Section Random Variables and Histograms
|
|
- Dwayne Walker
- 5 years ago
- Views:
Transcription
1 Section 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 define the random variable X to represent the number of times we get tails. Example 2: Suppose we roll a die until a 5 is facing up. Then we could define the random variable Y to represent the number of times we rolled the die. Example 3: Suppose a flashlight is left on until the battery runs out. Then we could define the random variable Z to represent the amount of time that passed. Types of Random Variables 1. A finite discrete random variable is one which can only take on a limited number of values that can be listed. 2. An infinite discrete random variable is one which can take on an unlimited number of values that can be listed in some sort of sequence. 3. A continuous random variable is one which takes on any of the infinite number of values in some interval of real numbers. (i.e. usually measurements) Example 4: Classify each of the following random variables as finite discrete, infinite discrete, or continuous and list/describe the possible values for the random variable. a) A drawer contains 15 red pens, 8 blue pens, and 5 black pens. An experiment consists of drawing pens out of the drawer with replacement until a red pen is drawn. Let X represent the number of pens drawn. b) An experiment consists of randomly selecting 3 pens without replacement from a conveyor belt. Let Y represent the total weight (in ounces) of the selected pens. c) An experiment consists of randomly selecting 3 pens with replacement from a drawer that contains 15 red pens, 8 blue pens, and 5 black pens. Let Z represent the number of red pens drawn. 1
2 Example 5: Referring to Example 1, find the probability distribution of the random variable X. Example 6: An experiment consists of randomly selecting a sample of 3 grapes out of a bowl that contains 20, of which 8 are rotten. Let X represent the number of rotten grapes in the sample. Find the probability distribution of X. Example 7: Blue Baker prides itself on having the best chocolate chips cookies in town. To be sure each cookie has the right number of chocolate chips, a few cookies are selected from each batch and the number of chocolate chips in each cookie is counted. This is done for several days and the following results were found: Number of Cookies Number of Chocolate Chips Identify the random variable X in this experiment and find the probability distribution of X. 2
3 We use histograms to represent the probability distributions of random variables. We place the possible values for the random variable X on the horizontal axis. We then center a bar around each x value and let its height be equal to the probability of that x value. Example 8: Referring to Example 7, a) Draw the histogram for the random variable X b) Find P(X 12) Definition: Given a sequence of n Bernoulli trials with the probability of success p the Binomial (Bernoulli) Distribution is given by P(X = r) = C(n,r)p r (1 p) (n r) for r = 0...n Example 9: A fair six-sided die is rolled 8 times. Let X represent the number of times the die lands on a number greater than 4. Find the probability distribution of X. Section 3.1 Homework Problems: 1-9 (odd), 13, 15, 19, 21, 23, 29 3
4 Section Measures of Central Tendency Example 1: Records kept by the cheif dietitian at the university caferteria over a 25-wk period show the following weekly consumption of milk (in gallons): Milk Weeks Find the average number of gallons of milk consumed per week in the cafeteria. Definition: Let X denote a random variable that assumes the values x 1,x 2,...,x n, with associated probabilites p 1, p 2,..., p n, respectively. Then the expected value of X, E(X), is given by: E(X) = x 1 p 1 + x 2 p 2 + x n p n Example 2: Referring to Example 1, let the random variable X denote the number of gallons of milk consumed in a week at the cafeteria. a) Find the probability distribution of X. b) Compute E(X) c) Draw a histogram for the random variable X Note: If we think of placing the histogram on a see-saw, the expected value occurs where we would put the fulcrum to balance it. 4
5 Example 3: In a lottery, 3000 tickets are sold for $1 each. One first prize of $5000, 1 second prize of $1000, 3 third prizes of $100, and 10 consolation prizes of $10 are to be awarded. What are the expected net earnings of a person who buys one ticket? Example 4: A woman purchased a $15,000 1-year term-life inusrance policy for $200. Assuming that the probability that she will live another year is 0.994, find the company s expected net gain. Definition: A fair game means that the expected value of the player s net winnings is zero. (E(X) = 0) Example 5: Mike and Bill play a card game with a standard deck of 52 cards. Mike selects a card from a well-shuffled deck and receives A dollars from Bill if the card selected is a diamond; otherwise, Mike pays Bill a dollar. Determine the value of A if the game is to be fair. 5
6 Example 6: If you roll a fair six-sided die 4 times, what is the expected number of times that the die will land with a two facing up? Definition: The expected value of a binomial distribution with n trials and probability of success p is Example 7: If you roll a fair six-sided die 100 times, what is the expected number of times that the die will land on a number less than three? Definition: The median is the middle value in a set of data arranged in increasing or decreasing order (when there is an odd number of entries). If there is an even number of entries, the median is the average of the two middle numbers. Definition The mode is the value that occurs most frequently in a set of data. Example 8: In each of the following cases, find the mean, median, and mode. Section 3.2 Homework Problems: 1, 5, 9, 11, 15, 17, 25, 29, 35, 37, 43 6
7 Section Measures of Spread Example 1: Draw the histograms for the random variables X and Y that have the following probability distributions: x P(X = x) y P(Y = y) Definition: Suppose a random variable X has the following probability distribution: x P(X = x) x 1 p 1 x 2 p 2 x n p n and expected value E(X) = µ. Then the variance of the random variable X is Var(X)=p 1 (x 1 µ) 2 + p 2 (x 2 µ) p n (x n µ) 2. Definition: The standard deviation of a random variable X, σ, is defined by: σ = Var(X) 7
8 Finding expected value and standard deviation using the calculator: Enter the x-values into L1 and the corresponding probabilities into L2 (STAT 1:Edit) On the homescreen type 1-Var Stats L 1, L 2 (STAT CALC 1:1-Var Stats) x is the expected value (mean) σx is the standard deviation To find variance, you would need to recall that Var(X) = σ 2 Example 2: Referring to Example 1, a) Find the variance and standard deviation of X. b) Find the variance and standard deviation of Y. Example 3: The percent of the voting age population who cast ballots in presidential election years from 1980 through 2000 are given below: Election Year Turnout Percentage Find the mean and standard deviation of the voter turnout. Example 4: If you roll a fair six-sided die 4 times, what is the variance and standard deviation of the number of times the die lands on a two? Definition: The variance of a binomial distribution with n trials and the probability of success equal to p is Example 5: If you roll a fair six-sided die 100 times, what is the variance and standard deviation of the number of times that the die will land on a number less than three? 8
9 Chebyshev s Inequality: Let X be a random variable with expected value µ and standard deviation σ. Then, the probability that a randomly chosen outcome of the experiment lies between µ kσ and µ + kσ is at least 1 1 k 2 ; that is: P(µ kσ X µ + kσ) 1 1 k 2 Example 6: A probability distribution has a mean of 10 and a standard deviation of 1.5. Use Chebyshev s inequality to estimate the probability that an outcome of the experiment lies between 7 and 13 Example 7: A probability distribution has a mean of 70 and a standard deviation of 3.2. Use Chebyshev s Inequality to find the value of c that guarantees that the probability is at least that an outcome of the experiment lies between 70 c and 70 + c. Section 3.3 Homework Problems: 1, 3, 7, 9, 13, 15, 21, 25, 33, 35, 37 9
10 Section The Normal Distribution Up until now, we have been dealing with finite discrete random variables. In finding the probability distribution, we could list the possible values in a table and represent it with a histogram. Definition: For a continuous random variable, a probability density function is defined to represent the probability distribution. Example 1: Note that the for a continous random variable, X, P(X x) = P(X < x) Definition: We concentrate on a special class of continuous probability distributions known as normal distributions. Each normal distribution is defined by µ and σ. Each normal distribution has the following characteristics: 1. The area under the curve is always The curve never crosses the x axis. 3. The peak occurs directly above µ 4. The curve is symmetric about a vertical line passing through the mean. Example 2: Definition: The standard normal variable usually denoted by Z has a normal probability distribution with µ = 0 and σ = 1. 10
11 Example 3: Find and sketch the following: a) P(Z 1.79) b) P(Z 3.49) c) P( 2 Z 1.79) Example 4: According to the data released by the Chamber of Commerce of a certain city, the weekly wages of factory workers are normally distributed with a mean of $600 and a standard deviation of $50. What is the probability that a worker selected at random from the city makes a weekly wage a) of less than $600? b) of more than $760? c) between $575 and $650? 11
12 Example 5: Let Z be the standard normal variable. Find the values of a if a satisfies: a) P(Z a) = b) P(Z a) = c) P( a Z a) = Example 6: The scores on an Econ exam were normally distributed with a mean of 72 and a standard deviation of 16. If the instructor assigns a grade of A to 8% of the class, a grade of B to 12% of the class, and a grade of C to 25% of the class, what is the lowest score a student may have and still obtain a C? Section 3.4 Homework Problems: 1, 5, 9, 13, 17, 21-33(odd) 12
Section 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 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 informationChapter 3: Probability Distributions and Statistics
Chapter 3: Probability Distributions and Statistics Section 3.-3.3 3. Random Variables and Histograms A is a rule that assigns precisely one real number to each outcome of an experiment. We usually denote
More informationSection 8.1 Distributions of Random Variables
Section 8.1 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 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 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 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 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 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 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 informationSection 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 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 informationName: Date: Pd: Quiz Review
Name: Date: Pd: Quiz Review 8.1-8.3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A die is cast repeatedly until a 1 falls uppermost. Let the 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 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 8 Homework Solutions Compiled by Joe Kahlig. speed(x) freq 25 x < x < x < x < x < x < 55 5
H homework problems, C-copyright Joe Kahlig Chapter Solutions, Page Chapter Homework Solutions Compiled by Joe Kahlig. (a) finite discrete (b) infinite discrete (c) continuous (d) finite discrete (e) 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 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 informationAP Statistics Review Ch. 6
AP Statistics Review Ch. 6 Name 1. Which of the following data sets is not continuous? a. The gallons of gasoline in a car. b. The time it takes to commute in a car. c. Number of goals scored by a hockey
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 informationStatistics, Measures of Central Tendency I
Statistics, Measures of Central Tendency I We are considering a random variable X with a probability distribution which has some parameters. We want to get an idea what these parameters are. We perfom
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 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 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 informationCH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
More informationMath Week in Review #10. Experiments with two outcomes ( success and failure ) are called Bernoulli or binomial trials.
Math 141 Spring 2006 c Heather Ramsey Page 1 Section 8.4 - Binomial Distribution Math 141 - Week in Review #10 Experiments with two outcomes ( success and failure ) are called Bernoulli or binomial trials.
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 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 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 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 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 informationProblem Set 07 Discrete Random Variables
Name Problem Set 07 Discrete Random Variables MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean of the random variable. 1) The random
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 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 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 informationSection 8.1 Distributions of Random Variables
Section 8.1 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 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 informationMath 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Ruomeng Lan Texas A&M University October 15, 2014 Ruomeng Lan (TAMU) Math 166 October 15, 2014 1 / 12 Mean, Median and Mode Definition: 1. The average or
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
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 informationFINAL REVIEW W/ANSWERS
FINAL REVIEW W/ANSWERS ( 03/15/08 - Sharon Coates) Concepts to review before answering the questions: A population consists of the entire group of people or objects of interest to an investigator, while
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 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 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 informationA.REPRESENTATION OF DATA
A.REPRESENTATION OF DATA (a) GRAPHS : PART I Q: Why do we need a graph paper? Ans: You need graph paper to draw: (i) Histogram (ii) Cumulative Frequency Curve (iii) Frequency Polygon (iv) Box-and-Whisker
More informationChapter 8 Homework Solutions Compiled by Joe Kahlig
homewk problems, B-copyright Joe Kahlig Chapter Solutions, Page Chapter omewk Solutions Compiled by Joe Kahlig 0. 0. 0. 0.. You are counting the number of games and there are a limited number of games
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 informationExample 1: Identify the following random variables as discrete or continuous: a) Weight of a package. b) Number of students in a first-grade classroom
Section 5-1 Probability Distributions I. Random Variables A variable x is a if the value that it assumes, corresponding to the of an experiment, is a or event. A random variable is if it potentially 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 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 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 informationHonors Statistics. Daily Agenda
Honors Statistics Aug 23-8:26 PM Daily Agenda Aug 23-8:31 PM 1 Write a program to generate random numbers. I've decided to give them free will. A Skip 4, 12, 16 Apr 25-10:55 AM Toss 4 times Suppose you
More informationHave you ever wondered whether it would be worth it to buy a lottery ticket every week, or pondered on questions such as If I were offered a choice
Section 8.5: Expected Value and Variance Have you ever wondered whether it would be worth it to buy a lottery ticket every week, or pondered on questions such as If I were offered a choice between a million
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 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 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 informationSection Introduction to Normal Distributions
Section 6.1-6.2 Introduction to Normal Distributions 2012 Pearson Education, Inc. All rights reserved. 1 of 105 Section 6.1-6.2 Objectives Interpret graphs of normal probability distributions Find areas
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 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 informationCounting Basics. Venn diagrams
Counting Basics Sets Ways of specifying sets Union and intersection Universal set and complements Empty set and disjoint sets Venn diagrams Counting Inclusion-exclusion Multiplication principle Addition
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 informationExpectations. Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or
Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or µ X, is E(X ) = µ X = x D x p(x) Definition Let X be a discrete
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 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 informationVIDEO 1. A random variable is a quantity whose value depends on chance, for example, the outcome when a die is rolled.
Part 1: Probability Distributions VIDEO 1 Name: 11-10 Probability and Binomial Distributions A random variable is a quantity whose value depends on chance, for example, the outcome when a die is rolled.
More informationPart 10: The Binomial Distribution
Part 10: The Binomial Distribution The binomial distribution is an important example of a probability distribution for a discrete random variable. It has wide ranging applications. One readily available
More informationFall 2015 Math 141:505 Exam 3 Form A
Fall 205 Math 4:505 Exam 3 Form A Last Name: First Name: Exam Seat #: UIN: On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work Signature: INSTRUCTIONS Part
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 informationIn a binomial experiment of n trials, where p = probability of success and q = probability of failure. mean variance standard deviation
Name In a binomial experiment of n trials, where p = probability of success and q = probability of failure mean variance standard deviation µ = n p σ = n p q σ = n p q Notation X ~ B(n, p) The probability
More informationAP Statistics Section 6.1 Day 1 Multiple Choice Practice. a) a random variable. b) a parameter. c) biased. d) a random sample. e) a statistic.
A Statistics Section 6.1 Day 1 ultiple Choice ractice Name: 1. A variable whose value is a numerical outcome of a random phenomenon is called a) a random variable. b) a parameter. c) biased. d) a random
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 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. 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 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 informationSTT315 Chapter 4 Random Variables & Probability Distributions AM KM
Before starting new chapter: brief Review from Algebra Combinations In how many ways can we select x objects out of n objects? In how many ways you can select 5 numbers out of 45 numbers ballot to win
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 informationMANAGEMENT PRINCIPLES AND STATISTICS (252 BE)
MANAGEMENT PRINCIPLES AND STATISTICS (252 BE) Normal and Binomial Distribution Applied to Construction Management Sampling and Confidence Intervals Sr Tan Liat Choon Email: tanliatchoon@gmail.com Mobile:
More informationChapter 8. Variables. Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 8 Random Variables Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc. 8.1 What is a Random Variable? Random Variable: assigns a number to each outcome of a random circumstance, or,
More informationAMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Summer 2014 1 / 26 Sampling Distributions!!!!!!
More informationSTOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics (Chap 5) Lecture 14: Sampling Distributions for Counts and Proportions 5/31/11 Lecture 14 1 Statistic & Its Sampling Distribution
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 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 informationExam II Math 1342 Capters 3-5 HCCS. Name
Exam II Math 1342 Capters 3-5 HCCS Name Date Provide an appropriate response. 1) A single six-sided die is rolled. Find the probability of rolling a number less than 3. A) 0.5 B) 0.1 C) 0.25 D 0.333 1)
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 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 informationHHH HHT HTH THH HTT THT TTH TTT
AP Statistics Name Unit 04 Probability Period Day 05 Notes Discrete & Continuous Random Variables Random Variable: Probability Distribution: Example: A probability model describes the possible outcomes
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 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 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 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 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 informationTest 7A AP Statistics Name: Directions: Work on these sheets.
Test 7A AP Statistics Name: Directions: Work on these sheets. Part 1: Multiple Choice. Circle the letter corresponding to the best answer. 1. Suppose X is a random variable with mean µ. Suppose we observe
More informationHonors Statistics. Daily Agenda
Honors Statistics Aug 23-8:26 PM Daily Agenda 1. Review OTL C6#4 Chapter 6.2 rules for means and variances Aug 23-8:31 PM 1 Nov 21-8:16 PM Working out Choose a person aged 19 to 25 years at random and
More informationEXERCISES FOR PRACTICE SESSION 2 OF STAT CAMP
EXERCISES FOR PRACTICE SESSION 2 OF STAT CAMP Note 1: The exercises below that are referenced by chapter number are taken or modified from the following open-source online textbook that was adapted by
More informationThe Binomial Distribution
MATH 382 The Binomial Distribution Dr. Neal, WKU Suppose there is a fixed probability p of having an occurrence (or success ) on any single attempt, and a sequence of n independent attempts is made. Then
More 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 informationChapter 7. Random Variables: 7.1: Discrete and Continuous. Random Variables. 7.2: Means and Variances of. Random Variables
Chapter 7 Random Variables In Chapter 6, we learned that a!random phenomenon" was one that was unpredictable in the short term, but displayed a predictable pattern in the long run. In Statistics, we are
More informationList of Online Quizzes: Quiz7: Basic Probability Quiz 8: Expectation and sigma. Quiz 9: Binomial Introduction Quiz 10: Binomial Probability
List of Online Homework: Homework 6: Random Variables and Discrete Variables Homework7: Expected Value and Standard Dev of a Variable Homework8: The Binomial Distribution List of Online Quizzes: Quiz7:
More informationThe Normal Distribution
5.1 Introduction to Normal Distributions and the Standard Normal Distribution Section Learning objectives: 1. How to interpret graphs of normal probability distributions 2. How to find areas under the
More information