MANAGEMENT PRINCIPLES AND STATISTICS (252 BE)
|
|
- Franklin Kennedy
- 6 years ago
- Views:
Transcription
1 MANAGEMENT PRINCIPLES AND STATISTICS (252 BE) Normal and Binomial Distribution Applied to Construction Management Sampling and Confidence Intervals Sr Tan Liat Choon Mobile:
2 Probability and Statistics 2
3 Probability and Statistics Probability : Population Sample Statistics : Population Sample In probability theory, it is assumed that properties and characteristics of a population are known. In statistics, those of a sample are obtainable. Those of the population are NOT known and they are exactly what we like to know and understand. Based on data, we investigate appropriate models, estimate them (parameters) and make scientific decisions ( Hypothesis testing ). Since a sample is a subset of a population, we might reach incorrect conclusions. It is necessary to quantify and control the uncertainties and errors 3
4 Supplements (1) Mode: The most frequently occurring value in a data set is called the mode. Ex1) { 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5} Mode = 5 Unimodal Ex2) {2, 2, 3, 4, 4, 5} There are two modes (2 and 4). Bimodal Midrange: (maximum + minimum)/2 Ex3) {20, 30, 34, 55, 10, 1} midrange = (1+55)/2 = 28 4
5 Supplements (2) Order Statistics: Sort a data set in ascending order. x, x,..., x x, x,..., x => x(1) x(2) x( n) 1 2 n (1) (2) ( n) (... ) Ex) Data = { 20, 14, 5, 6, 20 } x (1) = 5 {5, 6, 14, 20, 20} x x (2) (3) = 6 = 14 x (4) = 20 x (5) = 20 5
6 Supplements (3) 100p-th Percentiles: The 100p-th percentile is a number below which 100p % observations lie. If 100p% of data lie below a number, then the percentile rank of the number is 100p. How to find the 100p-th percentile: = X + 100p-th percentile (( n 1)* p) (n+1)*p-th order statistic 6
7 Supplements (4) 100p-th Percentiles: Example) {4, 6, 17, 8, 12, 16, 13, 7} Find the 70 th percentile of the data set! 7
8 100p-th Percentiles: Supplements (5) Example 1) {4, 6, 17, 8, 12, 16, 13, 7} 1. Sort the data set in ascending order. { 4, 6, 7, 8, 12, 13, 16, 17 } 2. Find p. p= Find (n+1)*p th order statistic 9*0.7 th order statistic = 6.3-th order statistic. We want to find 6.3th order statistic. 6 th order statistic is 13 and 7 th order statistic is 16. The total distance is 3 units, we wish to go 30% of that, which is 0.9. The 70 th percentile is
9 Supplements (6) 100p-th Percentiles: Example 2) {40, 6, 17, 80, 12, 7, 9} Find 80 th percentile of the data set. 9
10 Probability Definitions and Properties (1) Sample Space (S): The set of every possible outcomes of an experiment is called a sample space (S). Ex1) What is the sample space when you toss a coin S= {Head, Tail}={H,T} Ex2) What if you toss the coin twice? S=? Ex3) What about rolling a die? S=? 10
11 Definitions and Properties (2) Event : an event is a subset of a sample space. Ex1) List all events of S={H,T}: {}, {H}, {T}, {H, T}. Ex2) # of all subsets of a set 2^(# of data in the set} For S={H,T} 2^2 = 4 11
12 Definitions and Properties (3) Probability (p) (always between 0 and 1): a numerical quantity that expresses the likelihood of an event. The probability of an event E is written as P(E) or Pr(E). Ex1) Probability of a fair coin: S={H,T}. p({})=0 p({h})=1/2 P({T})=1/2 P({H,T})=1 Ex2) Probability of a fair die : S={1, 2, 3, 4, 5, 6} 12
13 Definitions and Properties (4-1) (Set Theory) Set operations - Union (or) (The set of all objects that are a member of A, or B, or both) - Intersection (and) (The set of all objects that are members of both) - Complement (not) (The set of all members of S that are not members of A) A B = { x : x A or x B} A B = { x : x A and x B} C c A = { x: x A} 13
14 Definitions and Properties (4-2) (Set Theory) Let A and B be two events (S : Sample space) (a) (b) (c) (d) (e) P( φ ) = P({}) = 0 c PA ( ) = 1 PA ( ) PA ( B) = PA ( ) + PB ( ) PA ( B) PS ( ) = 1 If A B, P( A) P( B) 14
15 Definitions and Properties (4-3) (Set Theory) Mutually exclusive If Independent If A B= φ = {} PA ( B) = PA ( ) PB ( ), then A and B are mutually exclusive., then A and B are independent. Ex) suppose that P(A)=1/2 and P(B)=2/3 and they are independent. PA ( B) Calculate! Ex) suppose that P(A)=1/2 and P(B)=1/3 and they are mutually exclusive. PA ( B) Calculate! 15
16 Definitions and Properties (4-4) Venn Diagram: (Set Theory) A visual representation of a set. Each event is represented as a circle and the sample space is represented as a rectangle Ex) A={1,2,3,4}, B={3,4,5,6}, S={1,2,3,4,5,6,7} Ex) P(A)=0.82, P(B)=0.31, P(A n B)= 0.19 Ex) Mutually exclusive events A and B 16
17 Definitions and Properties (4-5) Example: (Set Theory) In a certain residential suburb, 60% of all households subscribe to the metropolitan newspaper published in a nearby city, 80% subscribe to the local afternoon paper, and 50% of all households subscribe to both papers. If a household is selected at random, what is the probability that it subscribes to (1) at least one of the two newspapers and (2) exactly one of the two newspapers? 17
18 Definitions and Properties (5) Conditional Probabilities: Suppose that P(B) is not equal to 0, we define the conditional probability of A given that the event B has occurred, PAB ( ) = PA ( B) PB ( ) Ex) P(A n B) = 0.3, P(B)= 0.5 P(A B)=? Ex) Two events A and B are independent. P(A)=0.4, P(B)=0.5, then P(A B)=? 18
19 Definitions and Properties (5-1) Example: Suppose that of all individuals buying a certain personal computer, 60% include a word processing program in their purchase, 40% include a spreadsheet program, and 30% include both types of programs. Consider randomly selecting a purchaser and let A={ word processing program included } and B = { spreadsheet program included }. (1) Given that the selected individual included a spreadsheet program, the probability that a word processing program was also included? (2) Given that the selected individual included a word processing program, the probability that a spreadsheet program was also included? 19
20 Definitions and Properties (6) Multiplication Rule for P(A n B): PA ( B) = PAB ( ) PB ( ) Ex) P(A B) = 0.3, P(B)= 0.5 P(A n B)=? Ex) Two events A and B are independent. P(A)=0.4, P(B)=0.5, then P(A n B)=? If two events are independent then, P(A n B) = P(A) x P(B). 20
21 Definitions and Properties (6-1) Example: Four individuals have responded to a request by a blood bank for blood donations. None of them has donated before, so their blood types are unknown. Suppose only type A+ is desired and only one of the four actually has this type. If the potential donors are selected in random order for typing, what is the probability that at least three individuals must be typed to obtain the desired type? 21
22 Definitions and Properties (7) Probability Tree Diagram: a probability tree diagram provides a convenient way to break a problem into parts and to organize the information available. Ex) a fair coin is tossed twice. Show the tree diagram. Pr (heads on both tosses) =? Ex) In the Drosophila population, 30% of the flies are black and 70% are gray. Suppose that two flies are randomly chosen from the population. Find the probability that both flies are the same color. (Use the tree diagram!) 22
23 Probability Tree Example 1 Suppose that a student who is about to take a multiple choice test has only learned 40% of the material covered by the exam. Thus, there is a 40% chance that she will know the answer to a question. However, even if she does not know the answer to a question, she will has a 20% chance of getting the right answer by guessing. If we choose a question at random from the exam, what is the probability that she will get it right? 23
24 Probability Rules Example 2 : The relation ship between hair color and eye color for a group of 1770 German men. Hair color Brown Black Red Total Eye color Brown Blue Total ) P(black hair or red hair)=? 2) P(black hair)=? 3) P(blue eyes black hair)=? 4) P(red hair and brown eyes)=? 24
25 Density Curves Density Curves 1) Relative Frequency Histogram (discrete) Density curve (continuous) ( as class widths go to 0) 2) The density curve is entirely above the x-axis. 3) The area under an entire density curve must be equal to 1. 4) For any two numbers A and B Area under density curve between A and B = Proportion of X values between A and B = P(A X B) 25
26 Random Variables Random Variables 1) A variable that takes on numerical values that depend on the outcome of a chance operation. 2) A rule that associates a number with each outcome in a sample space. 3) A function that associates a number with each outcome in a sample space. X(s)=x (x is the value associated with the outcome s by the random variable X ) 26
27 Random Variables Example (discrete random variable): We toss a fair coin once. With S={H, T}, we can define an random variable X by X(H)=1 X(T)=0 If X=1, it means that we got heads If X=0, it means that we got tails P(X=1)=? P(X=0)=? 27
28 Random Variables Example (discrete random variable): 1) We toss a fair dice once. With S={A, B, C, D, E, F}, we can define an random variable X by X(A)=1, X(B)=2, X(C)=3, X(D)=4, X(E)=5, X(F)=6 P(X=1)=? P(X=2)=? 2) We toss a fair dice once. With S={1, 2, 3, 4, 5, 6}, we can define an random variable X by X(s)=s (s is an outcome) 28
29 Random Variables Example (continuous random variable): We measure and record the height of a man chosen randomly from a certain population. With S={s: 5 ft < s < 8 ft}, we can define an random variable X by X(s)=s (s is an outcome) 29
30 Random Variables Mean of a Random Variable: The mean of a discrete random variable X = µ = x PX ( = x) X i i = The expected value of X = E(X) 30
31 Random Variables Variance of a Random Variable: The variance of a discrete random variable X = σ = ( x µ ) PX ( = x) 2 2 X i X i = VAR(X) 31
32 Random Variables Mean of a Random Variable (Example): In preparation for an ecological study of centipedes, the floor of a beech woods is divided into a large number of one-foot squares. At a certain moment, the distribution of the number of centipedes in the squares, X, is as shown in the table. 1) The mean value of X =? 2) The variance of X=? Number of Centipedes Percent Frequency Total
33 Counting (supplement) X-factorial The factorial of a non-negative integer x, denoted by x!, is the product of all positive integers less than or equal to x. x! = xx ( 1)( x 2) (2)(1) 1! = 1 0! = 1 Ex1) 5!=? Ex2) 3!/5!=? 33
34 Counting (supplement) Permutations An ordered collection of distinct elements. The number of r-permutations (each of size r) from a set S with n elements (size n) is denoted by n n! Pr =, n r ( n r)! When counting groups of times (where the order inside the group changes the group), we use a permutation. 34
35 Counting (supplement) Example How many different ordered arrangements of 2 could be selected from the 3 items A, B, and C? AB, BA, AC, CA, BC, and CB 6 ordered arrangements are possible! Each arrangement is known as a permutation. 3! P2 = = = = (3 2)! 1!
36 Counting (supplement) Combinations An unordered collection of distinct elements. The number of r-combinations (each of size r) from a set S with n elements (size n) is denoted by n n n! Cr = =, n r r r!( n r)! When counting groups of times (where the order inside the group does not change the group), we use a combination. 36
37 Counting (supplement) Example How many different groups of 2 could be selected from the 3 items A, B, and C? AB, BA, AC, CA, BC, and CB 6 ordered arrangements are possible! But, here we don t care about the order!!! AB, BA AC, CA BC, CB should be counted once. (Therefore, 3 groups) 3! C2 = = = = 3 2!(3 2)! 2!1! 2 37
38 Counting (supplement) Example There are 20 kids who have applied to be captain of a kickball team. There are 4 total teams. In how many ways can we choose 4 kids to be captains? 20 20! C4 = = = = 4 4!(16)!
39 Counting (supplement) Example Jimmy and Tom are to be assigned to 3 different jobs, one to each job. How many different assignments are possible? 1 st Job 2 nd Job 3 rd Job Jimmy Tom Tom Jimmy Jimmy Tom Tom Jimmy 3! P2 = = = 6 (3 2)! 1! Jimmy Tom Tom Jimmy 39
40 Binomial Distribution Binomial Random Variables :A random variable, X= the number of successes among the n trials, that satisfies the following four conditions. (1) Binary outcomes : There are only two possible outcomes for each trial (success and failure). (2) Independent trials : The outcomes of the trials are independent of each other. (3) Fixed n : The number of trial, n, is fixed in advance. (4) Same value of p : The probability of a success on a single trial is the same for all trials. 40
41 Binomial Distribution Binomial Distribution Formula : For a binomial random variable X, the probability that n trials result in k success (and n-k failures) is given by the following formula. k n k P( k Successes) = P( X = k) = C p (1 p), 0 k n X ~ Bin( n, p) n k Ex) n=4, p=1/2, P(X=2)=? Ex) n=3, p=1/3, P(2 successes)=? Ex) X~Bin(4, 0.3), P(X=3)=? 41
42 Binomial Distribution Example Ex) n=4, p=1/2, P(X=2)=? PX ( 2) C (1/ 2) (1 1/ 2) = = 4 2 = = = Ex) n=3, p=1/3, P(2 successes)=? Ex) X~Bin(4, 0.3), P(X=3)=? 4! !2! ! !1! PX ( = 2) = 3C2 (1 / 3) (1 1 / 3) = = = 4! PX= = = = = 3!1! ( 3) 4C3 (0.3) (1 0.3)
43 Binomial Distribution Example Suppose we draw a random sample of five individuals from a large population in which 39% of the individuals are mutants. What is the probability of a sample containing 3 mutants and 2 nonmutants? 43
44 Binomial Distribution Answer n=5 and p=.39 X= the number of successes ( here, the number of mutants) P(3 mutants)=p(x=3) PX= = = ( 3) 5C3(.39) (1.39) 10 (.39) (.61).22 44
45 Binomial Distribution Example (n=5 and p=.39) Number of 0.4 Mutants (k) Non-mutants (nk) Probability Number of mutants
46 Binomial Distribution Mean and Variance of a Binomial When X~Bin(n,p), 1) The mean (that is, the average number of success) = E(X)=np 2) The variance for a binomial random variable = Var(X)=np(1-p) ==> the standard deviation for a binomial random variable = np(1 p) 46
47 Binomial Distribution Example In the United States, 37% of the population has type A blood. Consider taking a sample size 4. Let X denote the number of persons in the sample with type A blood. Find (a) P(X=0) (b) P(X=1) (c) P(0 X 1) (d) mean? (e) standard deviation? 47
48 Binomial Distribution Answer n=4, p=.37 X = the number of persons in the sample with type A blood k ( = ) = (.37) (.63) PX k 4C k k ( a) PX ( = 0) = C (.37) (.63) = ( b) PX ( = 1) = C (.37) (.63) = 4(.37)(.63) ( c) P(0 X 1) = PX ( = 0) + PX ( = 1) = (.37)(.63) 4 3 ( d) E( X ) = np = 4(.37) = 1.48 ( e) σ = Var( X ) = np(1 p) = 4(.37)(.63) = =
49 Confidence Intervals 49
50 Confidence Intervals This module explores the development and interpretation of confidence intervals, with a focus on confidence intervals for the population mean, based on the sample mean. 50
51 1. In the Chicago area, the price of new tires is normally distributed with a standard deviation of σ = $ A random sample of 64 tires indicates a mean selling price of x = $ Construct an 85% confidence interval for the mean selling price, µ of this new tire in the Chicago area. In order to estimate within $10.00 of the population mean, how large of a sample should be taken in order to be 95% confident of achieving this level of accuracy. x = 98.7 σ = n = S. E. = σ n = = 1.44 x ± Z(x ) SE 98.7 ± 2.06 = [96.630, ] 98.7 ±
52 2. Fifty electric bills from the apartment of a certain city apartment are chosen at random. The mean electric bill was x = $ with s = $ The electric bills have a normal distribution. Construct a 98% confidence interval for P. x = s = n = S. E. = σ n = = 3.07 x ± Z(x ) SE ± (2.33)(3.07) ± 7.15 = [ , ] 52
53 3. Thirty SAT scores were chosen at random from the records of seniors at a certain high school over the last 20 years Construct a 95% confidence interval for the population mean µ. x ± Z(x ) SE ± (1.96)(21.289) x = s = n = 30 σ S. E. = = = n ± = [590.63, ] 53
54 4. A random sample on n = 100 voters in a community produced x = 59 voters in favor of a candidate A. Estimate the fraction of the voting population favoring candidate A using a 95% and a 90% confidence interval. P ± Z(P) SE.59 ± (1.96)(.049) P ± Z(P).59 ±1.96 p(1 p) n (.59)(.41) 100 (.494,.686). 59 ± 1.645(.049) (.509,.677) 54
55 How many people must be asked if candidate A wants A 95% confidence interval with a margin of error of + 3 %? P =.59 Z(P) p(1 p) n ME 1.96 (.59)(.41) n n
56 5. A recent poll cited that 76 out of 180 randomly chosen Households watch at least 2 hours of public television per week. Find a 90% confidence interval for p, the proportion of households that watch at least 2 hours of public television per week. P ± Z(P) SE.42 ± (1.645)(.037) P ± Z(P) p(1 p) n (.359,.480).42 ±1.645 (.42)(.58)
57 How many people must be asked if the station wants a 90% confidence interval with a margin of error of + 4 %? P =.42 Z(P) p(1 p) n ME (.42)(.58) n n
58 6. A manufacturer of gunpowder claims to have developed a gun powder that is designed to produce a muzzle velocity of 3000 ft sec. The following data is collected in ft/sec Construct a 95% and a 85% confidence interval for µ. x ± t( x) SE x = ± s = n = 8, 7d. f [ , ] S. E. = s n = =
59 7. The profit for a car dealership for the past week was $210 $300 $120 $620 $450 $510 Construct a 90% confidence interval for the average profit x ± t( x) SE x = ± s = n = 6, 5d. f [$211, $525] S. E. = s n = =
60 THANKS YOU 60
Theoretical 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 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 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 informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More information4.2 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 informationSection Distributions of Random Variables
Section 8.1 - Distributions of Random Variables Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could
More informationChapter 3 - Lecture 5 The Binomial Probability Distribution
Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009 Experiment Examples Moments and moment generating function of a Binomial Random Variable Outline Experiment Examples A binomial experiment
More 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 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 informationShifting our focus. We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why?
Probability Introduction Shifting our focus We were studying statistics (data, displays, sampling...) The next few lectures focus on probability (randomness) Why? What is Probability? Probability is used
More 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 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 informationProbability and Sample space
Probability and Sample space We call a phenomenon random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. The probability of any outcome
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 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 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 informationProbability & Sampling The Practice of Statistics 4e Mostly Chpts 5 7
Probability & Sampling The Practice of Statistics 4e Mostly Chpts 5 7 Lew Davidson (Dr.D.) Mallard Creek High School Lewis.Davidson@cms.k12.nc.us 704-786-0470 Probability & Sampling The Practice of Statistics
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 informationBasic Procedure for Histograms
Basic Procedure for Histograms 1. Compute the range of observations (min. & max. value) 2. Choose an initial # of classes (most likely based on the range of values, try and find a number of classes that
More informationUnit 04 Review. Probability Rules
Unit 04 Review Probability Rules A sample space contains all the possible outcomes observed in a trial of an experiment, a survey, or some random phenomenon. The sum of the probabilities for all possible
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 informationPart 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going?
1 The Law of Averages The Expected Value & The Standard Error Where Are We Going? Sums of random numbers The law of averages Box models for generating random numbers Sums of draws: the Expected Value Standard
More 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 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 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 information***SECTION 8.1*** The Binomial Distributions
***SECTION 8.1*** The Binomial Distributions CHAPTER 8 ~ The Binomial and Geometric Distributions In practice, we frequently encounter random phenomenon where there are two outcomes of interest. For example,
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 informationChapter 3 Class Notes Intro to Probability
Chapter 3 Class Notes Intro to Probability Concept: role a fair die, then: what is the probability of getting a 3? Getting a 3 in one roll of a fair die is called an Event and denoted E. In general, Number
More informationProbability is the tool used for anticipating what the distribution of data should look like under a given model.
AP Statistics NAME: Exam Review: Strand 3: Anticipating Patterns Date: Block: III. Anticipating Patterns: Exploring random phenomena using probability and simulation (20%-30%) Probability is the tool used
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 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 informationPart 10: The Binomial Distribution
Part 10: The Binomial Distribution The binomial distribution is an important example of a probability distribution for a discrete random variable. It has wide ranging applications. One readily available
More 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 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 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 informationStatistics for Business and Economics: Random Variables:Continuous
Statistics for Business and Economics: Random Variables:Continuous STT 315: Section 107 Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides. Murray Bourne (interactive
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 6: Random Variables. Ch. 6-3: Binomial and Geometric Random Variables
Chapter : Random Variables Ch. -3: Binomial and Geometric Random Variables X 0 2 3 4 5 7 8 9 0 0 P(X) 3???????? 4 4 When the same chance process is repeated several times, we are often interested in whether
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 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 information6 If and then. (a) 0.6 (b) 0.9 (c) 2 (d) Which of these numbers can be a value of probability distribution of a discrete random variable
1. A number between 0 and 1 that is use to measure uncertainty is called: (a) Random variable (b) Trial (c) Simple event (d) Probability 2. Probability can be expressed as: (a) Rational (b) Fraction (c)
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 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 informationContents. The Binomial Distribution. The Binomial Distribution The Normal Approximation to the Binomial Left hander example
Contents The Binomial Distribution The Normal Approximation to the Binomial Left hander example The Binomial Distribution When you flip a coin there are only two possible outcomes - heads or tails. This
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 informationThe topics in this section are related and necessary topics for both course objectives.
2.5 Probability Distributions The topics in this section are related and necessary topics for both course objectives. A probability distribution indicates how the probabilities are distributed for outcomes
More informationFall 2015 Math 141:505 Exam 3 Form A
Fall 205 Math 4:505 Exam 3 Form A Last Name: First Name: Exam Seat #: UIN: On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work Signature: INSTRUCTIONS Part
More information6.1 Binomial Theorem
Unit 6 Probability AFM Valentine 6.1 Binomial Theorem Objective: I will be able to read and evaluate binomial coefficients. I will be able to expand binomials using binomial theorem. Vocabulary Binomial
More information4 Random Variables and Distributions
4 Random Variables and Distributions Random variables A random variable assigns each outcome in a sample space. e.g. called a realization of that variable to Note: We ll usually denote a random variable
More informationLecture 6 Probability
Faculty of Medicine Epidemiology and Biostatistics الوبائيات واإلحصاء الحيوي (31505204) Lecture 6 Probability By Hatim Jaber MD MPH JBCM PhD 3+4-7-2018 1 Presentation outline 3+4-7-2018 Time Introduction-
More informationLesson 97 - Binomial Distributions IBHL2 - SANTOWSKI
Lesson 97 - Binomial Distributions IBHL2 - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin 3 times where P(H) = / (b) THUS, find the probability
More informationOpening Exercise: Lesson 91 - Binomial Distributions IBHL2 - SANTOWSKI
08-0- Lesson 9 - Binomial Distributions IBHL - SANTOWSKI Opening Exercise: Example #: (a) Use a tree diagram to answer the following: You throwing a bent coin times where P(H) = / (b) THUS, find the probability
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 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 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 informationPROBABILITY and BAYES THEOREM
PROBABILITY and BAYES THEOREM From: http://ocw.metu.edu.tr/pluginfile.php/2277/mod_resource/content/0/ ocw_iam530/2.conditional%20probability%20and%20bayes%20theorem.pdf CONTINGENCY (CROSS- TABULATION)
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 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 informationMath 14 Lecture Notes Ch. 4.3
4.3 The Binomial Distribution Example 1: The former Sacramento King's DeMarcus Cousins makes 77% of his free throws. If he shoots 3 times, what is the probability that he will make exactly 0, 1, 2, or
More informationChapter 5: Probability
Chapter 5: These notes reflect material from our text, Exploring the Practice of Statistics, by Moore, McCabe, and Craig, published by Freeman, 2014. quantifies randomness. It is a formal framework with
More informationSection 3.1 Distributions of Random Variables
Section 3.1 Distributions of Random Variables Random Variable A random variable is a rule that assigns a number to each outcome of a chance experiment. There are three types of random variables: 1. Finite
More 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 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 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 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 informationChapter 5. Discrete Probability Distributions. Random Variables
Chapter 5 Discrete Probability Distributions Random Variables x is a random variable which is a numerical description of the outcome of an experiment. Discrete: If the possible values change by steps or
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 information30 Wyner Statistics Fall 2013
30 Wyner Statistics Fall 2013 CHAPTER FIVE: DISCRETE PROBABILITY DISTRIBUTIONS Summary, Terms, and Objectives A probability distribution shows the likelihood of each possible outcome. This chapter deals
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 informationSection 8.1 Distributions of Random Variables
Section 8.1 Distributions of Random Variables Random Variable A random variable is a rule that assigns a number to each outcome of a chance experiment. There are three types of random variables: 1. Finite
More informationStatistics and Probability
Statistics and Probability Continuous RVs (Normal); Confidence Intervals Outline Continuous random variables Normal distribution CLT Point estimation Confidence intervals http://www.isrec.isb-sib.ch/~darlene/geneve/
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 information***SECTION 7.1*** Discrete and Continuous Random Variables
***SECTION 7.1*** Discrete and Continuous Random Variables UNIT 6 ~ Random Variables Sample spaces need not consist of numbers; tossing coins yields H s and T s. However, in statistics we are most often
More informationRandom variables. Discrete random variables. Continuous random variables.
Random variables Discrete random variables. Continuous random variables. Discrete random variables. Denote a discrete random variable with X: It is a variable that takes values with some probability. Examples:
More informationProbability Distributions
Chapter 6 Discrete Probability Distributions Section 6-2 Probability Distributions Definitions Let S be the sample space of a probability experiment. A random variable X is a function from the set S into
More informationEvent p351 An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.
Chapter 12: From randomness to probability 350 Terminology Sample space p351 The sample space of a random phenomenon is the set of all possible outcomes. Example Toss a coin. Sample space: S = {H, T} Example:
More informationThese Statistics NOTES Belong to:
These Statistics NOTES Belong to: Topic Notes Questions Date 1 2 3 4 5 6 REVIEW DO EVERY QUESTION IN YOUR PROVINCIAL EXAM BINDER Important Calculator Functions to know for this chapter Normal Distributions
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 informationSection 8.1 Distributions of Random Variables
Section 8.1 Distributions of Random Variables Random Variable A random variable is a rule that assigns a number to each outcome of a chance experiment. There are three types of random variables: 1. Finite
More 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 information1/2 2. Mean & variance. Mean & standard deviation
Question # 1 of 10 ( Start time: 09:46:03 PM ) Total Marks: 1 The probability distribution of X is given below. x: 0 1 2 3 4 p(x): 0.73? 0.06 0.04 0.01 What is the value of missing probability? 0.54 0.16
More informationAMS7: WEEK 4. CLASS 3
AMS7: WEEK 4. CLASS 3 Sampling distributions and estimators. Central Limit Theorem Normal Approximation to the Binomial Distribution Friday April 24th, 2015 Sampling distributions and estimators REMEMBER:
More informationThe following content is provided under a Creative Commons license. Your support
MITOCW Recitation 6 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make
More informationTOPIC: PROBABILITY DISTRIBUTIONS
TOPIC: PROBABILITY DISTRIBUTIONS There are two types of random variables: A Discrete random variable can take on only specified, distinct values. A Continuous random variable can take on any value within
More 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 informationConsider the following examples: ex: let X = tossing a coin three times and counting the number of heads
Overview Both chapters and 6 deal with a similar concept probability distributions. The difference is that chapter concerns itself with discrete probability distribution while chapter 6 covers continuous
More informationIntroduction to 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 information2.) What is the set of outcomes that describes the event that at least one of the items selected is defective? {AD, DA, DD}
Math 361 Practice Exam 2 (Use this information for questions 1 3) At the end of a production run manufacturing rubber gaskets, items are sampled at random and inspected to determine if the item is Acceptable
More informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.3 Binomial and Geometric Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Binomial and Geometric Random
More information( ) P = = =
1. On a lunch counter, there are 5 oranges and 6 apples. If 3 pieces of fruit are selected, find the probability that 1 orange and apples are selected. Order does not matter Combinations: 5C1 (1 ) 6C P
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 informationBinomal and Geometric Distributions
Binomal and Geometric Distributions Sections 3.2 & 3.3 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 7-2311 Cathy Poliak, Ph.D. cathy@math.uh.edu
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 and Statistics. Copyright Cengage Learning. All rights reserved.
Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.3 Binomial Probability Copyright Cengage Learning. All rights reserved. Objectives Binomial Probability The Binomial Distribution
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 informationConsumer Guide Dealership Word of Mouth Internet
8.1 Graphing Data In this chapter, we will study techniques for graphing data. We will see the importance of visually displaying large sets of data so that meaningful interpretations of the data can be
More informationLecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
More informationPlease have out... - notebook - calculator
Please have out... - notebook - calculator May 6 8:36 PM 6.3 How can we find probabilities when each observation has two possible outcomes? 1 What are we learning today? John Doe claims to possess ESP.
More informationSimple Random Sample
Simple Random Sample A simple random sample (SRS) of size n consists of n elements from the population chosen in such a way that every set of n elements has an equal chance to be the sample actually selected.
More informationReview for Final Exam Spring 2014 Jeremy Orloff and Jonathan Bloom
Review for Final Exam 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom THANK YOU!!!! JON!! PETER!! RUTHI!! ERIKA!! ALL OF YOU!!!! Probability Counting Sets Inclusion-exclusion principle Rule of product
More information