TYPES OF RANDOM VARIABLES. Discrete Random Variable. Examples of discrete random. Two Characteristics of a PROBABLITY DISTRIBUTION OF A
|
|
- Ethan Nelson
- 5 years ago
- Views:
Transcription
1 TYPES OF RANDOM VARIABLES DISRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS We distinguish between two types of random variables: Discrete random variables ontinuous random variables Discrete Random Variable Definition A random variable that assumes finite or countable values is called a discrete random variable Eamples 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 4 PROBABLITY DISTRIBUTION OF A DISRETE RANDOM VARIABLE Definition The probability distribution of a discrete random variable lists all the possible values that the random variable can assume and their corresponding probabilities Two haracteristics of a Probability Distribution tion The probability distribution of a discrete random variable possesses the following two characteristics: P () for each value of P () = 5 6
2 Eample : the frequency and relative frequency distributions of the number of vehicles owned by families Eample : the frequency and relative frequency distributions of the number of vehicles owned by families (continued) 7 Number of Relative Frequency Vehicles Owned Frequency / = 5 47/ = 5 85/ = / = 45 6/ = 8 N = Sum = Now let be the number of vehicles owned by a randomly selected family Write the probability distribution of 8 Number of Vehicles Owned Probability ) ) = Graphical presentation of the probability distribution from previous Eample Eample : Each of the following tables lists certain values of and their probabilities ),45,4,5,,5,,5,,5 4 an you determine whether or not each table represents a valid probability distribution? a) ) b) ) c) ) Answer Key Eample a) NO: the respective probabilities DO NOT sum up to, though all are in the range of [,]; b) YES: both characteristics ti are satisfied; The following table lists the probability distribution of the number of breakdowns per week for a machine based on past data c) NO: the probability bilit of an event (when takes value equal to 9) cannot be negative, though they sum up to Breakdowns per week Probability 5 5
3 haracteristics: Probability p() or f() function and cumulative distribution function F() Eample (continued) a) Present this probability distribution graphically Number in tossing fair die once Sum of two numbers in tossing two fair dice once b) Find the probability that the number of breakdowns for this machine during a given week is: i eactly ii to iii more than iv at most 4 Eample (continued) Let denote the number of breakdowns for this machine during a given week; the following table lists the probability distribution of ) 5 5 ) = a) Eample (continued) ),4,5,,5, 5,5,,5 5 6 Eample (concluded) b) i P (eactly breakdowns) = P ( = ) = 5 ii P ( to breakdowns) = P ( ) = P ( = ) + P ( = ) + P ( = ) = = 7 iii P (more then breakdown) = P ( > ) = P ( =)+P P ( =) = 5 + = 65 Eample According to a survey, 6% of all students at a large university suffer from math aniety Two students are randomly selected from this university Let denote the number of students in this sample who suffer from math aniety Your task is to develop the probability bilit distribution of iv P (at most one breakdown) = P ( ) = P ( = ) + P ( = ) = 5 + = 5 7 8
4 Eample ontinued: Tree Diagram Eample (continued) Let us define the following two events: N = the student selected does not suffer from math aniety M = the student selected suffers from math aniety P ( = ) = NN) = 6 P ( = ) = NM or MN) = NM) + MN) = = 48 P ( = ) = MM) = 6 9 Eample (concluded) Probability distribution of the number of students with math aniety in a sample of two students is summarized in the following table: ) ) = MEAN OF A DISRETE RANDOM VARIABLE The mean of a discrete variable is the value that is epected to occur per repetition, on average, if an eperiment is repeated a large number of times It is denoted by µ and calculated as μ = P () The mean of a discrete random variable is also called its epected value and is denoted by E (); that is, μ = E () = P () Eample 4: omputing the Mean Recall the table from Eample, where represents the number of breakdowns for a machine during a given week, and P () is the probability of the corresponding value of Your task is to find the mean number (or the epected value) of breakdowns per week for this machine Eample 4: omputing the Mean (concluded) ) ) 5 (5) = () = 5 (5) = 7 () = 9 ) = 8 Thus the mean is The mean is μ = P () = 8 4
5 STANDARD DEVIATION OF A DISRETE RANDOM VARIABLE The standard deviation of a discrete random variable measures the spread of its probability distribution and is computed as P ( ) Eample 5 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 denote the number of defective computer parts in a shipment of 4 The following table gives the probability distribution of ompute the standard d deviation of ) 8 7 Eample 5: omputing the Standard Deviation ) ) ² ²) ) = P 4 5 defective computer parts in P 77 (5 ) 45 defective computer parts 4 ²) = 77 8 Eample 6 Loraine orporation is planning to market a new makeup product According to the analysis made by the financial department of the company, it will earn an annual profit of $45 million if this product has high sales and an annual profit of $ million if the sales are mediocre, and it will lose $ million a year if the sales are low The probabilities of these three scenarios are, 5 and 7 respectively Thus: a) Let be the profits (in millions of dollars) earned per annum by the company from this product Write the probability bilit distribution ib ti of b) alculate the mean and the standard deviation of Eample 6 (continued) Eample 6 (concluded): omputations to Find the Mean and Standard Deviation 9 a) The following table lists the probability distribution of ) ) ) ² ²) P P ) = 66 ²) = 87 $66 $4 million million 87 (66)
6 FATORIALS AND OMBINATIONS Factorials Factorials ombinations Using the Table of ombinations Definition The symbol n!, read as n factorial, represents the product of all the integers from n to In other words, n! = n(n - )(n )(n ) By definition,! = Eample 7 Solution Evaluate the following: a) 7! b)! c) ( 4)! d) (5 5)! a) 7! = = 54 b)! = =,68,8 c) ( 4)! = 8! = = 4, d) (5 5)! =! = 4 ombinations ombinations cont Definition ombinations give the number of ways elements can be selected from n elements The notation used to denote the total number of combinations is n which is read as the number of combinations of n elements selected at a time = n n denotes the total number of elements the number of combinations of n elements selected at a time denotes the number of elements selected per selection 5 6
7 ombinations (continued) Eample 8 The number of combinations for selecting from n distinct elements is given by the formula n n!!( n )! An ice cream parlor has si flavors of ice cream Kristen wants to buy two flavors of ice cream If she randomly selects two flavors out of si, how many combinations are there? 7 8 n = 6 = 6!!(6 )! Answer Key 6! !4! 4 6 Eample 9 Three members of a jury will be randomly selected from five people How many different combinations are possible? Thus, there are 5 ways for Kristin to select two ice cream flavors out of si 9 4 Solution 5! 5!!(5 )!!! 6 5 Eample : Using the Table of ombinations Marv & Sons advertised to hire a financial analyst The company has received applications from candidates who seem to be equally qualified The company manager has decided to call only of these candidates for an interview i If she randomly selects candidates from the, how many total selections are possible? 4 4
8 Eample : Determining the Value of THE BINOMIAL PROBABILITY DISTRIBUTION n = X = n 45 The Binomial Eperiment The Binomial Probability Distribution and binomial Formula Using the Table of Binomial Probabilities Probability of Success and the Shape of the Binomial Distribution The Binomial Probability Distribution and Binomial Formula For a binomial eperiment, the probability of eactly successes in n trials is given by the binomial formula where n p q n - ) n p q n n p = total number of trials = probability bilit of success = p = probability of failure = number of successes in n trials = number of failures in n trials q n 46 Binomial Distribution Number of Successes in a of n Observations (Trials) # Reds in 5 Spins of Roulette Wheel # Defective Items in a Batch of 5 Items # orrect on a Question Eam # ustomers Who Purchase Out of ustomers Who Enter Store # of Bush-heney supporters in survey of people Binomial Distribution-How tion to find it Sequence of n Identical Trials Each Trial Has Outcomes Success (Desired/specified Outcome) or Failure onstant Trial Probability 4 Trials Are Independent 5 # of successes in n trials is a binomial random variable??? Binomial??? Pick 6 students from this class Each flips a coin ount # of heads Pick 6 students from this class X= # of st year students selected Random digit dialing of numbers # of Bush-heney supporters Random digit dialing of numbers Sum of ages of respondents 47 48
9 Eample Tree diagram for selecting three VRs Five percent of all VRs manufactured by a large electronics company are defective A quality control inspector randomly selects three VRs from the production line What is the probability that eactly one of these three VRs are defective? 49 5 Eample : Answer Key EampleAns Eample : Answer erke Key (continued) Let D = a selected VR is defective G = a selected VR is good P (DGG ) = P (D )P (G )P (G ) = (5)(95)(95) = 45 P (GDG ) = P (G )P (D )P (G ) = (95)(5)(95) = 45 P (GGD ) = P (G )P (G )P (D ) = (95)(95)(5) = 45 Therefore, P ( VR is defective in ) = P (DGG or GDG or GGD ) = P (DGG ) + P (GDG ) + P (GGD ) = = EampleAns Eample : Answer erke Key (continued) EampleAns Eample : Answer erke Key (concluded) n = total number of trials = VRs = number of successes = number of defective VRs = n =- = p = P (success) = 5 q = P (failure) = p = 95 Therefore, the probability of selecting eactly one defective VR P ( ) (5) (95) ()(5)(95) 54 The probability 54 is slightly different from the earlier calculation 5 because of rounding 5 54
10 Eample Eample 55 At the Epress House Delivery Service, providing high-quality service to customers is the top priority of fthe 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, % of the packages mailed through this company do not arrive at their destinations within the specified time Suppose a corporation mails packages through Epress House Delivery Service on a certain day 56 a) Find the probability that eactly of these packages will not arrive at its destination within the specified time b) Find the probability that at most of these packages will not arrive at its destination within the specified time Eample : Answer Key EampleAns Eample : Answer erke Key (continued) n = total number of packages mailed = p = P (success) = q = P (failure) = = 98 a) ) = number of successes = n = number of failures = = 9 () (98) 9! () (98)!( )! ()()(874776) Eample : Answer Key (concluded) Eample b) ) ) ) () (98) () (98) ()()(8778) ()()(874776) According to an Allstate Survey, 56% of Baby Boomers have car loans and are making payments on these loans (USA TODAY, October 8, ) Assume that this result holds true for the current population p of all Baby Boomers Let denote the number in a random sample of three Baby Boomers who are making payments on their car loans Write the probability distribution of and draw a bar graph for this probability distribution 59 6
11 Eample : Answer Key Eample : Answer Key n = total Baby boomers in the sample = p = P (a Baby Boomer is making car loan payments) = 56 q = P (a Baby Boomer is not making car loan payments) = - 56 = 44 ) (56) (44) ()()(8584) 85 ) (56) (44) ()(56)(96) 5 ) (56) (44) ()(6)(44) 44 ) (56) (44) ()(7566)() The following table represents the probability distribution ib ti of P () Here we have a bar graph of the probability distribution of ),45 4,4,5,,5,,5,, Eample 4: Using the Table of Binomial Probabilities Eample 4 65 According to a study of college students by Harvard University s School of Public health, 9% of those included in the study abstained from drinking (USA TODAY, April, ) Suppose that t of all current college students t in the United States, % abstain from drinking A random sample of si college students is selected 66 Using Table IV of Appendi, answer the following a) Find the probability that eactly three college students in this sample abstain from drinking b) Find the probability that at most two college students in this sample abstain from drinking c) Find the probability that at least three college students in this sample abstain from drinking d) Find the probability that one to three college students in this sample abstain from drinking e) Let be the number of college students in this sample who abstain from drinking Write the probability distribution of and draw a bar graph for this probability distribution
12 Determining P ( = ) for n = 6 and p = p = p n 5 95 n = = P ( = ) = 89 Eample 4: Answers a) P ( = ) = 89 b) P (at most ) = P ( or or ) = P ( =)+P P ( =)+P P ( =) = = 9 c) P (at least ) = or 4 or 5 or 6) = P ( = ) + P ( = 4) + P ( =5) + P ( = 6) = = 989 d) P ( to ) = P ( = ) + P ( = ) + P ( = ) = = Probability Distribution of for n = 6 and p= Bar graph for the probability distribution of ) ),45,4,5,,5,,5,, Probability of Success and the Shape of the Binomial Distribution Bar graph from the probability distribution from previous table The binomial probability distribution is symmetric if p = 5 ),4 Probability Distribution of for n = 4 and p = 5 7 ) ,,, 4
13 Probability of Success and the Shape of the Binomial Distribution cont Bar graph for the probability distribution for the previous table The binomial probability distribution is skewed to the right if p is less than 5 ),5 7 Probability Distribution of for n = 4 and p = ) ,4,,, 4 Probability of Success and the Shape of the Binomial Distribution cont The binomial probability distribution is skewed to the left if p is greater than 5 Bar graph for the probability distribution from previous table ),5 4,4 75 Probability Distribution of for n =4 and p = 8 ) ,,, 4 Mean and Standard Deviation of the Binomial Distributiontion The mean and standard deviation of a binomial distribution are np and npq where n is the total number of trails, p is the probability of success, and q is the probability of failure General rules for mean (epected values) If X and Y are random variables, a and b are constants: ) Ea ( ) a ) EaX ( ) aex ( ) EaX ( b) aex ( ) b ) E ( X Y ) E ( X ) E ( Y ) EaX ( by) aex ( ) bey ( ) 4) E( XY) E( X) E( Y) if X and Y are independent random variables 77 78
14 Rules for Variance If X and Y are random variables, a and b are constants, the variance V(X), V(Y) have: V( a) V( ax) a V( X) ( ) ( ) V ax b a V X 4 V( ax by ) av( X ) bv( Y ) if X and Y are independent Eample 5 In a Martiz poll of adult drivers conducted in July, 45% said that they often or sometimes eat or drink while driving i (USATODAY TODAY, October, ) Assume that this result is true for the current population p of all adult drivers A sample of 4 adult drivers is selected Let be the number of drivers in this sample who often or sometimes eat or drink while driving Find the mean and standard deviation of the probability distribution of 79 8 Eample 5: Answer Key n = 4 p = 45, and q = 55 np 4(45) 45) 8 npq (4)( 4)(45)(55) 45)( 55) 46 THE HYPERGEOMETRI PROBABILITY DISTRIBUTION Let 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) = number of successes in n trials n = number of failures in n trials 8 8 THE HYPERGEOMETRI PROBABILITY DISTRIBUTION HYPERGEOMETRI vs BINOMIAL The probability of successes in n trials is given by number of ways successes can be selected from a total of r successes in the population ) r N r N n n number of ways n failures can be selected from a total of N r failures in the population The hypergeometric distribution is closely related to the binomial distribution However, for the hypergeometric distribution: the trials are not independent, and the probability bilit of success changes from trial to trial 8 number of ways a sample of size n can be selected from a population of size N 84
15 Hypergeometric limit distribution Hypergeometric limit distribution onsider a hypergeometric distribution with n trials and let p = (r/n) denote the probability of a success on the first trial If the population size is large, the term (N n)/(n ) approaches The epected value and variance can be written E() = np and Var() = np( p) Note that these are the epressions for the epected value and variance of a binomial distribution onsider a hypergeometric distribution with n trials and let p = (r/n) denote the probability of a success on the first trial If the population size is large, the term (N n)/(n ) approaches The epected ed value and variance a can be written E() = np and Var() = np( p)?? Binomial?? When the population size is large, a hypergeometric distribution can be approimated by a binomial distribution with n trials and a probability of success p = (r/n) Eample 6 Eample 6: Solution Brown Manufacturing makes auto parts that are sold to auto dealers Last week the company shipped 5 auto parts to a dealer Later on, it found out that five of those parts were defective By the time the company manager contacted t the dealer, four auto parts from that shipment have already been sold What is the probability that three of those four parts were good parts and one was defective? r N r n 5 P ( ) N n! 5!!( )!!(5 )! 5! 4!(5 4)! 5 (4)(5) 456,65 Thus, the probability that three of the four parts sold are good and one is defective is Eample 7 Eample 7: Answers Dawn orporation has employees who hold managerial positions Of them, seven are female and five are male The company is planning to send of these managers to a conference If managers are randomly selected out of, a) Find the probability that all of them are female b) Find the probability that at most of them is a female (a) r N r n 5 ) N n (5)() 7 59 Thus, the probability that all three of managers selected re female ae is
16 Eample 7: Answers THE POISSON PROBABILITY DISTRIBUTION (b) r N r n 5 ) P ( ) r N N r N n n n 7 ()() (7)() 8 ) ) ) Using the Table of Poisson probabilities Mean and Standard Deviation of the Poisson Probability Distribution 9 9 THE POISSON PROBABILITY DISTRIBUTION (continued) Eamples onditions to Apply the Poisson Probability Distribution The following three conditions must be satisfied to apply the Poisson probability distribution is a discrete random variable The occurrences are random The occurrences are independent The number of accidents that occur on a given highway during a one-week period The number of customers entering a grocery store during a one hour interval The number of television i sets sold at a department t store during a given week 9 94 THE POISSON PROBABILITY DISTRIBUTION (continued) Poisson Probability Distribution Formula According to the Poisson probability bilit distribution, ib ti the probability of occurrences in an interval is 95 P ( ) e! where is the mean number of occurrences in that interval and the value of e is approimately Poisson distribution-characteristicstion Poisson variables is used to count the number of successes within a specified time or region Eg number of customers arriving at a bank teller in the net 5 minutes Poisson variables possess the following properties: The number of successes occurred in any interval is independent of the number of successes occurred in any other interval The probability that a success will occur in an interval is the same for all intervals of equal size and is proportional to the size of the interval The probability that two or more successes will occur in an interval approaches zero as the interval becomes smaller
17 Poisson distribution-link tion to binomial Poisson variables can be used to approimate Binomial variables when: p in binomial is small (less than 5) and n is large (larger than ), then we can use np to approimate binomial distribution Eample-using Poisson instead of binomial Probability of a defective screw p = Probability bilit of more than defects in a lot of screws? Binomial distribution: = np = ()() = Since p <<, can use Poisson distribution to approimate solution j X ) F() f ( j ) e e! j j j! X ) X ) 8% 997!! Eample 8 Eample 8: Solution On average, a household receives 95 telemarketing phone calls per week Using the Poisson distribution formula, find the probability that a randomly selected household receives eactly si telemarketing phone calls during a given week e (95) 6 e 95 6)! 6! (75,9896)(7485) Eample 9 Solution to Eample 9 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 net month this machine will have ( a) () e (9)(497877) P ( ) 4! a) eactly two breakdowns b) at most one breakdown ( b) () e () e P ( ) P ( )!! ()(497877) ()(497877)
18 Eample Eample : Answer Key ynthia s Mail Order ompany provides free eamination of its products for seven days If not completely satisfied, a customer can return the product within that period and get a full refund According to past records of the company, an average of of every products sold by this company are returned for a refund Using the Poisson probability distribution formula, find the probability that eactly 6 of the 4 products sold by this company on a given day will be returned for a refund = 8 =6 e 6)! 6 (8) e 6! 8 (6,44)(546) 7 4 Eample : Using the Table of Poisson Probabilities Portion of Table of Poisson Probabilities for = 5 On average, two new accounts are opened per day at an Imperial Saving Bank branch Using the Poisson table, find the probability that on a given day the number of new accounts opened at this bank will be a) eactly 6 b) at most c) at least = = 6 P ( = 6) Eample : Answers Mean and Standard Deviation of the Poisson Probability bilit Distribution ib ti 7 a) P ( = 6) = b) P (at most ) =P ( = ) + P ( = ) + P ( = ) + P ( = ) = = 857 c) P (at least 7) = P ( = 7) + P ( = 8) + P ( = 9) = = 45 8
19 Eample Eample : Answers Probability bilit Distribution ib ti of for = 9 9 An auto salesperson sells an average of 9 car per day Let be the number of cars sold by this salesperson on any given day Using the Poisson probability distribution table, a) Write the probability distribution of b) Draw a graph of the probability distribution c) Find the mean, variance, and standard deviation P () Eample : Answers (continued) Bar graph for the probability distribution of previous table Eample : Answers (concluded) ),45,4,5,,5,,5, 5, car car
8.1 Binomial Distributions
8.1 Binomial Distributions The Binomial Setting The 4 Conditions of a Binomial Setting: 1.Each observation falls into 1 of 2 categories ( success or fail ) 2 2.There is a fixed # n of observations. 3.All
More informationChapter 5 Student Lecture Notes 5-1. Department of Quantitative Methods & Information Systems. Business Statistics
Chapter 5 Student Lecture Notes 5-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals
More informationx is a random variable which is a numerical description of the outcome of an experiment.
Chapter 5 Discrete Probability Distributions Random Variables is a random variable which is a numerical description of the outcome of an eperiment. Discrete: If the possible values change by steps or jumps.
More informationBusiness Statistics. Chapter 5 Discrete Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 5 Discrete Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationExercises for Chapter (5)
Exercises for Chapter (5) MULTILE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) 500 families were interviewed and the number of children per family was
More informationLean Six Sigma: Training/Certification Books and Resources
Lean Si Sigma Training/Certification Books and Resources Samples from MINITAB BOOK Quality and Si Sigma Tools using MINITAB Statistical Software A complete Guide to Si Sigma DMAIC Tools using MINITAB Prof.
More informationDiscrete Random Variables and Their Probability Distributions
58 Chapter 5 Discrete Random Variables and Their Probability Distributions Discrete Random Variables and Their Probability Distributions Chapter 5 Section 5.6 Example 5-18, pg. 213 Calculating a Binomial
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 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 informationCHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS
CHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer.. The weighted average of the possible
More information6. THE BINOMIAL DISTRIBUTION
6. THE BINOMIAL DISTRIBUTION Eg: For 1000 borrowers in the lowest risk category (FICO score between 800 and 850), what is the probability that at least 250 of them will default on their loan (thereby rendering
More informationStatistics for Managers Using Microsoft Excel 7 th Edition
Statistics for Managers Using Microsoft Excel 7 th Edition Chapter 5 Discrete Probability Distributions Statistics for Managers Using Microsoft Excel 7e Copyright 014 Pearson Education, Inc. Chap 5-1 Learning
More informationThe Central Limit Theorem
Section 6-5 The Central Limit Theorem I. Sampling Distribution of Sample Mean ( ) Eample 1: Population Distribution Table 2 4 6 8 P() 1/4 1/4 1/4 1/4 μ (a) Find the population mean and population standard
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 information5: Several Useful Discrete Distributions
: Several Useful Discrete Distributions. Follow the instructions in the My Personal Trainer section. The answers are shown in the tables below. The Problem k 0 6 7 P( k).000.00.0.0.9..7.9.000 List the
More informationProbability Distributions. Chapter 6
Probability Distributions Chapter 6 McGraw-Hill/Irwin The McGraw-Hill Companies, Inc. 2008 GOALS Define the terms probability distribution and random variable. Distinguish between discrete and continuous
More 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 informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic Probability Distributions: Binomial and Poisson Distributions Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College
More informationRandom Variable: Definition
Random Variables Random Variable: Definition A Random Variable is a numerical description of the outcome of an experiment Experiment Roll a die 10 times Inspect a shipment of 100 parts Open a gas station
More information5. In fact, any function of a random variable is also a random variable
Random Variables - Class 11 October 14, 2012 Debdeep Pati 1 Random variables 1.1 Expectation of a function of a random variable 1. Expectation of a function of a random variable 2. We know E(X) = x xp(x)
More informationDiscrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8)
3 Discrete Random Variables and Probability Distributions Stat 4570/5570 Based on Devore s book (Ed 8) Random Variables We can associate each single outcome of an experiment with a real number: We refer
More 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 informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 6 Learning Objectives Define terms random variable and probability distribution. Distinguish between discrete and continuous probability distributions. Calculate
More informationProbability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 2 Discrete Distributions The binomial distribution 1 Chapter 2 Discrete Distributions Bernoulli trials and the
More informationAP Statistics Test 5
AP Statistics Test 5 Name: Date: Period: ffl If X is a discrete random variable, the the mean of X and the variance of X are given by μ = E(X) = X xp (X = x); Var(X) = X (x μ) 2 P (X = x): ffl If X is
More informationStatistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More informationA probability distribution shows the possible outcomes of an experiment and the probability of each of these outcomes.
Introduction In the previous chapter we discussed the basic concepts of probability and described how the rules of addition and multiplication were used to compute probabilities. In this chapter we expand
More informationChapter 5. Discrete Probability Distributions. McGraw-Hill, Bluman, 7 th ed, Chapter 5 1
Chapter 5 Discrete Probability Distributions McGraw-Hill, Bluman, 7 th ed, Chapter 5 1 Chapter 5 Overview Introduction 5-1 Probability Distributions 5-2 Mean, Variance, Standard Deviation, and Expectation
More informationChapter 7: Random Variables and Discrete Probability Distributions
Chapter 7: Random Variables and Discrete Probability Distributions 7. Random Variables and Probability Distributions This section introduced the concept of a random variable, which assigns a numerical
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives After this lesson we will be able to: determine whether a probability
More informationPart V - Chance Variability
Part V - Chance Variability Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Part V - Chance Variability 1 / 78 Law of Averages In Chapter 13 we discussed the Kerrich coin-tossing experiment.
More 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 informationS = 1,2,3, 4,5,6 occurs
Chapter 5 Discrete Probability Distributions The observations generated by different statistical experiments have the same general type of behavior. Discrete random variables associated with these experiments
More 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 informationCIVL Learning Objectives. Definitions. Discrete Distributions
CIVL 3103 Discrete Distributions Learning Objectives Define discrete distributions, and identify common distributions applicable to engineering problems. Identify the appropriate distribution (i.e. binomial,
More informationII - Probability. Counting Techniques. three rules of counting. 1multiplication rules. 2permutations. 3combinations
II - Probability Counting Techniques three rules of counting 1multiplication rules 2permutations 3combinations Section 2 - Probability (1) II - Probability Counting Techniques 1multiplication rules In
More informationDiscrete Random Variables and Probability Distributions
Chapter 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables A quantity resulting from an experiment that, by chance, can assume different values. A random variable is a variable
More informationChapter 3. Discrete Probability Distributions
Chapter 3 Discrete Probability Distributions 1 Chapter 3 Overview Introduction 3-1 The Binomial Distribution 3-2 Other Types of Distributions 2 Chapter 3 Objectives Find the exact probability for X successes
More informationStats CH 6 Intro Activity 1
Stats CH 6 Intro Activit 1 1. Purpose can ou tell the difference between bottled water and tap water? You will drink water from 3 samples. 1 of these is bottled water.. You must test them in the following
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 informationDiscrete Random Variables and Their Probability Distributions
Chapter 5 Discrete Random Variables and Their Probability Distributions Mean and Standard Deviation of a Discrete Random Variable Computing the mean and standard deviation of a discrete random variable
More informationChapter 4 Probability Distributions
Slide 1 Chapter 4 Probability Distributions Slide 2 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 4-5
More informationCHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 4 DISCRETE PROBABILITY DISTRIBUTIONS A random variable is the description of the outcome of an experiment in words. The verbal description of a random variable tells you how to find or calculate
More informationGOALS. Discrete Probability Distributions. A Distribution. What is a Probability Distribution? Probability for Dice Toss. A Probability Distribution
GOALS Discrete Probability Distributions Chapter 6 Dr. Richard Jerz Define the terms probability distribution and random variable. Distinguish between discrete and continuous probability distributions.
More informationBinomial population distribution X ~ B(
Chapter 9 Binomial population distribution 9.1 Definition of a Binomial distributio If the random variable has a Binomial population distributio i.e., then its probability function is given by p n n (
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 informationDiscrete Probability Distributions Chapter 6 Dr. Richard Jerz
Discrete Probability Distributions Chapter 6 Dr. Richard Jerz 1 GOALS Define the terms probability distribution and random variable. Distinguish between discrete and continuous probability distributions.
More 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 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 informationSTUDY SET 1. Discrete Probability Distributions. x P(x) and x = 6.
STUDY SET 1 Discrete Probability Distributions 1. Consider the following probability distribution function. Compute the mean and standard deviation of. x 0 1 2 3 4 5 6 7 P(x) 0.05 0.16 0.19 0.24 0.18 0.11
More informationChapter 4 and 5 Note Guide: Probability Distributions
Chapter 4 and 5 Note Guide: Probability Distributions Probability Distributions for a Discrete Random Variable A discrete probability distribution function has two characteristics: Each probability is
More informationMathacle. PSet Stats, Concepts In Statistics Level Number Name: Date: Distribution Distribute in anyway but normal
Distribution Distribute in anyway but normal VI. DISTRIBUTION A probability distribution is a mathematical function that provides the probabilities of occurrence of all distinct outcomes in the sample
More informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 3: Special Discrete Random Variable Distributions Section 3.5 Discrete Uniform Section 3.6 Bernoulli and Binomial Others sections
More 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 informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 6 McGraw-Hill/Irwin Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved. GOALS 6-2 1. Define the terms probability distribution and random variable.
More informationOverview. Definitions. Definitions. Graphs. Chapter 4 Probability Distributions. probability distributions
Chapter 4 Probability Distributions 4-1 Overview 4-2 Random Variables 4-3 Binomial Probability Distributions 4-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 4-5 The Poisson Distribution
More 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 informationChapter 4 Discrete Random Variables
Chapter 4 Discrete Random Variables It is often the case that a number is naturally associated to the outcome of a random eperiment: the number of boys in a three-child family, the number of defective
More informationMath 227 Practice Test 2 Sec Name
Math 227 Practice Test 2 Sec 4.4-6.2 Name Find the indicated probability. ) A bin contains 64 light bulbs of which 0 are defective. If 5 light bulbs are randomly selected from the bin with replacement,
More informationClass 13. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700
Class 13 Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 017 by D.B. Rowe 1 Agenda: Recap Chapter 6.3 6.5 Lecture Chapter 7.1 7. Review Chapter 5 for Eam 3.
More informationDiscrete Probability Distributions
Discrete Probability Distributions Chapter 6 Copyright 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Learning
More informationCentral Limit Theorem (cont d) 7/28/2006
Central Limit Theorem (cont d) 7/28/2006 Central Limit Theorem for Binomial Distributions Theorem. For the binomial distribution b(n, p, j) we have lim npq b(n, p, np + x npq ) = φ(x), n where φ(x) is
More informationDiscrete Probability Distributions
Page 1 of 6 Discrete Probability Distributions In order to study inferential statistics, we need to combine the concepts from descriptive statistics and probability. This combination makes up the basics
More informationBinomial Random Variable - The count X of successes in a binomial setting
6.3.1 Binomial Settings and Binomial Random Variables What do the following scenarios have in common? Toss a coin 5 times. Count the number of heads. Spin a roulette wheel 8 times. Record how many times
More informationProbability and Statistics for Engineers
Probability and Statistics for Engineers Chapter 4 Probability Distributions ruochen Liu ruochenliu@xidian.edu.cn Institute of Intelligent Information Processing, Xidian University Outline Random variables
More informationRandom variables. Contents
Random variables Contents 1 Random Variable 2 1.1 Discrete Random Variable............................ 3 1.2 Continuous Random Variable........................... 5 1.3 Measures of Location...............................
More informationStatistics (This summary is for chapters 17, 28, 29 and section G of chapter 19)
Statistics (This summary is for chapters 17, 28, 29 and section G of chapter 19) Mean, Median, Mode Mode: most common value Median: middle value (when the values are in order) Mean = total how many = x
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 Distributions
. Binomial Distributions Essential Question How can you determine the frequency of each outcome of an event? Analyzing Histograms Work with a partner. The histograms show the results when n coins are flipped.
More informationProbability Models.S2 Discrete Random Variables
Probability Models.S2 Discrete Random Variables Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard Results of an experiment involving uncertainty are described by one or more random
More informationBinomial and multinomial distribution
1-Binomial distribution Binomial and multinomial distribution The binomial probability refers to the probability that a binomial experiment results in exactly "x" successes. The probability of an event
More information6.5: THE NORMAL APPROXIMATION TO THE BINOMIAL AND
CD6-12 6.5: THE NORMAL APPROIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS In the earlier sections of this chapter the normal probability distribution was discussed. In this section another useful aspect
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
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 informationLearning Objec0ves. Statistics for Business and Economics. Discrete Probability Distribu0ons
Statistics for Business and Economics Discrete Probability Distribu0ons Learning Objec0ves In this lecture, you learn: The proper0es of a probability distribu0on To compute the expected value and variance
More informationChapter 7. Sampling Distributions
Chapter 7 Sampling Distributions Section 7.1 Sampling Distributions and the Central Limit Theorem Sampling Distributions Sampling distribution The probability distribution of a sample statistic. Formed
More informationMAKING SENSE OF DATA Essentials series
MAKING SENSE OF DATA Essentials series THE NORMAL DISTRIBUTION Copyright by City of Bradford MDC Prerequisites Descriptive statistics Charts and graphs The normal distribution Surveys and sampling Correlation
More informationSTOR Lecture 7. Random Variables - I
STOR 435.001 Lecture 7 Random Variables - I Shankar Bhamidi UNC Chapel Hill 1 / 31 Example 1a: Suppose that our experiment consists of tossing 3 fair coins. Let Y denote the number of heads that appear.
More informationDiscrete Probability Distribution
1 Discrete Probability Distribution Key Definitions Discrete Random Variable: Has a countable number of values. This means that each data point is distinct and separate. Continuous Random Variable: Has
More informationOverview. Definitions. Definitions. Graphs. Chapter 5 Probability Distributions. probability distributions
Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 5-5 The Poisson Distribution
More 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 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 information12. THE BINOMIAL DISTRIBUTION
12. THE BINOMIAL DISTRIBUTION Eg: The top line on county ballots is supposed to be assigned by random drawing to either the Republican or Democratic candidate. The clerk of the county is supposed to make
More information12. THE BINOMIAL DISTRIBUTION
12. THE BINOMIAL DISTRIBUTION Eg: The top line on county ballots is supposed to be assigned by random drawing to either the Republican or Democratic candidate. The clerk of the county is supposed to make
More informationMAS187/AEF258. University of Newcastle upon Tyne
MAS187/AEF258 University of Newcastle upon Tyne 2005-6 Contents 1 Collecting and Presenting Data 5 1.1 Introduction...................................... 5 1.1.1 Examples...................................
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 informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability
More informationUnit 4 The Bernoulli and Binomial Distributions
PubHlth 540 Fall 2013 4. Bernoulli and Binomial Page 1 of 21 Unit 4 The Bernoulli and Binomial Distributions If you believe in miracles, head for the Keno lounge - Jimmy the Greek The Amherst Regional
More informationThe Binomial and Geometric Distributions. Chapter 8
The Binomial and Geometric Distributions Chapter 8 8.1 The Binomial Distribution A binomial experiment is statistical experiment that has the following properties: The experiment consists of n repeated
More informationProbability Distributions
4.1 Probability Distributions Random Variables A random variable x represents a numerical value associated with each outcome of a probability distribution. A random variable is discrete if it has a finite
More 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 informationChapter 5 Probability Distributions. Section 5-2 Random Variables. Random Variable Probability Distribution. Discrete and Continuous Random Variables
Chapter 5 Probability Distributions Section 5-2 Random Variables 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance and Standard Deviation for the Binomial Distribution Random
More informationThe binomial distribution
The binomial distribution The coin toss - three coins The coin toss - four coins The binomial probability distribution Rolling dice Using the TI nspire Graph of binomial distribution Mean & standard deviation
More informationLecture 3. Sampling distributions. Counts, Proportions, and sample mean.
Lecture 3 Sampling distributions. Counts, Proportions, and sample mean. Statistical Inference: Uses data and summary statistics (mean, variances, proportions, slopes) to draw conclusions about a population
More informationCHAPTER 5 SOME DISCRETE PROBABILITY DISTRIBUTIONS. 5.2 Binomial Distributions. 5.1 Uniform Discrete Distribution
CHAPTER 5 SOME DISCRETE PROBABILITY DISTRIBUTIONS As we had discussed, there are two main types of random variables, namely, discrete random variables and continuous random variables. In this chapter,
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 information(c) The probability that a randomly selected driver having a California drivers license
Statistics Test 2 Name: KEY 1 Classify each statement as an example of classical probability, empirical probability, or subjective probability (a An executive for the Krusty-O cereal factory makes an educated
More informationRandom Variables Handout. Xavier Vilà
Random Variables Handout Xavier Vilà Course 2004-2005 1 Discrete Random Variables. 1.1 Introduction 1.1.1 Definition of Random Variable A random variable X is a function that maps each possible outcome
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 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 information