Statistical Methods in Practice STAT/MATH 3379
|
|
- Milton Miles
- 6 years ago
- Views:
Transcription
1 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 Random Variables 6.2 Binomial Probability Distribution 6.3 Continuous Random Variables and the Normal Probability Distribution 6.4 Standard Normal Distribution 6.5 Applications of the Normal Distribution 6.1 Discrete Random Variables Objectives: By the end of this section, I will be able to 1) Identify random variables. 2) Explain what a discrete probability distribution is and construct probability distribution tables and graphs. 3) Calculate the mean, variance, and standard deviation of a discrete random variable. 1
2 Random Variables A variable whose values are determined by chance Chance in the definition of a random variable is crucial Example: Notation for random variables Suppose our experiment is to toss a single fair die, and we are interested in the number rolled. We define our random variable X to be the outcome of a single die roll. a. Why is the variable X a random variable? b. What are the possible values that the random variable X can take? c. What is the notation used for rolling a 5? d. Use random variable notation to express the probability of rolling a 5. Example continued Main types of random variables a) We don t know the value of X before we toss the die, which introduces an element of chance into the experiment b) Possible values for X: 1, 2, 3, 4, 5, and 6. c) When a 5 is rolled, then X equals the outcome 5, or X = 5. Discrete random variable - a finite or a countable number of values Continuous random variable can take infinitely many values d) Probability of rolling a 5 for a fair die is 1/6, thus P(X = 5) = 1/6. 2
3 Example - Discrete and continuous random variables For the following random variables, (i) determine whether they are discrete or continuous, and (ii) indicate the possible values they can take. a. The number of automobiles owned by a family b. The width of your desk in this classroom c. The number of games played in the next World Series d. The weight of model year 2007 SUVs Example continued a. Since the possible number of automobiles owned by a family is finite and may be written as a list of numbers, it represents a discrete random variable. The possible values are {0, 1, 2, 3, 4,...}. Example continued Example 6.3 continued b. Width is something that must be measured, not counted. Width can take infinitely many different possible values, with these values forming an interval on the number line. Thus, the width of your desk is a continuous random variable. The possible values might be 1 ft W 10 ft. c. The number of games played in the next World Series can be counted and thus represents a discrete random variable. The possible values are {4, 5, 6, 7}. 3
4 Example continued d. The weight of model year 2007 SUVs must be measured, not counted, and so represents a continuous random variable. Weight can take infinitely many different possible values in an interval: possible values 2500 lb Y 7000 lb. Discrete Probability Distributions Provides all the possible values that the random variable can assume Together with the probability associated with each value Can take the form of a table, graph, or formula Describe populations, not samples Example - Probability distribution table Construct the probability distribution table of the number of heads observed when tossing a fair coin twice. Example continued The probability distribution table given in Table 6.2 uses probabilities we found in Example 6.4 page 261. Table 6.2 Probability distribution of number of heads on two fair coin tosses 4
5 Requirements Probability Distribution of a Discrete Random Variable The sum of the probabilities of all the possible values of a discrete random variable must equal 1. That is, ΣP(X) = 1. The probability of each value of X must be between 0 and 1, inclusive. That is, 0 P(X ) 1. Mean of a Discrete Random Variable The mean μ of a discrete random variable represents the mean result when the experiment is repeated an indefinitely large number of times Also called the expected value or expectation of the random variable X. Denoted as E(X ) Holds for discrete and continuous random variables Finding the Mean of a Discrete Random Variable Multiply each possible value of X by its probability. Add the resulting products. X P X Example: An auto insurance company pays $5000 to the customer in case of a loss due to theft. If the risk of a loss is assessed to be 3 in every 500 and the insurance company is selling this policy for $100, what is expected profit from each customer? 5
6 Variability of a Discrete Random Variable Formulas for the Variance and Standard Deviation of a Discrete Random Variable Definition Formulas 2 2 X P X 2 X P X Computational Formulas X P X 2 2 X P X 6.2 Binomial Probability Distribution Objectives: By the end of this section, I will be able to 1) Explain what constitutes a binomial experiment. 2) Compute probabilities using the binomial probability formula. 3) Find probabilities using the binomial tables. 4) Calculate and interpret the mean, variance, and standard deviation of the binomial random variable. Binomial Experiment Four Requirements: 1) Each trial of the experiment has only two possible outcomes (success or failure) Notation Table 6.6 Notation for binomial experiments and the binomial distribution 2) Fixed number of trials 3) Experimental outcomes are independent of each other 4) Probability of observing a success remains the same from trial to trial 6
7 Formula for the Number of Combinations The number of combinations of X items chosen from n different items is given by n C X n! X! n X! Binomial Probability Distribution Formula The probability of observing exactly X successes in n trials of a binomial experiment is P(X) = ( n C X ) p X (1 - p) n-x where n! represents n factorial, which equals n(n - 1)(n - 2)... (2)(1); and 0! is defined to be 1. Binomial Distribution Tables n is the number of trials X is the number of successes p is the probability of observing a success Binomial Mean, Variance, and Standard Deviation Mean (or expected value): μ = n p Variance: σ 2 = n p (1 - p) Standard deviation: n p1 p FIGURE: Excerpt from the binomial tables. 7
8 Example - Mean, variance, and standard deviation of left-handed students Suppose we know that the population proportion p of left-handed students is a. In a sample of 200 students, how many would we expect to be left-handed? b. Would 40 left-handed students out of 200 be considered unusual? Example 6.18 continued The binomial random variable here is X = the number of left-handed students. a. Here, n = 200, and p = So the expected number of left-handed students in a sample of 200 is E(X) = μ = n p = (200)(0.10) = 20 Example continued b. To determine whether 40 lefties is unusual: 1. Find the standard deviation of X = the number of lefties in a sample of 200 students. 2. Calculate how many standard deviations 40 lies from the mean μ = 20. Example 6.18 continued The standard deviation of X is n p 1 p
9 Example continued How many standard deviations does 40 lie above the mean of 20? Simply take the difference between 40 and 20, and divide the result by the standard deviation : X Finding 40 lefties in a sample of 200 is unusual because this value lies 4.7 standard deviations above the mean. 6.4 Continuous Random Variables and the Normal Probability Distribution Objectives: By the end of this section, I will be able to 1) Identify a continuous probability distribution. 2) Explain the properties of the normal probability distribution. FIGURE Figure continued (a) Relatively small sample (n = 100) with large class widths (0.5 lb). (c) Very large sample (n = 400) with very small class widths (0.1 lb). (b) Large sample (n = 200) with smaller class widths (0.2 lb). (d) Eventually, theoretical histogram of entire population becomes smooth curve with class widths arbitrarily small. 9
10 Continuous Probability Distributions A graph that indicates on the horizontal axis the range of values that the continuous random variable X can take Density curve is drawn above the horizontal axis Must follow the Requirements for the Probability Distribution of a Continuous Random Variable Requirements 1) The total area under the density curve must equal 1 (this is the Law of Total Probability for Continuous Random Variables). 2) The vertical height of the density curve can never be negative. That is, the density curve never goes below the horizontal axis. Probability Probability for Continuous Distributions is represented by area under the curve above an interval. The Normal Probability Distribution Most important probability distribution in the world Population is said to be normally distributed, the data values follow a normal probability distribution Specific population mean μ Specific population standard deviation σ μ and σ are parameters of the normal distribution 10
11 The normal distribution is symmetric about its mean μ. 6.5 Standard Normal Distribution Objectives: By the end of this section, I will be able to 1) Find areas under the standard normal curve, given a Z-value. 2) Find the standard normal Z-value, given an area. The Standard Normal (Z) Distribution A normal distribution with mean μ = 0 and standard deviation σ = Applications of the Normal Distribution Objectives: By the end of this section, I will be able to 1) Compute probabilities for a given value of any normal random variable. 2) Find the appropriate value of any normal random variable, given an area or probability. 3) Calculate binomial probabilities using the normal approximation to the binomial distribution. 11
12 Standardizing a Normal Random Variable Any normal random variable X can be transformed into the standard normal random variable Z by standardizing using the formula x Z 12
Section 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
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 informationSection M Discrete Probability Distribution
Section M Discrete Probability Distribution A random variable is a numerical measure of the outcome of a probability experiment, so its value is determined by chance. Random variables are typically denoted
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 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 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 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 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 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 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 informationSection Random Variables and Histograms
Section 3.1 - Random Variables and Histograms 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 informationChapter 4 Probability Distributions
Slide 1 Chapter 4 Probability Distributions Slide 2 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 4-5
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability
More informationTheoretical Foundations
Theoretical Foundations Probabilities Monia Ranalli monia.ranalli@uniroma2.it Ranalli M. Theoretical Foundations - Probabilities 1 / 27 Objectives understand the probability basics quantify random phenomena
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 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 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 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 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 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 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 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 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 informationDiscrete Random Variables
Discrete Random Variables MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives During this lesson we will learn to: distinguish between discrete and continuous
More informationDiscrete Random Variables
Discrete Random Variables MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Objectives During this lesson we will learn to: distinguish between discrete and continuous
More informationOverview. Definitions. Definitions. Graphs. Chapter 4 Probability Distributions. probability distributions
Chapter 4 Probability Distributions 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 4-5 The Poisson Distribution
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 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 informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
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 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 informationDiscrete Random Variables
Discrete Random Variables In this chapter, we introduce a new concept that of a random variable or RV. A random variable is a model to help us describe the state of the world around us. Roughly, a RV can
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 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 informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
More informationStatistics 511 Additional Materials
Discrete Random Variables In this section, we introduce the concept of a random variable or RV. A random variable is a model to help us describe the state of the world around us. Roughly, a RV can be thought
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 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 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 informationDistributions in Excel
Distributions in Excel Functions Normal Inverse normal function Log normal Random Number Percentile functions Other distributions Probability Distributions A random variable is a numerical measure of the
More informationChapter 7 1. Random Variables
Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous
More informationContinuous Probability Distributions & Normal Distribution
Mathematical Methods Units 3/4 Student Learning Plan Continuous Probability Distributions & Normal Distribution 7 lessons Notes: Students need practice in recognising whether a problem involves a discrete
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 6 The Normal Distribution 2 Objectives Identify distributions as symmetrical or skewed. Identify the properties of the normal distribution. Find
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 information4: Probability. Notes: Range of possible probabilities: Probabilities can be no less than 0% and no more than 100% (of course).
4: Probability What is probability? The probability of an event is its relative frequency (proportion) in the population. An event that happens half the time (such as a head showing up on the flip of a
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 4 Random Variables & Probability Distributions Content 1. Two Types of Random Variables 2. Probability Distributions for Discrete Random Variables 3. The Binomial
More informationDiscrete Probability Distributions
Page 1 of 6 Discrete Probability Distributions In order to study inferential statistics, we need to combine the concepts from descriptive statistics and probability. This combination makes up the basics
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 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 informationChapter 5 Probability Distributions. Section 5-2 Random Variables. Random Variable Probability Distribution. Discrete and Continuous Random Variables
Chapter 5 Probability Distributions Section 5-2 Random Variables 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance and Standard Deviation for the Binomial Distribution Random
More informationChapter 4 and 5 Note Guide: Probability Distributions
Chapter 4 and 5 Note Guide: Probability Distributions Probability Distributions for a Discrete Random Variable A discrete probability distribution function has two characteristics: Each probability is
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 6 Normal Probability Distributions 6-1 Overview 6-2 The Standard Normal Distribution
More informationProbability Distributions
Chapter 6 Discrete Probability Distributions Section 6-2 Probability Distributions Definitions Let S be the sample space of a probability experiment. A random variable X is a function from the set S into
More informationProbability Distributions
4.1 Probability Distributions Random Variables A random variable x represents a numerical value associated with each outcome of a probability distribution. A random variable is discrete if it has a finite
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 informationChapter 6. The Normal Probability Distributions
Chapter 6 The Normal Probability Distributions 1 Chapter 6 Overview Introduction 6-1 Normal Probability Distributions 6-2 The Standard Normal Distribution 6-3 Applications of the Normal Distribution 6-5
More informationIOP 201-Q (Industrial Psychological Research) Tutorial 5
IOP 201-Q (Industrial Psychological Research) Tutorial 5 TRUE/FALSE [1 point each] Indicate whether the sentence or statement is true or false. 1. To establish a cause-and-effect relation between two variables,
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 information11.5: Normal Distributions
11.5: Normal Distributions 11.5.1 Up to now, we ve dealt with discrete random variables, variables that take on only a finite (or countably infinite we didn t do these) number of values. A continuous random
More information***SECTION 7.1*** Discrete and Continuous Random Variables
***SECTION 7.1*** Discrete and Continuous Random Variables UNIT 6 ~ Random Variables Sample spaces need not consist of numbers; tossing coins yields H s and T s. However, in statistics we are most often
More informationChapter 4. Section 4.1 Objectives. Random Variables. Random Variables. Chapter 4: Probability Distributions
Chapter 4: Probability s 4. Probability s 4. Binomial s Section 4. Objectives Distinguish between discrete random variables and continuous random variables Construct a discrete probability distribution
More information4: Probability. What is probability? Random variables (RVs)
4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random
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 informationMath 14 Lecture Notes Ch. 4.3
4.3 The Binomial Distribution Example 1: The former Sacramento King's DeMarcus Cousins makes 77% of his free throws. If he shoots 3 times, what is the probability that he will make exactly 0, 1, 2, or
More informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
More informationDiscrete Probability Distribution
1 Discrete Probability Distribution Key Definitions Discrete Random Variable: Has a countable number of values. This means that each data point is distinct and separate. Continuous Random Variable: Has
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 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 informationOverview. Definitions. Definitions. Graphs. Chapter 5 Probability Distributions. probability distributions
Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 5-5 The Poisson Distribution
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 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 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 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 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 informationMATH 264 Problem Homework I
MATH Problem Homework I Due to December 9, 00@:0 PROBLEMS & SOLUTIONS. A student answers a multiple-choice examination question that offers four possible answers. Suppose that the probability that the
More informationChapter 5 Discrete Probability Distributions. Random Variables Discrete Probability Distributions Expected Value and Variance
Chapter 5 Discrete Probability Distributions Random Variables Discrete Probability Distributions Expected Value and Variance.40.30.20.10 0 1 2 3 4 Random Variables A random variable is a numerical description
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 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 informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
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 Distribution Unit Review
Probability Distribution Unit Review Topics: Pascal's Triangle and Binomial Theorem Probability Distributions and Histograms Expected Values, Fair Games of chance Binomial Distributions Hypergeometric
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 information4 Random Variables and Distributions
4 Random Variables and Distributions Random variables A random variable assigns each outcome in a sample space. e.g. called a realization of that variable to Note: We ll usually denote a random variable
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 informationX = x p(x) 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6. x = 1 x = 2 x = 3 x = 4 x = 5 x = 6 values for the random variable X
Calculus II MAT 146 Integration Applications: Probability Calculating probabilities for discrete cases typically involves comparing the number of ways a chosen event can occur to the number of ways all
More information30 Wyner Statistics Fall 2013
30 Wyner Statistics Fall 2013 CHAPTER FIVE: DISCRETE PROBABILITY DISTRIBUTIONS Summary, Terms, and Objectives A probability distribution shows the likelihood of each possible outcome. This chapter deals
More informationIntroduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous 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 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 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 informationLecture 8. The Binomial Distribution. Binomial Distribution. Binomial Distribution. Probability Distributions: Normal and Binomial
Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed
More informationContents. The Binomial Distribution. The Binomial Distribution The Normal Approximation to the Binomial Left hander example
Contents The Binomial Distribution The Normal Approximation to the Binomial Left hander example The Binomial Distribution When you flip a coin there are only two possible outcomes - heads or tails. This
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
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 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 informationMath 160 Professor Busken Chapter 5 Worksheets
Math 160 Professor Busken Chapter 5 Worksheets Name: 1. Find the expected value. Suppose you play a Pick 4 Lotto where you pay 50 to select a sequence of four digits, such as 2118. If you select the same
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 informationSection 3.1 Distributions of Random Variables
Section 3.1 Distributions of Random Variables Random Variable A random variable is a rule that assigns a number to each outcome of a chance experiment. There are three types of random variables: 1. Finite
More informationRandom Variables and Probability Functions
University of Central Arkansas Random Variables and Probability Functions Directory Table of Contents. Begin Article. Stephen R. Addison Copyright c 001 saddison@mailaps.org Last Revision Date: February
More informationChapter 4. The Normal Distribution
Chapter 4 The Normal Distribution 1 Chapter 4 Overview Introduction 4-1 Normal Distributions 4-2 Applications of the Normal Distribution 4-3 The Central Limit Theorem 4-4 The Normal Approximation to the
More information