Chapter 5: Discrete Random Variables and Their Probability Distributions. Introduction. 5.1 Random Variables. Continuous Random Variable
|
|
- Gwen Grant
- 7 years ago
- Views:
Transcription
1 Chapter 5: Discrete Random Variables and Their Probability Distributions 5.1 Random Variables 5.2 Probability Distribution of a Discrete Random Variable 5.3 Mean and Standard Deviation of a Discrete Random Variable 5.4 The Binomial Probability Distribution 5.5 The Hypergeometric Probability Distribution 5.6 The Poisson Probability Distribution Introduction We discussed concepts and rules of probability in chapter 4. It helps in solving simple problem, but not complicated ones such as finding probability of getting at least 5 heads in 10 tosses of a fair coin. We want to study the probability mathematically, so we assign numerical values to experimental outcomes and define random variables. Study the probability characteristic of random variables the topic of chapters 5 & Random Variables Continuous Random Variable A random variable is a variable whose value is determined by the outcome of a random experiment Discrete Random Variable A random variable that assumes countable values is called a discrete random variable Examples of discrete random variables The number of cars sold at a dealership during a given month The number of houses in a certain block The number of fish caught on a fishing trip The number of complaints received at the office of an airline on a given day The number of customers who visit a bank during any given hour The number of heads obtained in three tosses of a coin A random variable that can assume any value contained in one or more intervals is called a continuous random variable Examples of continuous random variables The length of a room The time taken to commute from home to work The amount of milk in a gallon (note that we do not expect a gallon to contain exactly one gallon of milk but either slightly more or slightly less than one gallon) The weight of a letter The price of a house
2 5.2 Probability Distribution of a Discrete Random Variable The probability distribution of a discrete random variable lists all the possible values that the random variable can assume and their corresponding probabilities Two Characteristics of a Probability Distribution 0 P(x) 1 for each value of x ΣP(x) = 1 Example of tossing 2 coins, X = # of heads Example 5-1 Recall the frequency and relative frequency distributions of the number of vehicles owned by families given in Table 5.1. That table is reproduced below as Table 5.2. Let x be the number of vehicles owned by a randomly selected family. Write the probability distribution of x. X P Example 5-2 Example 5-3 Each of the following tables lists certain values of x and their probabilities. Determine whether or not each table represents a valid probability distribution The following table lists the probability distribution of the number of breakdowns per week (x) for a machine based on past data a) No, since the sum of all probabilities is not equal to 1.0. b) Yes c) No since one of the probabilities is negative. Present this probability distribution graphically. Find the probability that the number of breakdowns for this machine during a given week is exactly 2 0 to 2 more than 1 at most
3 Example 5-3: Solution Example 5-4 Graphical presentation Finding probabilities P(exactly 2 breakdowns) = P(x = 2) =.35 P(0 to 2 breakdowns) = P(0 x 2) = P(x = 0) + P(x = 1) + P(x = 2) = =.70 P(more then 1 breakdown) = P(x > 1) = P(x = 2) + P(x = 3) = =.65 P(at most one breakdown) = P(x 1) = P(x = 0) + P(x = 1) = =.35 According to a survey, 60% of all students at a large university suffer from math anxiety. Two students are randomly selected from this university. Let x denote the number of students in this sample who suffer from math anxiety. Develop the probability distribution of x. Let us define the following two events: N = the student selected does not suffer from math anxiety M = the student selected suffers from math anxiety Then P(x = 0) = P(NN) =.16 P(x = 1) = P(NM or MN) = P(NM) + P(MN) = =.48 P(x = 2) = P(MM) = Mean and Standard Deviation of a Discrete Random Variable The mean of a discrete variable x is the value that is expected to occur per repetition, on average, if an experiment is repeated a large number of times. It is denoted by µ and calculated as µ = Σ x P(x) The mean of a discrete random variable x is also called its expected value and is denoted by E(x); that is, E(x) = Σ x P(x) Example of tossing a coin twice, x = # of heads Example 5-5 Recall Example 5-3 of Section 5-2. The probability distribution Table 5.4 from that example is reproduced on the next slide. In this table, x represents the number of breakdowns for a machine during a given week, and P(x) is the probability of the corresponding value of x. Find the mean number of breakdown per week for this machine. X P 1/4 2/4 ¼ xp 0 2/4 2/4 µ = Σ x P(x) = 1 The mean is µ = Σx P(x) =
4 Standard Deviation of a Discrete Random Variable The standard deviation of a discrete random variable, denoted by σ, measures the spread of its probability distribution. A higher value for the standard deviation of a discrete random variable indicates that x can assume values over a larger range about the mean. A smaller value for the standard deviation indicates that most of the values that x can assume are clustered closely about the mean. Definition of variance - 2 Deviation Standard deviation = square root of variance Book s definition of σ Interpretation of the Standard Deviation same way as Section 3.4 of Chapter x P( x) Example 5-6 Baier s Electronics manufactures computer parts that are supplied to many computer companies. Despite the fact that two quality control inspectors at Baier s Electronics check every part for defects before it is shipped to another company, a few defective parts do pass through these inspections undetected. Let x denote the number of defective computer parts in a shipment of 400. The following table gives the probability distribution of x. Compute the standard deviation of x Book s Method to Find the Standard Deviation Recommended Method Computations to Find the Mean and Standard Deviation 2 = x 2 P(x) - 2 = = 1.45 => = X P(x) xp(x) (X-) 2 = (X - ) 2 (X-) 2 P(x) = 1.45 σ
5 Easy Example The following table gives the probability distribution for a random Variable X, the number of DVDs that were returned late in a local Blockbuster per week. X P ? Find the probability that one or two DVDs were returned late. 2. Find the probability that at least one DVD was returned late. 3. Find X's mean 4. Find X's variance 2 5. Find X's standard deviation Why not use Excel? 5.4 The Binomial Probability Distribution Binomial Experiment: an experiment satisfying five conditions There are fixed n identical trials Each trial has only two outcomes, success & failure Probability of success p remains constant for each trial The trials are independent X = the number of successes in n trials X is called a binomial random variable and its distribution called BPD X ~ B(n, p) Examples of binomial experiment Tossing a coin 10 times Example 5-8 Example 5-9: 5% of all DVD players made by a large electronics company are defective and 3 DVD players are randomly selected Rolling a die (trial, not experiment, has 2 outcomes) Random guess for answers of a multiple-choice test/quiz Notation and Formula Example 5-10 Notation n = total number of trials p = probability of success q = 1 p = probability of failure x = number of successes in n trials n x =number of failures in n trials Formula for X ~ B(n, p) x n x P( x) C p q, x = 0, 1,, n n x For a problem, first check if it is binomial experiment by using the five conditions If answer is a yes, then identify n, p, x. Use formula to obtain binomial probability distribution Five percent of all DVD players manufactured by a large electronics company are defective. A quality control inspector randomly selects three DVD player from the production line. What is the probability that exactly one of these three DVD players is defective?
6 Example 5-18: Solution Let D = a selected DVD player is defective P(D)=.05 G = a selected DVD player is good P(G)=.95 P(DGG) = P(D)P(G)P(G) = (.05)(.95)(.95) =.0451 P(GDG) = P(G)P(D)P(G) = (.95)(.05)(.95) =.0451 P(GGD) = P(G)P(G)P(D) = (.95)(.95)(.05) =.0451 P(1 DVD player in 3 is defective) = P(DGG or GDG or GGD) = P(DGG)+P(GDG)+P(GGD) = =.1353 Formula way n = 3, p = 0.05, q = 0.95, x = 1 Example 5 11 At the Express House Delivery Service, providing highquality service to customers is the top priority of the management. The company guarantees a refund of all charges if a package it is delivering does not arrive at its destination by the specified time. It is known from past data that despite all efforts, 2% of the packages mailed through this company do not arrive at their destinations within the specified time. Suppose a corporation mails 10 packages through Express House Delivery Service on a certain day. Find the probability that exactly one of these 10 packages will not arrive at its destination within the specified time. Find the probability that at most one of these 10 packages will not arrive at its destination within the specified time Example 5 12 In a Pew Research Center nationwide telephone survey conducted in March through April 2011, 74% of college graduates said that college provided them intellectual growth (Time, May 30, 2011). Assume that this result holds true for the current population of college graduates. Let x denote the number in a random sample of three college graduates who hold this opinion. Write the probability distribution of x and draw a bar graph for this probability distribution. 0 3 Px ( 0) 3C0(.74) (.26) (1)(1)( ) Px ( 1) 3C1(.74) (.26) (3)(.74)(.0676) Px ( 2) 3C2(.74) (.26) (3)(.5476)(.26) Px ( 3) C(.74) (.26) (1)( )(1) Automatic Way to Find Binomial Probabilities Table I in Appendix C, the table of binomial probabilities. List the probabilities of x for n = 1 to n = 25. List the probabilities of x for selected values of p Hence the table is very limited Using Excel Binom.dist(number_s, trials, probability_s, cumulative) = binomdist(x, n, p, cumulative) Cumulative = 0, false for cumulative, it gives P(x) Cumulative = 1, true for cumulative, it gives cumulative probabilities from 0 to x, i.e., sum of P(0) through P(x). Calculator TI 83: 2 nd => DISTR => 0 (A) binompdf(n, p, x): gives P(x). binomcdf(n, p, x): gives cumulative probabilities from 0 to x, i.e., binomcdf(n, p, x) = P(0) + P(1) + + P(x)
7 Example 5 13 In an NPD Group survey of adults, 30% of 50-year-old or older (let us call them 50-plus) adult Americans said that they would be willing to pay more for healthier options at restaurants (USA TODAY, 2011). Suppose this result holds true for the current population of 50-plus adult Americans. A random sample of six 50-plus adult Americans is selected. Answer the following. Find the probability that exactly 3 persons in this sample hold the said opinion. Find the probability that at most two persons in this sample hold the said opinion. Find the probability that at least three persons in this sample hold the said opinion. Find the probability that one to three persons in this sample hold the said opinion. Let x be the number of persons in this sample who hold the said opinion. Write the probability distribution of x, and draw a bar graph for this probability distribution Determining P(x = 3) for n = 6 and p =.30 Table way Excel way In a cell of Excel type in binom.dist(3, 6, 0.3, 0) and hit enter key P(x = 3) = binom.dist(3, 6, 0.3, 0) = Probability of Success and the Shape of the Binomial Distribution The binomial probability distribution is symmetric if p =.50 For any n, it gives a symmetric bell-shape The binomial probability distribution is skewed to the right if p is less than.50. The binomial probability distribution is skewed to the left if p is greater than.50. For large n, any p, it gives rough bell-shaped Using Excel to show such feature Mean and Standard Deviation of the Binomial Distribution General formula Mean Variance Special formula = n p 2 = n p q Examples Examples from the book Tossing a fair coin twice, X = # of heads Tossing a fair coin 10 times, X = # of heads Finding any kind of probability Example 5-14 at page
8 5.5 The Hypergeometric Probability Distribution Notations N = total number of elements in the population r = number of successes in the population N r = number of failures in the population n = number of trials (sample size) x = number of successes in n trials n x = number of failures in n trials The probability of x successes in n trials is given by rcx NrCnx Px ( ) C N 5-29 n Example 5-15 Brown Manufacturing makes auto parts that are sold to auto dealers. Last week the company shipped 25 auto parts to a dealer. Later, it found out that 5 of those parts were defective. By the time the company manager contacted the dealer, 4 auto parts from that shipment had already been sold. What is the probability that 3 of those 4 parts were good parts and 1 was defective? : N = 25, r = 20, N r = 5, n = 4, x = 3, n x = 1 20! 5! r Cx C N r nx 20C3 5C1 3!(20 3)! 1!(5 1)! P( x 3) C 25! N n 25C4 4!(25 4)! (1140)(5) , The Poisson Probability Distribution The following three conditions must be satisfied to apply the Poisson probability distribution. x is a discrete random variable. The occurrences are random. The occurrences are independent. Examples of Poisson Probability Distribution The number of accidents that occur on a given highway during a 1- week period. The number of customers entering a grocery store during a 1 hour interval. The number of television sets sold at a department store during a given week. Poisson Probability Distribution Formula According to the Poisson probability distribution, the probability of x occurrences in an interval is x e P ( x) x! where λ (pronounced lambda) is the mean number of occurrences in that interval and the value of e is approximately Mean and Standard Deviation
9 Example 5-17 On average, a household receives 9.5 telemarketing phone calls per week. Using the Poisson distribution formula, find the probability that a randomly selected household receives exactly 6 telemarketing phone calls during a given week. x e (9.5) e Formula way P( x 6) x! 6! Excel way: POISSON.DIST(6, 9.5, 0) = (735, )( ) Example 5-18 A washing machine in a laundromat breaks down an average of three times per month. Using the Poisson probability distribution formula, find the probability that during the next month this machine will have exactly two breakdowns at most one breakdown Formula way Excel way Calculator way TI-84 calculator way poissonpdf(9.5, 6) = Example 5-20 Example 5-21 On average, two new accounts are opened per day at an Imperial Saving Bank branch. Using Table III of Appendix C, find the probability that on a given day the number of new accounts opened at this bank will be exactly 6 at most 3 at least 7 Three ways An auto salesperson sells an average of.9 car per day. Let x be the number of cars sold by this salesperson on any given day. Find the mean, variance, and standard deviation. s car car
Probability Distributions
CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution
More informationYou 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 informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
More informationThe 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 informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,
More information4.1 4.2 Probability Distribution for Discrete Random Variables
4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.
More informationProbability Distributions
CHAPTER 6 Probability Distributions Calculator Note 6A: Computing Expected Value, Variance, and Standard Deviation from a Probability Distribution Table Using Lists to Compute Expected Value, Variance,
More informationSection 6.1 Discrete Random variables Probability Distribution
Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationChapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationWHERE 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 information2. Discrete random variables
2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be
More informationCh5: 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 informationBinomial Probability Distribution
Binomial Probability Distribution In a binomial setting, we can compute probabilities of certain outcomes. This used to be done with tables, but with graphing calculator technology, these problems are
More informationST 371 (IV): Discrete Random Variables
ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
STATISTICS/GRACEY PRACTICE TEST/EXAM 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous.
More informationChapter 5. Discrete Probability Distributions
Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable
More informationAn Introduction to Basic Statistics and Probability
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.
More informationSample 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 informationBusiness Statistics, 9e (Groebner/Shannon/Fry) Chapter 5 Discrete Probability Distributions
Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 5 Discrete Probability Distributions 1) A random variable is generated when a variableʹs value is determined by using classical probability. Answer:
More informationSection 5-3 Binomial Probability Distributions
Section 5-3 Binomial Probability Distributions Key Concept This section presents a basic definition of a binomial distribution along with notation, and methods for finding probability values. Binomial
More informationChapter 5 - Practice Problems 1
Chapter 5 - Practice Problems 1 Identify the given random variable as being discrete or continuous. 1) The number of oil spills occurring off the Alaskan coast 1) A) Continuous B) Discrete 2) The ph level
More informationCharacteristics 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 informationChapter 9 Monté Carlo Simulation
MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?
More informationSTAT 3502. x 0 < x < 1
Solution - Assignment # STAT 350 Total mark=100 1. A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous
More informationReview for Test 2. Chapters 4, 5 and 6
Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair six-sided die. Find the probability of each event: a. Event A: rolling a 3 1/6 b. Event B: rolling a 7 0 c. Event C: rolling a number less than
More informationMBA 611 STATISTICS AND QUANTITATIVE METHODS
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
More informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More informationSection 5 Part 2. Probability Distributions for Discrete Random Variables
Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability
More informationStats on the TI 83 and TI 84 Calculator
Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationBinomial random variables (Review)
Poisson / Empirical Rule Approximations / Hypergeometric Solutions STAT-UB.3 Statistics for Business Control and Regression Models Binomial random variables (Review. Suppose that you are rolling a die
More informationBINOMIAL DISTRIBUTION
MODULE IV BINOMIAL DISTRIBUTION A random variable X is said to follow binomial distribution with parameters n & p if P ( X ) = nc x p x q n x where x = 0, 1,2,3..n, p is the probability of success & q
More informationMAT 155. Key Concept. September 22, 2010. 155S5.3_3 Binomial Probability Distributions. 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 informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover
More informationUniversity of California, Los Angeles Department of Statistics. Random variables
University of California, Los Angeles Department of Statistics Statistics Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.
More information5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.
The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution
More informationSolutions 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 informationLecture 5 : The Poisson Distribution
Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Ch. 4 Discrete Probability Distributions 4.1 Probability Distributions 1 Decide if a Random Variable is Discrete or Continuous 1) State whether the variable is discrete or continuous. The number of cups
More informationNormal Probability Distribution
Normal Probability Distribution The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use
More informationRandom variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.
Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()
More informationSome special discrete probability distributions
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that
More informationSOLUTIONS: 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 informationDETERMINE whether the conditions for a binomial setting are met. COMPUTE and INTERPRET probabilities involving binomial random variables
1 Section 7.B Learning Objectives After this section, you should be able to DETERMINE whether the conditions for a binomial setting are met COMPUTE and INTERPRET probabilities involving binomial random
More informationMULTIPLE 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. 1) If two events are mutually exclusive, what is the probability that one or the other occurs? A)
More informationChapter 5: Discrete Probability Distributions
Chapter 5: Discrete Probability Distributions Section 5.1: Basics of Probability Distributions As a reminder, a variable or what will be called the random variable from now on, is represented by the letter
More informationProbability and Statistics Vocabulary List (Definitions for Middle School Teachers)
Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More informationDistributions. GOALS When you have completed this chapter, you will be able to: 1 Define the terms probability distribution and random variable.
6 Discrete GOALS When you have completed this chapter, you will be able to: Probability 1 Define the terms probability distribution and random variable. 2 Distinguish between discrete and continuous probability
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
More informationQuestion: 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 informationACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers
ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationMATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem
MATH 10: Elementary Statistics and Probability Chapter 7: The Central Limit Theorem Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you
More informationExploratory Data Analysis
Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
More informationChapter 5 Discrete Probability Distribution. Learning objectives
Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able
More informationProbability Distributions
Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.
More informationECON1003: Analysis of Economic Data Fall 2003 Answers to Quiz #2 11:40a.m. 12:25p.m. (45 minutes) Tuesday, October 28, 2003
ECON1003: Analysis of Economic Data Fall 2003 Answers to Quiz #2 11:40a.m. 12:25p.m. (45 minutes) Tuesday, October 28, 2003 1. (4 points) The number of claims for missing baggage for a well-known airline
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce
More information8 6 X 2 Test for a Variance or Standard Deviation
Section 8 6 x 2 Test for a Variance or Standard Deviation 437 This test uses the P-value method. Therefore, it is not necessary to enter a significance level. 1. Select MegaStat>Hypothesis Tests>Proportion
More informationBinomial random variables
Binomial and Poisson Random Variables Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Binomial random variables 1. A certain coin has a 5% of landing heads, and a 75% chance
More informationCHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
More informationPractice Problems #4
Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice
More informationMath 461 Fall 2006 Test 2 Solutions
Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two
More information6 POISSON DISTRIBUTIONS
6 POISSON DISTRIBUTIONS Chapter 6 Poisson Distributions Objectives After studying this chapter you should be able to recognise when to use the Poisson distribution; be able to apply the Poisson distribution
More informationACMS 10140 Section 02 Elements of Statistics October 28, 2010. Midterm Examination II
ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages of the examination
More informationThe Normal Distribution
Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More information2 Binomial, Poisson, Normal Distribution
2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability
More informationNotes on Continuous Random Variables
Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes
More informationSection 6-5 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More informationUNIT I: RANDOM VARIABLES PART- A -TWO MARKS
UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0
More informationHomework 4 - KEY. Jeff Brenion. June 16, 2004. Note: Many problems can be solved in more than one way; we present only a single solution here.
Homework 4 - KEY Jeff Brenion June 16, 2004 Note: Many problems can be solved in more than one way; we present only a single solution here. 1 Problem 2-1 Since there can be anywhere from 0 to 4 aces, the
More informationAP Statistics 7!3! 6!
Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!
More informationMATH 103/GRACEY PRACTICE EXAM/CHAPTERS 2-3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MATH 3/GRACEY PRACTICE EXAM/CHAPTERS 2-3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The frequency distribution
More informationRandom Variables. Chapter 2. Random Variables 1
Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets
More informationSums of Independent Random Variables
Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables
More informationConstruct 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 informationInterpreting Data in Normal Distributions
Interpreting Data in Normal Distributions This curve is kind of a big deal. It shows the distribution of a set of test scores, the results of rolling a die a million times, the heights of people on Earth,
More informationNormal and Binomial. Distributions
Normal and Binomial Distributions Library, Teaching and Learning 14 By now, you know about averages means in particular and are familiar with words like data, standard deviation, variance, probability,
More informationMind on Statistics. Chapter 8
Mind on Statistics Chapter 8 Sections 8.1-8.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable
More informationThe right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median
CONDENSED LESSON 2.1 Box Plots In this lesson you will create and interpret box plots for sets of data use the interquartile range (IQR) to identify potential outliers and graph them on a modified box
More informationProbability. Distribution. Outline
7 The Normal Probability Distribution Outline 7.1 Properties of the Normal Distribution 7.2 The Standard Normal Distribution 7.3 Applications of the Normal Distribution 7.4 Assessing Normality 7.5 The
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More informationThe mathematical branch of probability has its
ACTIVITIES for students Matthew A. Carlton and Mary V. Mortlock Teaching Probability and Statistics through Game Shows The mathematical branch of probability has its origins in games and gambling. And
More informationDepartment of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
More informationPaper No 19. FINALTERM EXAMINATION Fall 2009 MTH302- Business Mathematics & Statistics (Session - 2) Ref No: Time: 120 min Marks: 80
Paper No 19 FINALTERM EXAMINATION Fall 2009 MTH302- Business Mathematics & Statistics (Session - 2) Ref No: Time: 120 min Marks: 80 Question No: 1 ( Marks: 1 ) - Please choose one Scatterplots are used
More informationPROBABILITY SECOND EDITION
PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All
More information3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.
3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately
More informationBNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I
BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential
More informationCA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction
CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous
More informationMath 202-0 Quizzes Winter 2009
Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile
More informationThe Normal Distribution
The Normal Distribution Continuous Distributions A continuous random variable is a variable whose possible values form some interval of numbers. Typically, a continuous variable involves a measurement
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Certain derivations presented in this course have been somewhat heavy on algebra. For example, determining the expectation of the Binomial distribution (page 5.1 turned
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More information