Note: A probability distribution with a small (large) spread about its mean will have a small (large) variance.

Size: px
Start display at page:

Download "Note: A probability distribution with a small (large) spread about its mean will have a small (large) variance."

Transcription

1 Section 8.3: Variance and Standard Deviation The Variance of a random variable X is the measure of degree of dispersion, or spread, of a probability distribution about its mean (i.e. how much on average each of the values of X deviates from the mean). Note: A probability distribution with a small (large) spread about its mean will have a small (large) variance. Variance of a Random Variable X Suppose a random variable has the probability distribution ( = ) and expected value ( ) =. Then the variance of the random variable X is ( )= ( ) + ( ) + + ( ) Note: We square each since some may be negative. Standard Deviation measures the same thing as the variance. The standard deviation of a Random Variable X is = Var( )= ( ) + ( ) + + ( ) Example 1: Compute the mean, variance and standard deviation of the random variable X with probability distribution as follows: X P(X=x)

2 Example 2: An investor is considering two business ventures. The anticipated returns (in thousands of dollars) of each venture are described by the following probability distributions: Venture 1 Venture 2 Earnings Probability Earning Probability a. Compute the mean and variance for each venture. b. Which investment would provide the investor with the higher expected return (the greater mean)? c. Which investment would the element of risk be less (that is, which probability distribution has the smaller variance)? Popper 1: The probability distribution of a random variable X is given below. Given the mean = 5.2 X P(X=x) 1/3 1/10 2/15 7/30 1/5 Find the variance. a b. 42 c d

3 Chebychev s Inequality Let X be a random variable with expected value and standard deviation. Then, the probability that a randomly chosen outcome of the experiment lies between and + is at least 1 ; that is, ( + ) 1 Let s see what this is saying: Example 3: A probability distribution has a mean 20 and a standard deviation of 3. Use Chebychev s Inequality to estimate the probability that an outcome of the experiment lies between 12 and 28. 3

4 Example 4: A light bulb has an expected life of 200 hours and a standard deviation of 2 hours. Use Chebychev s Inequality to estimate the probability that one of these light bulbs will last between 190 and 210 hours? Popper 2: A light bulb at an art museum has an expected life of 300 hours and a standard deviation of 12 hours. Use Chebychev's Inequality to estimate the probability that one of these light bulbs will last between 280 and 320 hours of use. a b c d

5 Math 1313 Section 8.4 Section 8.4: The Binomial Distribution A binomial experiment has the following properties: 1. Number of trials is fixed. 2. There are 2 outcomes of the experiment. Success, probability denoted by, and failure, probability denoted by. Note + =1 3. The probability of success in each trial is the same. 4. The trials are independent of each other. Experiments with two outcomes are called Bernoulli trials or Binomial trials. Finding the Probability of an Event of a Binomial Experiment: In a binomial experiment in which the probability of success in any trial is p, the probability of exactly x successes in n independent trials is given by = =, X is called a binomial random variable and its probability distribution is called a binomial probability distribution. Example 1 on page 505 derives this formula. Example 1: Consider the following binomial experiment. A fair die is cast four times. Compute the probability of obtaining exactly one 6 in the four throws. 1

6 Math 1313 Section 8.4 Example 2: Let the random variable X denote the number of girls in a five-child family. If the probability of a female birth is 0.5, construct the binomial distribution associated with this experiment. What is the probability of having at least one girl? Example 3: Consider the following binomial experiment. If the probability that a marriage will end in divorce within 20 years after its start is 0.6, what is the probability that out of 6 couples just married, in the next 20 years all will be divorced? a. None will be divorced? b. Exactly two couples will be divorced? c. At least two couples will be divorced? 2

Math 1313 The Binomial Distribution. Binomial Experiments

Math 1313 The Binomial Distribution. Binomial Experiments Math 1313 The Binomial Distribution Binomial Experiments The remainder of the course material will be presentation of two very important probability distributions and the types of problems that we can

More information

a. next 20 years all will be divorced? b. None will be divorced? c. Exactly two couples will be divorced? d. At least two couples will be divorced?

a. next 20 years all will be divorced? b. None will be divorced? c. Exactly two couples will be divorced? d. At least two couples will be divorced? Math 1313 Section 8.4 Example 3: Consider the following binomial experiment. If the probability that a marriage will end in divorce within 20 years after its start is 0.6, what is the probability that

More information

Probability Distributions and Statistics

Probability Distributions and Statistics 8 Probability Distributions and Statistics Distributions of Random Variables Expected Value Variance and Standard Deviation Binomial Distribution Normal Distribution Applications of the Normal Distribution

More information

Math 1313 Finite Math Test 4 Supplemental Review

Math 1313 Finite Math Test 4 Supplemental Review Math 1313 Finite Math Test 4 Supplemental Review 1. Let A and B be events in a sample space S. Suppose that P(A) =.45, P(B) =.38, and P( A B) =. 21. Find each of the following: a. P( A B) b. P( B A) c

More information

WHERE DOES THE 10% CONDITION COME FROM?

WHERE DOES THE 10% CONDITION COME FROM? 1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

More information

Week in Review #6 ( , )

Week in Review #6 ( , ) Math 66 Week-in-Review - S. Nite 0/3/0 Page of 6 Week in Review #6 (.-.4, 3.-3.4) n( E) In general, the probability of an event is P ( E) =. n( S) The Multiplication Principle: Suppose there are m ways

More information

Stat 104 Lecture 13. Statistics 104 Lecture 13 (IPS 4.3, 4.4 & 5.1) Random variables

Stat 104 Lecture 13. Statistics 104 Lecture 13 (IPS 4.3, 4.4 & 5.1) Random variables Statistics 104 Lecture 13 (IPS 4.3, 4.4 & 5.1) Random variables A random variable is a variable whose value is a numerical outcome of a random phenomenon Usually denoted by capital letters, Y or Z Example:

More information

Random Variable: A variable whose value is the numerical outcome of an experiment or random phenomenon.

Random Variable: A variable whose value is the numerical outcome of an experiment or random phenomenon. STAT 515 -- Chapter 4: Discrete Random Variables Random Variable: A variable whose value is the numerical outcome of an experiment or random phenomenon. Discrete Random Variable : A numerical r.v. that

More information

M 1313 Review Test 4 1

M 1313 Review Test 4 1 M 1313 Review Test 4 1 Review for test 4: 1. Let E and F be two events of an experiment, P (E) =. 3 and P (F) =. 2, and P (E F) =.35. Find the following probabilities: a. P(E F) b. P(E c F) c. P (E F)

More information

Chapter 6 The Binomial Probability Distribution and Related Topics

Chapter 6 The Binomial Probability Distribution and Related Topics Chapter 6 The Binomial Probability Distribution and Related Topics Statistical Experiments and Random Variables Statistical Experiments any process by which measurements are obtained. A quantitative variable,

More information

CS 171 Lecture Outline March 22, 2010

CS 171 Lecture Outline March 22, 2010 CS 7 Lecture Outline March 22, 200 Variance We are interested in calculating how much a random variable deviates from its mean. This measure is called variance. Formally, for a random variable X we are

More information

Math 1313 Section 7.5 Section 7.5: Conditional Probability

Math 1313 Section 7.5 Section 7.5: Conditional Probability Section 7.5: Conditional Probability Example 1: Two cards are drawn without replacement in succession from a well-shuffled deck of 52 playing cards. What is the probability that the second card drawn is

More information

STAT 155 Introductory Statistics. Lecture 15 &16: Sampling Distributions for Counts and Proportions

STAT 155 Introductory Statistics. Lecture 15 &16: Sampling Distributions for Counts and Proportions The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 15 &16: Sampling Distributions for Counts and Proportions 10/26/06 Lecture 15&16 1 Statistic & Its Sampling Distribution

More information

A continuous random variable can take on any value in a specified interval or range

A continuous random variable can take on any value in a specified interval or range Continuous Probability Distributions A continuous random variable can take on any value in a specified interval or range Example: Let X be a random variable indicating the blood level of serum triglycerides,

More information

5.3: The Binomial Probability Distribution

5.3: The Binomial Probability Distribution 5.3: The Binomial Probability Distribution 5.3.1 Bernoulli trials: A Bernoulli trial is an experiment with exactly two possible outcomes. We refer to one of the outcomes as a success (S) and to the other

More information

Math 150 Sample Exam #2

Math 150 Sample Exam #2 Problem 1. (16 points) TRUE or FALSE. a. 3 die are rolled, there are 1 possible outcomes. b. If two events are complementary, then they are mutually exclusive events. c. If A and B are two independent

More information

RANDOM VARIABLES, EXPECTATION, AND VARIANCE

RANDOM VARIABLES, EXPECTATION, AND VARIANCE RANDOM VARIABLES, EXPECTATION, AND VARIANCE MATH 70 This write-up was originally created when we were using a different textbook in Math 70. It s optional, but the slightly different style may help you

More information

Practice Problems for Chapter 5

Practice Problems for Chapter 5 Basic Statistics ECON 1302 Practice Problems for Chapter 5 Section I: Multiple-Choice In a certain town 60% of the households own mutual funds, 40% own individual stocks, and 20% own both mutual funds

More information

Section 5-2 Random Variables

Section 5-2 Random Variables Section 5-2 Random Variables 5.1-1 Combining Descriptive Methods and Probabilities In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies

More information

8.1 Distributions of Random Variables

8.1 Distributions of Random Variables 8.1 Distributions of Random Variables A random variable is a rule that assigns a number to each outcome of an experiment. We usually denote a random variable by X. There are 3 types of random variables:

More information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

More information

Discrete Probability Distribution discrete continuous

Discrete Probability Distribution discrete continuous CHAPTER 5 Discrete Probability Distribution Objectives Construct a probability distribution for a random variable. Find the mean, variance, and expected value for a discrete random variable. Find the exact

More information

Name: Class: Date: 3. What is the probability of a sum of 5 resulting from the roll of two six-sided dice?

Name: Class: Date: 3. What is the probability of a sum of 5 resulting from the roll of two six-sided dice? Name: Class: Date: Sample Distributions Multiple Choice Identify the choice that best completes the statement or answers the question.. Which of the following is not an example of a discrete random variable?

More information

You flip a fair coin four times, what is the probability that you obtain three heads.

You flip a fair coin four times, what is the probability that you obtain three heads. Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

More information

The MEAN is the numerical average of the data set.

The MEAN is the numerical average of the data set. Notes Unit 8: Mean, Median, Standard Deviation I. Mean and Median The MEAN is the numerical average of the data set. The mean is found by adding all the values in the set, then dividing the sum by the

More information

AP Stats Chapter 8. What are the parameters of a binomial distribution? How can you abbreviate this information?

AP Stats Chapter 8. What are the parameters of a binomial distribution? How can you abbreviate this information? AP Stats Chapter 8 8.1: The Binomial Distribution What are the conditions for a binomial setting? 1. 2. 3. 4. What is a binomial random variable? What are the possible values of a binomial random variable?

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MATH 1342 TEST 3 REVIEW SUMMER 2015 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the standard deviation of the random variable. 1) The

More information

Chapter 3 - Lecture 6 Hypergeometric and Negative Binomial D. Distributions

Chapter 3 - Lecture 6 Hypergeometric and Negative Binomial D. Distributions Chapter 3 - Lecture 6 and Distributions October 14th, 2009 Chapter 3 - Lecture 6 and D experiment random variable distribution Moments Experiment Random Variable Distribution Moments and moment generating

More information

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions

SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions 1. The following table contains a probability distribution for a random variable X. a. Find the expected value (mean) of X. x 1 2

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. MA 116 - Chapter 5 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine the possible values of the random variable. 1) Suppose that two

More information

Discrete Random Variables

Discrete Random Variables Chapter 4 Discrete Random Variables 4.1 Discrete Random Variables 1 4.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize and understand discrete probability

More information

Math Exam 3 Review

Math Exam 3 Review Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 3 Review NOTE: For reviews of the other sections on Exam 3, refer to the first page of WIR #7 and #8. Exam 3 covers Sections 7.5, 7.6, and 8.1-8.6.

More information

MBA Fall 2015 Math Diagnostic Name. Address. I. Solve = 5 (1 +.05

MBA Fall 2015 Math Diagnostic Name.  Address. I. Solve = 5 (1 +.05 MBA Fall 2015 Math Diagnostic Name Email Address I. Solve 1. 1 x 1 (x 8) = 3x 2 3 2. 5 4(x + 4) = 7x 22 3. x 4 x 3 = x+2 5 10 5 4. 10 = 5 (1 +.05 x )15 II. Answer the following. 1. Mark works strictly

More information

Chapter 5: Normal Probability Distributions

Chapter 5: Normal Probability Distributions Chapter 5: Normal Probability Distributions 5.2 Normal Distributions: Finding Probabilities If you are given that a random variable X has a normal distribution, finding probabilities corresponds to finding

More information

6.2: The Binomial Probability Distribution

6.2: The Binomial Probability Distribution 6.2: The Binomial Probability Distribution 6.2.1 Bernoulli trials: A Bernoulli trial is an experiment with exactly two possible outcomes. We refer to one of the outcomes as a success (S) and to the other

More information

2. Discrete Random Variables and Expectation

2. Discrete Random Variables and Expectation 2. Discrete Random Variables and Expectation In tossing two dice we are often interested in the sum of the dice rather than their separate values The sample space in tossing two dice consists of 36 events

More information

Discrete Probability Distributions GOALS. What is a Probability Distribution? 2/21/16. Chapter 6 Dr. Richard Jerz

Discrete Probability Distributions GOALS. What is a Probability Distribution? 2/21/16. 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 information

HOLLOMAN S PROBABILITY AND STATISTICS BRASE CHAPTER 6, PAGE 1 OF 5

HOLLOMAN S PROBABILITY AND STATISTICS BRASE CHAPTER 6, PAGE 1 OF 5 6 The Binomial Probability Distribution and Related Topics Now that we are able to calculate probabilities, we can turn our attention to a related, and probably more important topic. 6.1 Introduction to

More information

PROBABILITY. Chapter Overview Conditional Probability

PROBABILITY. Chapter Overview Conditional Probability PROBABILITY Chapter. Overview.. Conditional Probability If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Practice Question Ch 5 and Ch 6 for Quiz 2 Name Identify the given random variable as being discrete or continuous. 1) The number of oil spills occurring off the Alaskan coast 1) A) Discrete B) Continuous

More information

Characteristics of Binomial Distributions

Characteristics of Binomial Distributions Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation

More information

Expected values and variance

Expected values and variance 3.5 3.6 Expected values and variance Prof. Tesler Math 186 January 27, 214 Prof. Tesler 3.5 3.6 Expected values and variance Math 186 / January 27, 214 1 / 23 Expected Winnings in a Game Setup A simple

More information

6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0.

6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0. Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.

More information

The Binomial Probability Distribution

The Binomial Probability Distribution The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

More information

Identify the given random variable as being discrete or continuous. 1) The cost of a randomly selected orange 1) A) Continuous B) Discrete

Identify the given random variable as being discrete or continuous. 1) The cost of a randomly selected orange 1) A) Continuous B) Discrete m227(ch5 practice test) Identify the given random variable as being discrete or continuous. 1) The cost of a randomly selected orange 1) A) Continuous B) Discrete 2) The ph level in a shampoo 2) A) Continuous

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. 1) Focus groups of 12 people are randomly selected to discuss products

More information

Yesterday s lab. Mean, Median, Mode. Section 2.3. Measures of Central Tendency. What are the units?

Yesterday s lab. Mean, Median, Mode. Section 2.3. Measures of Central Tendency. What are the units? Yesterday s lab Section 2.3 What are the units? Measures of Central Tendency Measures of Central Tendency Mean, Median, Mode Mean: The sum of all data values divided by the number of values For a population:

More information

Def: A random variable, x, represents a numerical value, determined by chance, assigned to an outcome of a probability experiment.

Def: A random variable, x, represents a numerical value, determined by chance, assigned to an outcome of a probability experiment. Lecture #5 chapter 5 Discrete Probability Distributions 5-2 Random Variables Def: A random variable, x, represents a numerical value, determined by chance, assigned to an outcome of a probability experiment.

More information

III. Poisson Probability Distribution (Chapter 5.3)

III. Poisson Probability Distribution (Chapter 5.3) III. Poisson Probability Distribution (Chapter 5.3) The Poisson Probability Distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval. The interval

More information

Ch. 8 Review. IB Statistics

Ch. 8 Review. IB Statistics Ch. 8 Review IB Statistics 1. In a large population of college students, 20% of the students have experienced feelings of math anxiety. If you take a random sample of 10 students from this population,

More information

Math 408, Spring 2005 Final Exam Solutions

Math 408, Spring 2005 Final Exam Solutions Math 408, Spring 005 Final Exam Solutions. Assume A and B are independent events with P A) = 0. and P B) = 0.. Let C be the event that neither A nor B occurs, let D be the event that exactly one of A or

More information

Discrete Random Variables

Discrete Random Variables Discrete Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Random Variables 2 1.1 Introduction........................................ 2 1.2 Cumulative Distribution Function...........................

More information

Tree Diagrams. on time. The man. by subway. From above tree diagram, we can get

Tree Diagrams. on time. The man. by subway. From above tree diagram, we can get 1 Tree Diagrams Example: A man takes either a bus or the subway to work with probabilities 0.3 and 0.7, respectively. When he takes the bus, he is late 30% of the days. When he takes the subway, he is

More information

Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

More information

5.1 Probability Distributions

5.1 Probability Distributions 5/3/03 Discrete Probability Distributions C H 5A P T E R Outline 5 Probability Distributions 5 Mean, Variance, Standard Deviation, and Expectation 5 3 The Binomial Distribution C H 5A P T E R Objectives

More information

Hypergeometric Distribution

Hypergeometric Distribution Assume we are drawing cards from a deck of well-shulffed cards with replacement, one card per each draw. We do this 5 times and record whether the outcome is or not. Then this is a binomial experiment.

More information

Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data.

Chapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Chapter 5 Section 5.1: Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Example 1: The test scores for a test were: 78, 81, 82, 76,

More information

MCQ BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS

MCQ BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS MCQ BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS MCQ 8.1 A Bernoulli trial has: (a) At least two outcomes (c) Two outcomes (b) At most two outcomes (d) Fewer than two outcomes MCQ 8.2 The two mutually exclusive

More information

Class : XII Mathematics - PROBABILITY. 1. If A and B are two independent events, find P(B) when P(A U B ) = 0.60 and P(A) = 0.35.

Class : XII Mathematics - PROBABILITY. 1. If A and B are two independent events, find P(B) when P(A U B ) = 0.60 and P(A) = 0.35. Class : XII Mathematics - PROBABILITY 1. If A and B are two independent events, find P(B) when P(A U B ) = 0.60 and P(A) = 0.35. 2. A card is drawn from a well shuffled pack of 52 cards. The outcome is

More information

7.1: Discrete and Continuous Random Variables

7.1: Discrete and Continuous Random Variables 7.1: Discrete and Continuous Random Variables RANDOM VARIABLE A random variable is a variable whose value is a numerical outcome of a random phenomenon. DISCRETE RANDOM VARIABLE A discrete random variable

More information

Solutions for Review Problems for Exam 2 Math 1040 1 1. You roll two fair dice. (a) Draw a tree diagram for this experiment.

Solutions for Review Problems for Exam 2 Math 1040 1 1. You roll two fair dice. (a) Draw a tree diagram for this experiment. Solutions for Review Problems for Exam 2 Math 1040 1 1. You roll two fair dice. (a) Draw a tree diagram for this experiment. 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2

More information

Math C067 Practice Questions

Math C067 Practice Questions Math C067 Practice Questions Richard Beigel March, 06 Note: you will be expected to show your work on the test. If you use a calculator, write down the formulas that you entered into the calculator. I

More information

M 225 Test 2 A Name SHOW YOUR WORK FOR FULL CREDIT!

M 225 Test 2 A Name SHOW YOUR WORK FOR FULL CREDIT! M 225 Test 2 A Name (1 point) SHOW YOUR WORK FOR FULL CREDIT! Problem Max. Points Your Points 1-10 10 11 10 12 6 13 6 14 5 15 12 16 4 17 6 18 6 19 3 20 3 21 3 22 6 23 13 24 6 Total 100 1 Multiple choice

More information

The number of phone calls to the attendance office of a high school on any given school day A) continuous B) discrete

The number of phone calls to the attendance office of a high school on any given school day A) continuous B) discrete Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) State whether the variable is discrete or continuous.

More information

Binomial Probabilities

Binomial Probabilities Binomial Probabilities Suppose that an experiment has two outcomes; success or failure. probability p and a failure happens with probability 1 p. Assume that a success happens with Examples of such experiments

More information

STATS 8: Introduction to Biostatistics. Estimation. Babak Shahbaba Department of Statistics, UCI

STATS 8: Introduction to Biostatistics. Estimation. Babak Shahbaba Department of Statistics, UCI STATS 8: Introduction to Biostatistics Estimation Babak Shahbaba Department of Statistics, UCI Parameter estimation We are interested in population mean and population variance, denoted as µ and σ 2 respectively,

More information

PROBABILITY DISTRIBUTIO S

PROBABILITY DISTRIBUTIO S CHAPTER 3 PROBABILITY DISTRIBUTIO S 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 information

Chapter 20 & 23 - Chance Error for Sampling PART

Chapter 20 & 23 - Chance Error for Sampling PART Chapter 20 & 23 - Chance Error for Sampling PART VI : SAMPLING Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 20 & 23 - Chance Error for Sampling 1 / 35 Sampling and Accuracy

More information

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Statistics Homework Ch 6 Name Provide an appropriate response. 1) State whether the variable is discrete or continuous. 1) The number of cups of coffee sold in a cafeteria during lunch A) continuous B)

More information

MAT S5.1 2_3 Random Variables. February 09, Review and Preview. Chapter 5 Probability Distributions. Preview

MAT S5.1 2_3 Random Variables. February 09, Review and Preview. Chapter 5 Probability Distributions. Preview MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance and Standard

More information

Outline. Chapter 3: Random Sampling, Probability, and the Binomial Distribution. Random Sampling Model. Random Variables. Types of Random Variables

Outline. Chapter 3: Random Sampling, Probability, and the Binomial Distribution. Random Sampling Model. Random Variables. Types of Random Variables Outline Chapter 3: Random Sampling,, and the Part II Density Curves Random Variables The Fitting a to Data Eric D. Nordmoe Math 261 Department of Mathematics and Computer Science Kalamazoo College Spring

More information

Lecture 11- The Binomial Distribution

Lecture 11- The Binomial Distribution Lecture - The Binomial Distribution Now that we have seen the basic ideas behind a discrete probability distribution we are going to see our first example: the binomial distribution. The binomial distribution

More information

Six Sigma Black Belt Study Guides

Six Sigma Black Belt Study Guides Six Sigma Black Belt Study Guides 1 www.pmtutor.org Powered by POeT Solvers Limited. Analytical statistics helps to draw conclusions about a population by analyzing the data collected from a sample of

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 24 STATISTICS I 1. Probability 2. Conditional probability 3. Combinations and permutations 4. Random variables 5. Mean,

More information

MAT 155. Key Concept. 155S5.3_3 Binomial Probability Distributions. September 21, Chapter 5 Probability Distributions

MAT 155. Key Concept. 155S5.3_3 Binomial Probability Distributions. September 21, Chapter 5 Probability Distributions MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance, and Standard

More information

Chapter 6: Random Variables and the Normal Distribution. 6.1 Discrete Random Variables. 6.2 Binomial Probability Distribution

Chapter 6: Random Variables and the Normal Distribution. 6.1 Discrete Random Variables. 6.2 Binomial Probability Distribution Chapter 6: Random Variables and the Normal Distribution 6.1 Discrete Random Variables 6.2 Binomial Probability Distribution 6.3 Continuous Random Variables and the Normal Probability Distribution 6.1 Discrete

More information

Probability Distribution

Probability Distribution Lecture 3 Probability Distribution Discrete Case Definition: A r.v. Y is said to be discrete if it assumes only a finite or countable number of distinct values. Definition: The probability that Y takes

More information

Stats Review Chapters 5-6

Stats Review Chapters 5-6 Stats Review Chapters 5-6 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

More information

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014 STAT511 Spring 2014 Lecture Notes 1 Chapter 3: Discrete Random Variable and Probability Distribution January 28, 2014 3 Discrete Random Variables Chapter Overview Random Variable (r.v. Definition Discrete

More information

Quiz 6.1C AP Statistics Name:

Quiz 6.1C AP Statistics Name: Quiz 6.1C AP Statistics Name: Joey says that if every student in his AP Statistics class of 18 students picked a number at random from 1 to 226, and did this very many times, that about 50% of the time

More information

Maximum Likelihood & Method of Moments Estimation

Maximum Likelihood & Method of Moments Estimation Maximum Likelihood & Method of Moments Estimation Patrick Zheng 01/30/14 1 Introduction Goal: Find a good POINT estimation of population parameter Data: We begin with a random sample of size n taken from

More information

Contents. Articles. References

Contents. Articles. References Probability MATH 105 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sun, 04 Mar 2012 22:07:20 PST Contents Articles About 1 Lesson

More information

Discrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8)

Discrete 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 information

Construct and Interpret Binomial Distributions

Construct and Interpret Binomial Distributions CH 6.2 Distribution.notebook A random variable is a variable whose values are determined by the outcome of the experiment. 1 CH 6.2 Distribution.notebook A probability distribution is a function which

More information

AP Statistics - Random Variables (Multiple Choice)

AP Statistics - Random Variables (Multiple Choice) Name: Class: _ Date: _ AP Statistics - Random Variables (Multiple Choice) Multiple Choice ( 1.5 minutes each, 4 points each) Identify the choice that best completes the statement or answers the question.

More information

A rule that assigns a number to each outcome of an experiment is called a random variable.

A rule that assigns a number to each outcome of an experiment is called a random variable. Math 1313 Section 8.1 Section 8.1: Distributions of Random Variables A rule that assigns a number to each outcome of an experiment is called a random variable. We can construct the probability distribution

More information

What can be said about the number of outliers for this data set? (A) 0 (B) At least 1 (C) No more than 1 (D) At least 2 (E) No more than 2

What can be said about the number of outliers for this data set? (A) 0 (B) At least 1 (C) No more than 1 (D) At least 2 (E) No more than 2 AP Stat- Unit 1 Review (Chapters 2-6) Multiple Choice Questions 1. A random sample of 25 birthweights (in ounces) is taken yielding the following summary statistics: Variable N Mean Median TrMean StDev

More information

Chapter 6: Random Variables and the Normal Distribution. 6.1 Discrete Random Variables. 6.2 Binomial Probability Distribution

Chapter 6: Random Variables and the Normal Distribution. 6.1 Discrete Random Variables. 6.2 Binomial Probability Distribution Chapter 6: Random Variables and the Normal Distribution 6.1 Discrete Random Variables 6.2 Binomial Probability Distribution 6.3 Continuous Random Variables and the Normal Probability Distribution 6.4 Standard

More information

Binomial Probability Distribution

Binomial Probability Distribution Binomial Probability Distribution Binomial Distribution The binomial distribution is the probability distribution that results from doing a binomial experiment. Binomial experiments have the following

More information

Desirable Statistical Properties of Estimators. 1. Two Categories of Statistical Properties

Desirable Statistical Properties of Estimators. 1. Two Categories of Statistical Properties Desirable Statistical Properties of Estimators 1. Two Categories of Statistical Properties There are two categories of statistical properties of estimators. (1) Small-sample, or finite-sample, properties

More information

Probability Distributions

Probability Distributions 9//5 : (Discrete) Random Variables For a given sample space S of some experiment, a random variable (RV) is any rule that associates a number with each outcome of S. In mathematical language, a random

More information

The Central Limit Theorem

The Central Limit Theorem Chapter 7 The Central Limit Theorem 7.1 The Central Limit Theorem 1 7.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the Central Limit Theorem problems.

More information

Discrete Random Variables

Discrete Random Variables 1 Discrete Random Variables 4.1 Probability Distributions for Discrete Random Variables This chapter will deal with the construction of discrete probability distributions by combining the methods of descriptive

More information

Lecture.7 Poisson Distributions - properties, Normal Distributions- properties. Theoretical Distributions. Discrete distribution

Lecture.7 Poisson Distributions - properties, Normal Distributions- properties. Theoretical Distributions. Discrete distribution Lecture.7 Poisson Distributions - properties, Normal Distributions- properties Theoretical distributions are Theoretical Distributions 1. Binomial distribution 2. Poisson distribution Discrete distribution

More information

Unit 4 The Bernoulli and Binomial Distributions

Unit 4 The Bernoulli and Binomial Distributions BIOSTATS 540 Fall 2015 4. Bernoulli and Binomial Page 1 of 22 Unit 4 The Bernoulli and Binomial Distributions If you believe in miracles, head for the Keno lounge - Jimmy the Greek The Amherst Regional

More information

Lecture 6: Special Probability Distributions

Lecture 6: Special Probability Distributions Lecture 6: Special Probability Distributions Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 The Discrete Uniform Distribution

More information

Below you will find a series of examples. For each example, you should provide the following information:

Below you will find a series of examples. For each example, you should provide the following information: Math 58B Spring Jo Hardin when do I use what?? Below you will find a series of examples. For each example, you should provide the following information: What units are being measured (or observed)? What

More information

Sample Questions for Mastery #5

Sample Questions for Mastery #5 Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could

More information