2011 Pearson Education, Inc
|
|
- Melanie Shepherd
- 5 years ago
- Views:
Transcription
1
2 Statistics for Business and Economics Chapter 4 Random Variables & Probability Distributions
3 Content 1. Two Types of Random Variables 2. Probability Distributions for Discrete Random Variables 3. The Binomial Distribution 4. Poisson and Hypergeometric Distributions 5. Probability Distributions for Continuous Random Variables 6. The Normal Distribution
4 Content (continued) 7. Approximating a Binomial Distribution with a Normal Distribution 8. Sampling Distributions 9. The Sampling Distribution of a Sample Mean and the Central Limit Theorem
5 Learning Objectives 1. Develop the notion of a random variable 2. Learn that numerical data are observed values of either discrete or continuous random variables 3. Study two important types of random variables and their probability models: the binomial and normal model 4. Define a sampling distribution as the probability of a sample statistic 5. Learn that the sampling distribution of x follows a normal model
6 Thinking Challenge You re taking a 33 question multiple choice test. Each question has 4 choices. Clueless on 1 question, you decide to guess. What s the chance you ll get it right? If you guessed on all 33 questions, what would be your grade? Would you pass?
7 4.1 Two Types of Random Variables
8 Random Variable A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point.
9 Discrete Random Variable Random variables that can assume a countable number (finite or infinite) of values are called discrete.
10 Discrete Random Variable Examples Experiment Random Variable Possible Values Make 100 Sales Calls # Sales 0, 1, 2,..., 100 Inspect 70 Radios # Defective 0, 1, 2,..., 70 Answer 33 Questions # Correct 0, 1, 2,..., 33 Count Cars at Toll Between 11:00 & 1:00 # Cars Arriving 0, 1, 2,...,
11 Continuous Random Variable Random variables that can assume values corresponding to any of the points contained in one or more intervals (i.e., values that are infinite and uncountable) are called continuous.
12 Continuous Random Variable Examples Experiment Random Variable Possible Values Weight 100 People Weight 45.1, 78,... Measure Part Life Hours 900, 875.9,... Amount spent on food $ amount 54.12, 42,... Measure Time Between Arrivals Inter-Arrival Time 0, 1.3, 2.78,...
13 4.2 Probability Distributions for Discrete Random Variables
14 Discrete Probability Distribution The probability distribution of a discrete random variable is a graph, table, or formula that specifies the probability associated with each possible value the random variable can assume.
15 Requirements for the Probability Distribution of a Discrete Random Variable x 1. p(x) 0 for all values of x 2. p(x) = 1 where the summation of p(x) is over all possible values of x.
16 Discrete Probability Distribution Example Experiment: Toss 2 coins. Count number of tails. Probability Distribution Values, x Probabilities, p(x) 0 1/4 = /4 = /4 = T/Maker Co.
17 Visualizing Discrete Probability Distributions Listing { (0,.25), (1,.50), (2,.25) } Graph p(x) x p ( x ) # Tails Table f(x) Count Formula p(x) n! = p x!(n x)! x (1 p) n x
18 Summary Measures 1. Expected Value (Mean of probability distribution) Weighted average of all possible values = E(x) = x p(x) 2. Variance Weighted average of squared deviation about mean 2 = E[(x 2 (x 2 p(x) 3. Standard Deviation 2
19 Summary Measures Calculation Table x p(x) x p(x) x (x 2 (x 2 p(x) Total x p(x) (x 2 p(x)
20 Thinking Challenge You toss 2 coins. You re interested in the number of tails. What are the expected value, variance, and standard deviation of this random variable, number of tails? T/Maker Co.
21 Expected Value & Variance Solution* x p(x) x p(x) x (x 2 (x 2 p(x) =
22 Probability Rules for Discrete Random Variables Let x be a discrete random variable with probability distribution p(x), mean µ, and standard deviation. Then, depending on the shape of p(x), the following probability statements can be made: Chebyshev s Rule Empirical Rule P x x µ 0.68 P x 2 x µ P x 3 x µ
23 4.3 The Binomial Distribution
24 Binomial Distribution Number of successes in a sample of n observations (trials) Number of reds in 15 spins of roulette wheel Number of defective items in a batch of 5 items Number correct on a 33 question exam Number of customers who purchase out of 100 customers who enter store (each customer is equally likely to pyrchase)
25 Binomial Probability Characteristics of a Binomial Experiment 1. The experiment consists of n identical trials. 2. There are only two possible outcomes on each trial. We will denote one outcome by S (for success) and the other by F (for failure). 3. The probability of S remains the same from trial to trial. This probability is denoted by p, and the probability of F is denoted by q. Note that q = 1 p. 4. The trials are independent. 5. The binomial random variable x is the number of S s in n trials.
26 Binomial Probability Distribution n n! p( x) p q p (1 p) x x! ( n x)! x n x x n x p(x) = Probability of x Successes p = Probability of a Success on a single trial q = 1 p n = Number of trials x = Number of Successes in n trials (x = 0, 1, 2,..., n) n x = Number of failures in n trials
27 Binomial Probability Distribution Example Experiment: Toss 1 coin 5 times in a row. Note number of tails. What s the probability of 3 tails? n! x p( x) p (1 p) x!( n x)! n x T/Maker Co. 5! p(3).5 (1.5) 3!(5 3)!
28 Binomial Probability Table n = 5 (Portion) k Cumulative Probabilities p(x 3) p(x 2) = =.312 p
29 Binomial Distribution Characteristics Mean E(x) np P(X) n = 5 p = 0.1 Standard Deviation npq P(X) n = 5 p = X X
30 Binomial Distribution Thinking Challenge You re a telemarketer selling service contracts for Macy s. You ve sold 20 in your last 100 calls (p =.20). If you call 12 people tonight, what s the probability of A. No sales? B. Exactly 2 sales? C. At most 2 sales? D. At least 2 sales?
31 Binomial Distribution Solution* n = 12, p =.20 A. p(0) =.0687 B. p(2) =.2835 C. p(at most 2) = p(0) + p(1) + p(2) = =.5584 D. p(at least 2) = p(2) + p(3)...+ p(12) = 1 [p(0) + p(1)] = =.7251
32 4.4 Other Discrete Distributions: Poisson and Hypergeometric
33 Poisson Distribution 1. Number of events that occur in an interval events per unit Time, Length, Area, Space 2. Examples Number of customers arriving in 20 minutes Number of strikes per year in the U.S. Number of defects per lot (group) of DVD s
34 Characteristics of a Poisson Random Variable 1. Consists of counting number of times an event occurs during a given unit of time or in a given area or volume (any unit of measurement). 2. The probability that an event occurs in a given unit of time, area, or volume is the same for all units. 3. The number of events that occur in one unit of time, area, or volume is independent of the number that occur in any other mutually exclusive unit. 4. The mean number of events in each unit is denoted by.
35 Poisson Probability Distribution Function 2 p(x) = Probability of x given = Mean (expected) number of events in unit e x p ( x ) x x e (x = 0, 1, 2, 3,...) = (base of natural logarithm) = Number of events per unit!
36 Poisson Probability Distribution Function = 0.5 Mean E(x) P(X) X Standard Deviation P(X) 2 4 = X
37 Poisson Distribution Example Customers arrive at a rate of 72 per hour. What is the probability of 4 customers arriving in 3 minutes? 1995 Corel Corp.
38 Poisson Distribution Solution 72 Per Hr. = 1.2 Per Min. = 3.6 Per 3 Min. Interval px ( ) x e - x! e p(4) !
39 Poisson Probability Table (Portion) x : : : : : : : : : : : : : : Cumulative Probabilities p(x 4) p(x 3) = =.191
40 Thinking Challenge You work in Quality Assurance for an investment firm. A clerk enters 75 words per minute with 6 errors per hour. What is the probability of 0 errors in a 255-word bond transaction? T/Maker Co.
41 Poisson Distribution Solution: Finding * 75 words/min = (75 words/min)(60 min/hr) = 4500 words/hr 6 errors/hr = 6 errors/4500 words = errors/word In a 255-word transaction (interval): = ( errors/word )(255 words) =.34 errors/255-word transaction
42 Poisson Distribution Solution: Finding p(0)* px ( ) x e - x! e p(0) !
43 Characteristics of a Hypergeometric Random Variable 1. The experiment consists of randomly drawing n elements without replacement from a set of N elements, r of which are S s (for success) and (N r) of which are F s (for failure). 2. The hypergeometric random variable x is the number of S s in the draw of n elements.
44 Hypergeometric Probability Distribution Function p x r N r x n x N n [x = Maximum [0, n (N r),, Minimum (r, n)] µ nr N 2 r N r N 2 n N n N 1 where...
45 Hypergeometric Probability Distribution Function N = Total number of elements r = Number of S s in the N elements n = Number of elements drawn x = Number of S s drawn in the n elements
46 4.5 Probability Distributions for Continuous Random Variables
47 Continuous Probability Density Function The graphical form of the probability distribution for a continuous random variable x is a smooth curve
48 Continuous Probability Density Function This curve, a function of x, is denoted by the symbol f(x) and is variously called a probability density function (pdf), a frequency function, or a probability distribution. The areas under a probability distribution correspond to probabilities for x. The area A beneath the curve between two points a and b is the probability that x assumes a value between a and b.
49 4.6 The Normal Distribution
50 Importance of Normal Distribution 1. Describes many random processes or continuous phenomena 2. Can be used to approximate discrete probability distributions Example: binomial 3. Basis for classical statistical inference
51 Normal Distribution 1. Bell-shaped & symmetrical f ( x ) 2. Mean, median, mode are equal x Mean Median Mode
52 Probability Density Function f (x) 1 2 e 1 2 x 2 where µ = Mean of the normal random variable x = Standard deviation π = e = P(x < a) is obtained from a table of normal probabilities
53 Effect of Varying Parameters ( & )
54 Normal Distribution Probability Probability is area under curve! P(c x d) c d f (x) dx? f ( x ) c d x
55 Standard Normal Distribution The standard normal distribution is a normal distribution with µ = 0 and = 1. A random variable with a standard normal distribution, denoted by the symbol z, is called a standard normal random variable.
56 The Standard Normal Table: P(0 < z < 1.96) Standardized Normal Probability Table (Portion) Z = 0 Probabilities = z Shaded area exaggerated
57 The Standard Normal Table: P( 1.26 z 1.26) Standardized Normal Distribution = z = 0 Shaded area exaggerated P( 1.26 z 1.26) = =.7924
58 The Standard Normal Table: P(z > 1.26) Standardized Normal Distribution = = 0 z P(z > 1.26) = =.1038
59 The Standard Normal Table: P( 2.78 z 2.00) Standardized Normal Distribution = z = 0 P( 2.78 z 2.00) = =.0201 Shaded area exaggerated
60 The Standard Normal Table: P(z > 2.13) Standardized Normal Distribution = z = 0 Shaded area exaggerated P(z > 2.13) = =.9834
61 Non-standard Normal Distribution Normal distributions differ by mean & standard deviation. Each distribution would require its own table. f(x) x That s an infinite number of tables!
62 Property of Normal Distribution If x is a normal random variable with mean μ and standard deviation, then the random variable z, defined by the formula z x µ has a standard normal distribution. The value z describes the number of standard deviations between x and µ.
63 Standardize the Normal Distribution Normal Distribution z x Standardized Normal Distribution = 1 x = 0 z One table!
64 Finding a Probability Corresponding to a Normal Random Variable 1. Sketch normal distribution, indicate mean, and shade the area corresponding to the probability you want. 2. Convert the boundaries of the shaded area from x values to standard normal random variable z z x µ Show the z values under corresponding x values. 3. Use Table IV in Appendix A to find the areas corresponding to the z values. Use symmetry when necessary.
65 Non-standard Normal μ = 5, σ = 10: Normal Distribution = 10 P(5 < x < 6.2) z x Standardized Normal Distribution = = x = 0.12 z Shaded area exaggerated
66 Non-standard Normal μ = 5, σ = 10: Normal Distribution = 10 P(3.8 x 5) z x Standardized Normal Distribution = = 5 x Shaded area exaggerated -.12 = 0 z
67 Non-standard Normal μ = 5, σ = 10: z x Normal Distribution = 10 P(2.9 x 7.1).21 z x Standardized Normal Distribution = x z Shaded area exaggerated
68 Non-standard Normal μ = 5, σ = 10: Normal Distribution = 10 P(x 8) z x Standardized Normal Distribution = = 5 8 x = 0.30 z Shaded area exaggerated
69 Non-standard Normal μ = 5, σ = 10: z x Normal Distribution = 10 P(7.1 X 8).21 z x Standardized Normal Distribution = = x = z Shaded area exaggerated
70 Normal Distribution Thinking Challenge You work in Quality Control for GE. Light bulb life has a normal distribution with = 2000 hours and = 200 hours. What s the probability that a bulb will last A. between 2000 and 2400 hours? B. less than 1470 hours?
71 Solution* P(2000 x 2400) Normal Distribution z x Standardized Normal Distribution = 200 = = x = z
72 Solution* P(x 1470) Normal Distribution = 200 z x Standardized Normal Distribution = = 2000 x 2.65 = 0 z
73 Finding z-values for Known Probabilities What is Z, given P(z) =.1217? Standardized Normal Probability Table (Portion).1217 = 1 Z Shaded area exaggerated = 0?.31 z
74 Finding x Values for Known Probabilities Normal Distribution = 10 Standardized Normal Distribution = = 5 8.1? x = 0.31 z x z Shaded areas exaggerated
75 4.8 Approximating a Binomial Distribution with a Normal Distribution
76 Normal Approximation of Binomial Distribution 1. Useful because not all binomial tables exist 2. Requires large sample size 3. Gives approximate probability only 4. Need correction for continuity n = 10 p = 0.50 p(x) x
77 Why Probability Is Approximate p(x) Binomial Probability: Bar Height Probability Added by Normal Curve Probability Lost by Normal Curve x Normal Probability: Area Under Curve from 3.5 to 4.5
78 Correction for Continuity 1. A 1/2 unit adjustment to discrete variable 2. Used when approximating a discrete distribution with a continuous distribution 3. Improves accuracy (4.5) 4.5 (4 +.5)
79 Using a Normal Distribution to Approximate Binomial Probabilities 1. Determine n and p for the binomial distribution, then calculate the interval: 3 np 3 np 1 p If interval lies in the range 0 to n, the normal distribution will provide a reasonable approximation to the probabilities of most binomial events.
80 Using a Normal Distribution to Approximate Binomial Probabilities 2. Express the binomial probability to be approximated by the form P x a For example, or P x b P x a P x 3 P x 2 P x 5 1 P x 4 P 7 x 10 P x 10 P x 6
81 Using a Normal Distribution to Approximate Binomial Probabilities 3. For each value of interest a, the correction for continuity is (a +.5), and the corresponding standard normal z-value is z a.5 µ
82 Using a Normal Distribution to Approximate Binomial Probabilities 4. Sketch the approximating normal distribution and shade the area corresponding to the event of interest. Using Table IV and the z-value (step 3), find the shaded area. This is the approximate probability of the binomial event.
83 Normal Approximation Example What is the normal approximation of p(x = 4) given n = 10, and p = 0.5? P(x) x
84 Normal Approximation Solution 1. Calculate the interval: np 3 np 1 p , 9.74 Interval lies in range 0 to 10, so normal approximation can be used 2. Express binomial probability in form: P x 4 P x 4 P x 3
85 Normal Approximation Solution 3. Compute standard normal z values: z a.5 n p n p 1 p z a.5 n p n p 1 p
86 Normal Approximation Solution 4. Sketch the approximate normal distribution: = 0 = z
87 Normal Approximation Solution 5. The exact probability from the binomial formula is (versus.2034) p(x) x
88 4.10 Sampling Distributions
89 Parameter & Statistic A parameter is a numerical descriptive measure of a population. Because it is based on all the observations in the population, its value is almost always unknown. A sample statistic is a numerical descriptive measure of a sample. It is calculated from the observations in the sample.
90 Common Statistics & Parameters Sample Statistic Population Parameter Mean x Standard Deviation s Variance s 2 2 Binomial Proportion ^ p p
91 Sampling Distribution The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic.
92 Developing Sampling Distributions Suppose There s a Population... Population size, N = 4 Random variable, x Values of x: 1, 2, 3, 4 Uniform distribution T/Maker Co.
93 Population Characteristics Summary Measure Population Distribution N x i i 1 N 2.5 P(x) x
94 All Possible Samples of Size n = 2 16 Samples 16 Sample Means 1st Obs 2nd Observation st Obs 2nd Observation ,1 1,2 1,3 1, ,1 2,2 2,3 2, ,1 3,2 3,3 3, ,1 4,2 4,3 4, Sample with replacement
95 Sampling Distribution of All Sample Means 1st Obs Sample Means Sampling Distribution of the Sample Mean 2nd Observation P(x) x
96 Summary Measure of All Sample Means X N x i i 1 N
97 Comparison Population Sampling Distribution P(x) x P(x) x 2.5 x 2.5
98 4.11 The Sampling Distribution of a Sample Mean and the Central Limit Theorem
99 Properties of the Sampling Distribution of x 1. Mean of the sampling distribution equals mean of sampled population*, that is, x E x. 2. Standard deviation of the sampling distribution equals Standard deviation of sampled population Square root of sample size That is, x n.
100 Standard Error of the Mean The standard deviation x is often referred to as the standard error of the mean.
101 Theorem 4.1 If a random sample of n observations is selected from a population with a normal distribution, the sampling distribution of x will be a normal distribution.
102 Sampling from Normal Populations Central Tendency Dispersion x x n Sampling with replacement n = 4 x = 5 Population Distribution = 50 = 10 Sampling Distribution x n =16 x = 2.5 x - = 50 x
103 Standardizing the Sampling Distribution of x z x x x x Sampling Distribution x n Standardized Normal Distribution = 1 x x = 0 z
104 Thinking Challenge You re an operations analyst for AT&T. Long-distance telephone calls are normally distributed with = 8 min. and = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes? T/Maker Co.
105 Sampling Distribution Solution* Sampling Distribution x =.4 z x n 25 z x n Standardized Normal Distribution = x z
106 Sampling from Non-Normal Populations Central Tendency Dispersion x x n Sampling with replacement Population Distribution n = 4 x = 5 = 50 = 10 Sampling Distribution x n =30 x = 1.8 x - = 50 x
107 Central Limit Theorem Consider a random sample of n observations selected from a population (any probability distribution) with mean μ and standard deviation. Then, when n is sufficiently large, the sampling distribution of x will be approximately a normal distribution with mean x and standard deviation x n. The larger the sample size, the better will be the normal approximation to the sampling distribution of x.
108 Central Limit Theorem As sample size gets large enough (n 30)... x n sampling distribution becomes almost normal. x x
109 Central Limit Theorem Example The amount of soda in cans of a particular brand has a mean of 12 oz and a standard deviation of.2 oz. If you select random samples of 50 cans, what percentage of the sample means would be less than oz? SODA
110 Central Limit Theorem Solution* Sampling Distribution x =.03 z x n Standardized Normal Distribution.0384 = x z Shaded area exaggerated.4616
111 Key Ideas Properties of Probability Distributions Discrete Distributions 1. p(x) 0 2. p x 1 all x Continuous Distributions 1. P(x = a) = 0 2. P(a < x < b) = area under curve between a and b
112 Key Ideas Normal Approximation to Binomial x is binomial (n, p) P x a P z a.5 µ
113 Key Ideas Methods for Assessing Normality 1. Histogram
114 Key Ideas Methods for Assessing Normality 2. Stem-and-leaf display
115 Key Ideas Methods for Assessing Normality 3. (IQR)/S Normal probability plot
116 Key Ideas Generating the Sampling Distribution of x
Statistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More informationSection Introduction to Normal Distributions
Section 6.1-6.2 Introduction to Normal Distributions 2012 Pearson Education, Inc. All rights reserved. 1 of 105 Section 6.1-6.2 Objectives Interpret graphs of normal probability distributions Find areas
More informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
More informationChapter 6 Continuous Probability Distributions. Learning objectives
Chapter 6 Continuous s Slide 1 Learning objectives 1. Understand continuous probability distributions 2. Understand Uniform distribution 3. Understand Normal distribution 3.1. Understand Standard normal
More informationSection 7.5 The Normal Distribution. Section 7.6 Application of the Normal Distribution
Section 7.6 Application of the Normal Distribution A random variable that may take on infinitely many values is called a continuous random variable. A continuous probability distribution is defined by
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More 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 4 Random Variables & Probability. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random variable =
More informationBayes s Rule Example. defective. An MP3 player is selected at random and found to be defective. What is the probability it came from Factory I?
Bayes s Rule Example A company manufactures MP3 players at two factories. Factory I produces 60% of the MP3 players and Factory II produces 40%. Two percent of the MP3 players produced at Factory I are
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 informationTheoretical Foundations
Theoretical Foundations Probabilities Monia Ranalli monia.ranalli@uniroma2.it Ranalli M. Theoretical Foundations - Probabilities 1 / 27 Objectives understand the probability basics quantify random phenomena
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 6 The Normal Distribution And Other Continuous Distributions
Statistics for Managers Using Microsoft Excel/SPSS Chapter 6 The Normal Distribution And Other Continuous Distributions 1999 Prentice-Hall, Inc. Chap. 6-1 Chapter Topics The Normal Distribution The Standard
More informationIntroduction to Business Statistics QM 120 Chapter 6
DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 When a RV x is discrete, we can
More 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 informationIntroduction to Statistical Data Analysis II
Introduction to Statistical Data Analysis II JULY 2011 Afsaneh Yazdani Preface Major branches of Statistics: - Descriptive Statistics - Inferential Statistics Preface What is Inferential Statistics? Preface
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter7 Probability Distributions and Statistics Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number of boys in
More 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 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 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 informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
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 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 informationPROBABILITY DISTRIBUTIONS
CHAPTER 3 PROBABILITY DISTRIBUTIONS Page Contents 3.1 Introduction to Probability Distributions 51 3.2 The Normal Distribution 56 3.3 The Binomial Distribution 60 3.4 The Poisson Distribution 64 Exercise
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
More informationProbability Distribution Unit Review
Probability Distribution Unit Review Topics: Pascal's Triangle and Binomial Theorem Probability Distributions and Histograms Expected Values, Fair Games of chance Binomial Distributions Hypergeometric
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 informationThe Normal Distribution
5.1 Introduction to Normal Distributions and the Standard Normal Distribution Section Learning objectives: 1. How to interpret graphs of normal probability distributions 2. How to find areas under the
More informationProbability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
More 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 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 informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number
More 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 informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 6 Normal Probability Distributions 6-1 Overview 6-2 The Standard Normal Distribution
More informationStatistics, Measures of Central Tendency I
Statistics, Measures of Central Tendency I We are considering a random variable X with a probability distribution which has some parameters. We want to get an idea what these parameters are. We perfom
More informationRandom Variables and Probability Functions
University of Central Arkansas Random Variables and Probability Functions Directory Table of Contents. Begin Article. Stephen R. Addison Copyright c 001 saddison@mailaps.org Last Revision Date: February
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 informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationChapter 7 1. Random Variables
Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous
More informationA continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x)
Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density
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 informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
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 informationHomework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a
Homework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a Announcements: There are some office hour changes for Nov 5, 8, 9 on website Week 5 quiz begins after class today and ends at
More informationSTAT Chapter 5: Continuous Distributions. Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s.
STAT 515 -- Chapter 5: Continuous Distributions Probability distributions are used a bit differently for continuous r.v. s than for discrete r.v. s. Continuous distributions typically are represented by
More informationStatistical Tables Compiled by Alan J. Terry
Statistical Tables Compiled by Alan J. Terry School of Science and Sport University of the West of Scotland Paisley, Scotland Contents Table 1: Cumulative binomial probabilities Page 1 Table 2: Cumulative
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 informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More 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 informationChapter 5 Discrete Probability Distributions. Random Variables Discrete Probability Distributions Expected Value and Variance
Chapter 5 Discrete Probability Distributions Random Variables Discrete Probability Distributions Expected Value and Variance.40.30.20.10 0 1 2 3 4 Random Variables A random variable is a numerical description
More informationWeek 1 Variables: Exploration, Familiarisation and Description. Descriptive Statistics.
Week 1 Variables: Exploration, Familiarisation and Description. Descriptive Statistics. Convergent validity: the degree to which results/evidence from different tests/sources, converge on the same conclusion.
More 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 informationCounting Basics. Venn diagrams
Counting Basics Sets Ways of specifying sets Union and intersection Universal set and complements Empty set and disjoint sets Venn diagrams Counting Inclusion-exclusion Multiplication principle Addition
More 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 informationLecture 12. Some Useful Continuous Distributions. The most important continuous probability distribution in entire field of statistics.
ENM 207 Lecture 12 Some Useful Continuous Distributions Normal Distribution The most important continuous probability distribution in entire field of statistics. Its graph, called the normal curve, is
More informationDATA SUMMARIZATION AND VISUALIZATION
APPENDIX DATA SUMMARIZATION AND VISUALIZATION PART 1 SUMMARIZATION 1: BUILDING BLOCKS OF DATA ANALYSIS 294 PART 2 PART 3 PART 4 VISUALIZATION: GRAPHS AND TABLES FOR SUMMARIZING AND ORGANIZING DATA 296
More informationExample. Chapter 8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables
Chapter 8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables You are dealt a hand of 5 cards. Find the probability distribution table for the number of hearts. Graph
More informationChapter ! Bell Shaped
Chapter 6 6-1 Business Statistics: A First Course 5 th Edition Chapter 7 Continuous Probability Distributions Learning Objectives In this chapter, you learn:! To compute probabilities from the normal distribution!
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 informationThe Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
More informationWeek 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
More informationDepartment of Quantitative Methods & Information Systems. Business Statistics. Chapter 6 Normal Probability Distribution QMIS 120. Dr.
Department of Quantitative Methods & Information Systems Business Statistics Chapter 6 Normal Probability Distribution QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationIntroduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous random variable)
More information. 13. The maximum error (margin of error) of the estimate for μ (based on known σ) is:
Statistics Sample Exam 3 Solution Chapters 6 & 7: Normal Probability Distributions & Estimates 1. What percent of normally distributed data value lie within 2 standard deviations to either side of the
More informationNormal distribution. We say that a random variable X follows the normal distribution if the probability density function of X is given by
Normal distribution The normal distribution is the most important distribution. It describes well the distribution of random variables that arise in practice, such as the heights or weights of people,
More informationChapter 8. Variables. Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Chapter 8 Random Variables Copyright 2004 Brooks/Cole, a division of Thomson Learning, Inc. 8.1 What is a Random Variable? Random Variable: assigns a number to each outcome of a random circumstance, or,
More information11.5: Normal Distributions
11.5: Normal Distributions 11.5.1 Up to now, we ve dealt with discrete random variables, variables that take on only a finite (or countably infinite we didn t do these) number of values. A continuous random
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More 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 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 informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10
ECO220Y Continuous Probability Distributions: Normal Readings: Chapter 9, section 9.10 Fall 2011 Lecture 8 Part 2 (Fall 2011) Probability Distributions Lecture 8 Part 2 1 / 23 Normal Density Function f
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 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 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 informationChapter 6. The Normal Probability Distributions
Chapter 6 The Normal Probability Distributions 1 Chapter 6 Overview Introduction 6-1 Normal Probability Distributions 6-2 The Standard Normal Distribution 6-3 Applications of the Normal Distribution 6-5
More informationMATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION
MATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION We have examined discrete random variables, those random variables for which we can list the possible values. We will now look at continuous random variables.
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 27 Continuous
More informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More 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 informationContinuous Probability Distributions & Normal Distribution
Mathematical Methods Units 3/4 Student Learning Plan Continuous Probability Distributions & Normal Distribution 7 lessons Notes: Students need practice in recognising whether a problem involves a discrete
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More informationAP STATISTICS FALL SEMESTSER FINAL EXAM STUDY GUIDE
AP STATISTICS Name: FALL SEMESTSER FINAL EXAM STUDY GUIDE Period: *Go over Vocabulary Notecards! *This is not a comprehensive review you still should look over your past notes, homework/practice, Quizzes,
More informationSection Random Variables and Histograms
Section 3.1 - Random Variables and Histograms Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationMath 2311 Bekki George Office Hours: MW 11am to 12:45pm in 639 PGH Online Thursdays 4-5:30pm And by appointment
Math 2311 Bekki George bekki@math.uh.edu Office Hours: MW 11am to 12:45pm in 639 PGH Online Thursdays 4-5:30pm And by appointment Class webpage: http://www.math.uh.edu/~bekki/math2311.html Math 2311 Class
More information8.1 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 informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationBinomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
More informationHomework: Due Wed, Feb 20 th. Chapter 8, # 60a + 62a (count together as 1), 74, 82
Announcements: Week 5 quiz begins at 4pm today and ends at 3pm on Wed If you take more than 20 minutes to complete your quiz, you will only receive partial credit. (It doesn t cut you off.) Today: Sections
More information2017 Fall QMS102 Tip Sheet 2
Chapter 5: Basic Probability 2017 Fall QMS102 Tip Sheet 2 (Covering Chapters 5 to 8) EVENTS -- Each possible outcome of a variable is an event, including 3 types. 1. Simple event = Described by a single
More informationData Analytics (CS40003) Practice Set IV (Topic: Probability and Sampling Distribution)
Data Analytics (CS40003) Practice Set IV (Topic: Probability and Sampling Distribution) I. Concept Questions 1. Give an example of a random variable in the context of Drawing a card from a deck of cards.
More informationChapter 4: Commonly Used Distributions. Statistics for Engineers and Scientists Fourth Edition William Navidi
Chapter 4: Commonly Used Distributions Statistics for Engineers and Scientists Fourth Edition William Navidi 2014 by Education. This is proprietary material solely for authorized instructor use. Not authorized
More informationSTAT 201 Chapter 6. Distribution
STAT 201 Chapter 6 Distribution 1 Random Variable We know variable Random Variable: a numerical measurement of the outcome of a random phenomena Capital letter refer to the random variable Lower case letters
More informationData that can be any numerical value are called continuous. These are usually things that are measured, such as height, length, time, speed, etc.
Chapter 8 Measures of Center Data that can be any numerical value are called continuous. These are usually things that are measured, such as height, length, time, speed, etc. Data that can only be integer
More informationA.REPRESENTATION OF DATA
A.REPRESENTATION OF DATA (a) GRAPHS : PART I Q: Why do we need a graph paper? Ans: You need graph paper to draw: (i) Histogram (ii) Cumulative Frequency Curve (iii) Frequency Polygon (iv) Box-and-Whisker
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 28 One more
More informationModel Paper Statistics Objective. Paper Code Time Allowed: 20 minutes
Model Paper Statistics Objective Intermediate Part I (11 th Class) Examination Session 2012-2013 and onward Total marks: 17 Paper Code Time Allowed: 20 minutes Note:- You have four choices for each objective
More informationChapter 4 Probability and Probability Distributions. Sections
Chapter 4 Probabilit and Probabilit Distributions Sections 4.6-4.10 Sec 4.6 - Variables Variable: takes on different values (or attributes) Random variable: cannot be predicted with certaint Random Variables
More informationChapter 5 Normal Probability Distributions
Chapter 5 Normal Probability Distributions Section 5-1 Introduction to Normal Distributions and the Standard Normal Distribution A The normal distribution is the most important of the continuous probability
More information