Introduction to Normal Distributions and the Standard Normal Distribution A The normal distribution is the most important of the continuous probability
More information