LESSON 9: BINOMIAL DISTRIBUTION
|
|
- Douglas Gibson
- 5 years ago
- Views:
Transcription
1 LESSON 9: Outline The context The properties Notation Formula Use of table Use of Excel Mean and variance 1 THE CONTEXT An important property of the binomial distribution: An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure. Example: Suppose that a production lot contains 100 items. The producer and a buyer agree that if at most 2 out of a sample of 10 items are defective, then all the remaining 90 items in the production lot will be purchased without further testing. Note that each item can be defective or non defective which are two mutually exclusive outcomes of testing. Given the probability that an item is defective, what is the probability that the 90 items will be purchased without further testing? 2 1
2 THE CONTEXT Trial Flip a coin Apply for a job Answer a Multiple choice question Two Mut. Excl. and exhaustive outcomes Head / Tail Get the job / not get the job Correct / Incorrect 3 THE ROERTIES The binomial distribution has the following properties: 1. The experiment consists of a finite number of trials. The number of trials is denoted by n. 2. An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure. 3. The probability of success stays the same for each trial. The probability of success is denoted by p. 4. The trials are independent. 4 2
3 THE NOTATION Notation n : the number of trials r : the number of observed successes p : the probability of success on each trial Note: n-r : the number of observed failures 1- p : the probability of failure on each trial 5 THE ROBABILITY DISTRIBUTION The binomial probability distribution gives the probability of getting exactly r successes out of a total of n trials. The probability of getting exactly r successes out of a total of n trials is as follows: b r; n, π = R = r = ( ) ( ) n r n r C π ( 1 π ) r n! ( ) r ( ) n = π 1 π r r! n r! n Note: In the above C r gives the number of different ways of choosing r objects out of a total of n objects 6 3
4 THE ROBABILITY DISTRIBUTION Example 1: If you toss a fair coin twice, what is the probability of getting one head and one tail? Use the binomial probability distribution formula. 7 THE ROBABILITY DISTRIBUTION Example 2: Redo Example 1 with a probability tree and verify if the probability tree gives the same answer. 8 4
5 THE CUMULATIVE ROBABILITY The cumulative probability gives the probability of getting at most r successes out of a total of n trials. The probability of getting at most r successes out of a total of n trials is as follows: B = ( r; n, π ) = ( R r) r x= 0 ( x; n, π ) Note: An uppercase B(r) is used to distinguish the cumulative probability distribution function from the probability mass function b(r) b 9 THE CUMULATIVE ROBABILITY Example 3: If you toss a fair coin three times, what is the probability of getting at most one head (at least two tails)? 10 5
6 THE CUMULATIVE ROBABILITY Example 4: Redo Example 3 with a probability tree and verify if the probability tree gives the same answer. 11 NECESSITY OF A TABLE OR SOFTWARE Example 5 (do not solve): If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Do not solve this problem, but discuss the computation required by the binomial probability distribution formula. 12 6
7 USE OF TABLE B( r;n,π ) ( R r) Table A, Appendix A, pp gives the probability of getting at most r successes out of a total of n trials, for probability of success in each trial p. The table can be used to find the probability of ( ) ( ) ( ( 1) ) exactly r successes: R = r = R r R r at least r successes: ( R r) = 1 ( R ( r 1) ) successes between a and b: ( a R b) = ( R r ) ( R ( r 1) ) 13 Example 6: Find the following using Table A: ( R 1 n = 5, π = 0.30) ( R 2 n = 5, π = 0.30) ( R 4 n = 10, π = 0.30) Example 7: Find the following using above values ( R = 2 n = 5, π = 0.30) ( R 3 n = 5, π = 0.30) ( 2 R 4 n = 5, π = 0.30) USE OF TABLE 14 7
8 USE OF TABLE Example 8: If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Solve this problem using the Table A. 15 USE OF EXCEL ( ) ( ) The Excel function BINOMDIST gives R r and R = r It takes four arguments. The first 3 arguments are r,n,p The last one is TRUE for ( R r) and FALSE for R = r ( ) Example 9: If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Solve this problem using Excel. Verify if Excel gives the same answer as it is given by Table A in Example 8. Answer: ( R 20 n = 50, π = 0.50) =BINOMDIST(20,50,0.5,TRUE) 16 8
9 MEAN AND VARIANCE The expected value and variance for the number of successes R may be computed as follows: E ( R) = nπ ( R) = nπ ( 1 π ) Var E(R) is the mean or expected value of R Var(X) is the variance of R n is the number of trials p is the probability of success on each trial The probability of failure on each trial = 1- p 17 MEAN AND VARIANCE Example 10: Let R be a random variable that gives number of heads when a fair coin is tossed 4 times. Compute E(R) and Var(R). 18 9
10 READING AND EXERCISES Lesson 9 Reading: Section 7-3, pp Exercises: 7-22, 7-24,
Statistics 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 informationx is a random variable which is a numerical description of the outcome of an experiment.
Chapter 5 Discrete Probability Distributions Random Variables is a random variable which is a numerical description of the outcome of an eperiment. Discrete: If the possible values change by steps or jumps.
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 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 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 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 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 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 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 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 informationDiscrete Probability Distributions
Discrete Probability Distributions Discrete Probability Distribution Are used to model outcomes that only have a finite number of possible values. For example, the number of congenitally missing third
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 informationLECTURE CHAPTER 3 DESCRETE RANDOM VARIABLE
LECTURE CHAPTER 3 DESCRETE RANDOM VARIABLE MSc Đào Việt Hùng Email: hungdv@tlu.edu.vn Random Variable A random variable is a function that assigns a real number to each outcome in the sample space of a
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 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 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 informationProbability mass function; cumulative distribution function
PHP 2510 Random variables; some discrete distributions Random variables - what are they? Probability mass function; cumulative distribution function Some discrete random variable models: Bernoulli Binomial
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 informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 3: Special Discrete Random Variable Distributions Section 3.5 Discrete Uniform Section 3.6 Bernoulli and Binomial Others sections
More informationLearning Objec0ves. Statistics for Business and Economics. Discrete Probability Distribu0ons
Statistics for Business and Economics Discrete Probability Distribu0ons Learning Objec0ves In this lecture, you learn: The proper0es of a probability distribu0on To compute the expected value and variance
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 informationThe binomial distribution p314
The binomial distribution p314 Example: A biased coin (P(H) = p = 0.6) ) is tossed 5 times. Let X be the number of H s. Fine P(X = 2). This X is a binomial r. v. The binomial setting p314 1. There are
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 5. Discrete Probability Distributions. Random Variables
Chapter 5 Discrete Probability Distributions Random Variables x is a random variable which is a numerical description of the outcome of an experiment. Discrete: If the possible values change by steps or
More informationguessing Bluman, Chapter 5 2
Bluman, Chapter 5 1 guessing Suppose there is multiple choice quiz on a subject you don t know anything about. 15 th Century Russian Literature; Nuclear physics etc. You have to guess on every question.
More informationProbability Distributions. Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution
Probability Distributions Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution Definitions Random Variable: a variable that has a single numerical value
More informationBinomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
More information15.063: Communicating with Data Summer Recitation 3 Probability II
15.063: Communicating with Data Summer 2003 Recitation 3 Probability II Today s Goal Binomial Random Variables (RV) Covariance and Correlation Sums of RV Normal RV 15.063, Summer '03 2 Random Variables
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 informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 4: Special Discrete Random Variable Distributions Sections 3.7 & 3.8 Geometric, Negative Binomial, Hypergeometric NOTE: The discrete
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 informationCHAPTER 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
CHAPTER 8 PROBABILITY DISTRIBUTIONS AND STATISTICS 8.1 Distribution of Random Variables Random Variable Probability Distribution of Random Variables 8.2 Expected Value Mean Mean is the average value of
More informationSTAT Mathematical Statistics
STAT 6201 - Mathematical Statistics Chapter 3 : Random variables 5, Event, Prc ) Random variables and distributions Let S be the sample space associated with a probability experiment Assume that we have
More informationLesson 97 - Binomial Distributions IBHL2 - SANTOWSKI
Lesson 97 - Binomial Distributions IBHL2 - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin 3 times where P(H) = / (b) THUS, find the probability
More informationOpening Exercise: Lesson 91 - Binomial Distributions IBHL2 - SANTOWSKI
08-0- Lesson 9 - Binomial Distributions IBHL - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin times where P(H) = / (b) THUS, find the probability
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 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 informationDiscrete Probability Distributions
5 Discrete Probability Distributions 5-3 Binomial Probability Distributions 5-5 Poisson Probability Distributions 52 Chapter 5: Discrete Probability Distributions 5-3 Binomial Probability Distributions
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 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 informationThe Multistep Binomial Model
Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test Friday 9th March - 12pm Examples Sheet 1 4 (not qu 3 or qu 5 on sheet 4) Lectures 1-9 10.1 A Discrete Model for Stock Price Reminder: The
More information6 If and then. (a) 0.6 (b) 0.9 (c) 2 (d) Which of these numbers can be a value of probability distribution of a discrete random variable
1. A number between 0 and 1 that is use to measure uncertainty is called: (a) Random variable (b) Trial (c) Simple event (d) Probability 2. Probability can be expressed as: (a) Rational (b) Fraction (c)
More information300 total 50 left handed right handed = 250
Probability Rules 1. There are 300 students at a certain school. All students indicated they were either right handed or left handed but not both. Fifty of the students are left handed. How many students
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 informationBinomial and multinomial distribution
1-Binomial distribution Binomial and multinomial distribution The binomial probability refers to the probability that a binomial experiment results in exactly "x" successes. The probability of an event
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 informationStat511 Additional Materials
Binomial Random Variable Stat511 Additional Materials The first discrete RV that we will discuss is the binomial random variable. The binomial random variable is a result of observing the outcomes from
More informationCentral Limit Theorem 11/08/2005
Central Limit Theorem 11/08/2005 A More General Central Limit Theorem Theorem. Let X 1, X 2,..., X n,... be a sequence of independent discrete random variables, and let S n = X 1 + X 2 + + X n. For each
More informationCD Appendix F Hypergeometric Distribution
D Appendix F Hypergeometric Distribution A hypergeometric experiment is an experiment where a sample of n items is taen without replacement from a finite population of items, each of which is classified
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 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 informationDiscrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8)
3 Discrete Random Variables and Probability Distributions Stat 4570/5570 Based on Devore s book (Ed 8) Random Variables We can associate each single outcome of an experiment with a real number: We refer
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 informationLecture 7 Random Variables
Lecture 7 Random Variables Definition: A random variable is a variable whose value is a numerical outcome of a random phenomenon, so its values are determined by chance. We shall use letters such as X
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 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 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 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 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 information6.3: The Binomial Model
6.3: The Binomial Model The Normal distribution is a good model for many situations involving a continuous random variable. For experiments involving a discrete random variable, where the outcome of the
More information6. THE BINOMIAL DISTRIBUTION
6. THE BINOMIAL DISTRIBUTION Eg: For 1000 borrowers in the lowest risk category (FICO score between 800 and 850), what is the probability that at least 250 of them will default on their loan (thereby rendering
More informationSection Random Variables
Section 6.2 - Random Variables According to the Bureau of the Census, the latest family data pertaining to family size for a small midwestern town, Nomore, is shown in Table 6.. If a family from this town
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 information1.15 (5) a b c d e. ?? (5) a b c d e. ?? (5) a b c d e. ?? (5) a b c d e FOR GRADER S USE ONLY: DEF T/F ?? M.C.
Name: M339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin The Prerequisite In-Term Exam Instructor: Milica Čudina Notes: This is a closed book and closed notes exam.
More informationOCR Statistics 1. Discrete random variables. Section 2: The binomial and geometric distributions. When to use the binomial distribution
Discrete random variables Section 2: The binomial and geometric distributions Notes and Examples These notes contain subsections on: When to use the binomial distribution Binomial coefficients Worked examples
More informationAP Statistics Ch 8 The Binomial and Geometric Distributions
Ch 8.1 The Binomial Distributions The Binomial Setting A situation where these four conditions are satisfied is called a binomial setting. 1. Each observation falls into one of just two categories, which
More informationThe Binomial Distribution
Patrick Breheny September 13 Patrick Breheny University of Iowa Biostatistical Methods I (BIOS 5710) 1 / 16 Outcomes and summary statistics Random variables Distributions So far, we have discussed the
More informationMATH 118 Class Notes For Chapter 5 By: Maan Omran
MATH 118 Class Notes For Chapter 5 By: Maan Omran Section 5.1 Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Ex1: The test scores
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 informationAP Statistics Test 5
AP Statistics Test 5 Name: Date: Period: ffl If X is a discrete random variable, the the mean of X and the variance of X are given by μ = E(X) = X xp (X = x); Var(X) = X (x μ) 2 P (X = x): ffl If X is
More informationChapter 3 Class Notes Intro to Probability
Chapter 3 Class Notes Intro to Probability Concept: role a fair die, then: what is the probability of getting a 3? Getting a 3 in one roll of a fair die is called an Event and denoted E. In general, Number
More information23.1 Probability Distributions
3.1 Probability Distributions Essential Question: What is a probability distribution for a discrete random variable, and how can it be displayed? Explore Using Simulation to Obtain an Empirical Probability
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 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 informationProbability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
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 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 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 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 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 informationExperimental Probability - probability measured by performing an experiment for a number of n trials and recording the number of outcomes
MDM 4U Probability Review Properties of Probability Experimental Probability - probability measured by performing an experiment for a number of n trials and recording the number of outcomes Theoretical
More informationExpected Value and Variance
Expected Value and Variance MATH 472 Financial Mathematics J Robert Buchanan 2018 Objectives In this lesson we will learn: the definition of expected value, how to calculate the expected value of a random
More informationSTOR Lecture 7. Random Variables - I
STOR 435.001 Lecture 7 Random Variables - I Shankar Bhamidi UNC Chapel Hill 1 / 31 Example 1a: Suppose that our experiment consists of tossing 3 fair coins. Let Y denote the number of heads that appear.
More informationChapter 3. Discrete Probability Distributions
Chapter 3 Discrete Probability Distributions 1 Chapter 3 Overview Introduction 3-1 The Binomial Distribution 3-2 Other Types of Distributions 2 Chapter 3 Objectives Find the exact probability for X successes
More informationDiscrete Random Variables and Probability Distributions
Chapter 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables A quantity resulting from an experiment that, by chance, can assume different values. A random variable is a variable
More informationS = 1,2,3, 4,5,6 occurs
Chapter 5 Discrete Probability Distributions The observations generated by different statistical experiments have the same general type of behavior. Discrete random variables associated with these experiments
More informationProbability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 2 Discrete Distributions The binomial distribution 1 Chapter 2 Discrete Distributions Bernoulli trials and the
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 informationLaw of Large Numbers, Central Limit Theorem
November 14, 2017 November 15 18 Ribet in Providence on AMS business. No SLC office hour tomorrow. Thursday s class conducted by Teddy Zhu. November 21 Class on hypothesis testing and p-values December
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 informationBinomial distribution
Binomial distribution Jon Michael Gran Department of Biostatistics, UiO MF9130 Introductory course in statistics Tuesday 24.05.2010 1 / 28 Overview Binomial distribution (Aalen chapter 4, Kirkwood and
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 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 informationMath489/889 Stochastic Processes and Advanced Mathematical Finance Homework 5
Math489/889 Stochastic Processes and Advanced Mathematical Finance Homework 5 Steve Dunbar Due Fri, October 9, 7. Calculate the m.g.f. of the random variable with uniform distribution on [, ] and then
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 information7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4
7. For the table that follows, answer the following questions: x y 1-1/4 2-1/2 3-3/4 4 - Would the correlation between x and y in the table above be positive or negative? The correlation is negative. -
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 informationINF FALL NATURAL LANGUAGE PROCESSING. Jan Tore Lønning, Lecture 3, 1.9
INF5830 015 FALL NATURAL LANGUAGE PROCESSING Jan Tore Lønning, Lecture 3, 1.9 Today: More statistics Binomial distribution Continuous random variables/distributions Normal distribution Sampling and sampling
More informationEconomics 430 Handout on Rational Expectations: Part I. Review of Statistics: Notation and Definitions
Economics 430 Chris Georges Handout on Rational Expectations: Part I Review of Statistics: Notation and Definitions Consider two random variables X and Y defined over m distinct possible events. Event
More information