Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions
|
|
- Austen Ramsey
- 5 years ago
- Views:
Transcription
1 Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions 1999 Prentice-Hall, Inc. Chap. 4-1
2 Chapter Topics Basic Probability Concepts: Sample Spaces and Events, Simple Probability, and Joint Probability, Conditional Probability Bayes Theorem The Probability of a Discrete Random Variable Binomial, Poisson, and Hypergeometric Distributions Covariance and its Applications in Finance 1999 Prentice-Hall, Inc. Chap. 4-2
3 Sample Spaces Collection of all Possible Outcomes e.g. All 6 faces of a die: e.g. All 52 cards of a bridge deck: 1999 Prentice-Hall, Inc. Chap. 4-3
4 Events Simple Event: Outcome from a Sample Space with 1 Characteristic e.g. A Red Card from a deck of cards. Joint or Compound Event: Involves 2 Outcomes Simultaneously e.g. An Ace which is also a Red Card from a deck of cards. An Ace given that it is a Red Card Prentice-Hall, Inc. Chap. 4-4
5 Visualizing Events Contingency Tables Ace Not Ace Total Black Red Total Tree Diagrams 1999 Prentice-Hall, Inc. Chap. 4-5
6 Simple Events The Event of a Happy Face There are 5 happy faces in this collection of 18 objects 1999 Prentice-Hall, Inc. Chap. 4-6
7 Joint Events The Event of a Happy Face AND Light Colored 3 Happy Faces which are light in color 1999 Prentice-Hall, Inc. Chap. 4-7
8 Compound Events The Event of Happy Face OR Light Colored 12 Items, 5 happy faces and 7 other light objects 1999 Prentice-Hall, Inc. Chap. 4-8
9 Special Events Null Event Club & Diamond on 1 Card Draw Null Event Complement of Event For Event A, All Events Not In A: A 1999 Prentice-Hall, Inc. Chap. 4-9
10 Dependent or Independent Events The Event of a Happy Face GIVEN it is Light Colored E = Happy Face Light Color 3 Items, 3 Happy Faces Given they are Light Colored 1999 Prentice-Hall, Inc. Chap. 4-10
11 Contingency Table Red Ace Red Black Total A Deck of 52 Cards Ace Not an Ace Total 1999 Prentice-Hall, Inc. Chap Sample Space
12 Tree Diagram Event Possibilities Full Deck of Cards Red Cards Black Cards Ace Not an Ace Ace Not an Ace 1999 Prentice-Hall, Inc. Chap. 4-12
13 Probability Probability is the numerical measure of the likelihood that the event will occur. Value is between 0 and 1. Sum of the probabilities of all mutually exclusive events is Certain Impossible 1999 Prentice-Hall, Inc. Chap. 4-13
14 Computing Probability The Probability of an Event, E: P(E) Number of Event Outcomes Total Outcomes in Sample Space X T e.g. P( ) = 2/36 (There are 2 ways to get one 6 and the other 4) Each of the Outcome in the Sample Space equally likely to occur Prentice-Hall, Inc. Chap. 4-14
15 Computing Joint Probability The Probability of a Joint Event, A and B: P(A and B) = Number of Event Outcomes from both A and B Total Outcomes in Sample Space e.g. P(Red Card and Ace) = 52 2 Re d Aces Total Number of Cards Prentice-Hall, Inc. Chap. 4-15
16 Joint Probability Using Contingency Table Event A 1 Event B 1 B 2 P(A 1 and B 1 ) P(A 1 and B 2 ) Total P(A 1 ) A 2 P(A 2 and B 1 ) P(A 2 and B 2 ) P(A 2 ) Total P(B 1 ) P(B 2 ) 1 Joint Probability Marginal (Simple) Probability 1999 Prentice-Hall, Inc. Chap. 4-16
17 Computing Compound Probability The Probability of a Compound Event, A or B: P( A or B ) Number of Event Outcomes from either Total Outcomes in Sample Space A or B e.g. P(Red Card or Ace) 4 Aces 26 Red Cards 2 Red Aces 52 Total NumberofCards Prentice-Hall, Inc. Chap. 4-17
18 Compound Probability Addition Rule P(A 1 or B 1 ) = P(A 1 ) +P(B 1 ) - P(A 1 and B 1 ) Event A 1 Event B 1 B 2 Total P(A 1 and B 1 ) P(A 1 and B 2 ) P(A 1 ) A 2 P(A 2 and B 1 ) P(A 2 and B 2 ) P(A 2 ) Total P(B 1 ) P(B 2 ) 1 For Mutually Exclusive Events: P(A or B) = P(A) + P(B) 1999 Prentice-Hall, Inc. Chap. 4-18
19 Computing Conditional Probability The Probability of the Event: Event A given that Event B has occurred P(A B) = P( A and P( B ) B ) e.g. P(Red Card given that it is an Ace) = 2 Red Aces 4 Aces Prentice-Hall, Inc. Chap. 4-19
20 Conditional Probability Using Contingency Table Conditional Event: Draw 1 Card. Note Kind & Color Color Type Red Black Total Ace Non-Ace Revised Sample Space Total P(Ace Red) = P(Ace AND P(Red) Red) 2 26 / / Prentice-Hall, Inc. Chap. 4-20
21 Conditional Probability and Statistical Independence Conditional Probability: P(A B) = P( A and P( B ) B ) Multiplication Rule: Events are Independent: P(A and B) = P(A B) P(B) P(A B) = P(A) Or, P(A and B) = P(A) P(B) Events A and B are Independent when the probability of one event, A is not affected by another event, B Prentice-Hall, Inc. Chap. 4-21
22 Bayes Theorem P(B i A) = P( AB 1 ) P(B P( AB ) P(B ) 1 i ) i P( AB k ) P(B k ) Same Event P(B anda) i P( A) Adding up the parts of A in all the B s 1999 Prentice-Hall, Inc. Chap. 4-22
23 Bayes Theorem: Contingency Table What are the chances of repaying a loan, given a college education? Loan Status Education Repay Default Prob. College ??? No College Prob.?? 1 P(Collegeand Re pay) P(Collegeand Re pay) P(Collegeand Default) P(Repay. 08 College) = 1999 Prentice-Hall, Inc. Chap. 4-23
24 Discrete Random Variable Random Variable: represents outcomes of an experiment. e.g. Throw a die twice: Count the number of times 4 comes up (0, 1, or 2 times) Discrete Random Variable: Obtained by Counting (0, 1, 2, 3, etc.) Usually finite by number of values e.g. Toss a coin 5 times. Count the number of tails. (0, 1, 2, 3, 4, or 5 times) 1999 Prentice-Hall, Inc. Chap. 4-24
25 Discrete Probability Distribution Example Event: Toss 2 Coins. Count # Tails. T T T T Probability Distribution Values Probability 0 1/4 = /4 = /4 = Prentice-Hall, Inc. Chap. 4-25
26 Discrete Probability Distribution List of All Possible [ X i, P(X i ) ] Pairs X i = Value of Random Variable (Outcome) P(X i ) = Probability Associated with Value Mutually Exclusive (No Overlap) Collectively Exhaustive (Nothing Left Out) 0 P(X i ) 1 S P(X i ) = Prentice-Hall, Inc. Chap. 4-26
27 Discrete Random Variable Expected Value Summary Measures The Mean of the Probability Distribution Weighted Average m = E(X) = X i P(X i ) e.g. Toss 2 coins, Count tails, Compute Expected Value: m = = 1 Variance Number of Tails Weighted Average Squared Deviation about Mean s 2 = E [ (X i - m ) 2 ]=S (X i - m ) 2 P(X i ) e.g. Toss 2 coins, Count tails, Compute Variance: s 2 = (0-1) 2 (.25) + (1-1) 2 (.50) + (2-1) 2 (.25) = Prentice-Hall, Inc. Chap. 4-27
28 Important Discrete Probability Distribution Models Discrete Probability Distributions Binomial Hypergeometric Poisson 1999 Prentice-Hall, Inc. Chap. 4-28
29 Binomial Probability Distributions n Identical Trials, e.g. 15 tosses of a coin, 10 light bulbs taken from a warehouse 2 Mutually Exclusive Outcomes, e.g. heads or tails in each toss of a coin, defective or not defective light bulbs Constant Probability for each Trial, e.g. probability of getting a tail is the same each time we toss the coin and each light bulb has the same probability of being defective 1999 Prentice-Hall, Inc. Chap. 4-29
30 Binomial Probability Distributions 2 Sampling Methods: Infinite Population Without Replacement Finite Population With Replacement Trials are Independent: The Outcome of One Trial Does Not Affect the Outcome of Another 1999 Prentice-Hall, Inc. Chap. 4-30
31 Binomial Probability Distribution Function P(X) n! X! n X p ( 1 ( )! p ) X n X P(X) = probability that X successes given a knowledge of n and p X = number of successes in sample, (X = 0, 1, 2,..., n) p = probability of success n = sample size Tails in 2 Toss of Coin X P(X) 0 1/4 = /4 = /4 = Prentice-Hall, Inc. Chap. 4-31
32 Binomial Distribution Characteristics Mean m Standard Deviation s np E ( X ) np e.g. m = 5 (.1) =.5 ( p ) e.g. s = 5(.5)(1 -.5) = P(X) P(X) n = 5 p = n = 5 p = X X 1999 Prentice-Hall, Inc. Chap. 4-32
33 Poisson Distribution Poisson Process: Discrete Events in an Interval The Probability of One Success in Interval is Stable The Probability of More than One Success in this Interval is 0 Probability of Success is Independent from Interval to Interval e.g. # Customers Arriving in 15 min. # Defects Per Case of Light Bulbs. P ( X x l l e Prentice-Hall, Inc. Chap x! l x
34 Poisson Probability Distribution Function P ( X ) e X P(X ) = probability of X successes given l l = expected (mean) number of successes e = (base of natural logs) X = number of successes per unit l X l! e.g. Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6. P(X) = e ! 4 = Prentice-Hall, Inc. Chap. 4-34
35 Poisson Distribution Characteristics Mean m E ( X ) l Standard Deviation s N i 1 l X P ( X ) i i P(X) P(X) l = l = X X 1999 Prentice-Hall, Inc. Chap. 4-35
36 Hypergeometric Distribution n Trials in a Sample Taken From a Finite Population of size N Sample taken Without Replacement Trials are Dependent Concerned With Finding the Probability of X Successes in the Sample where there are A Successes in the Population 1999 Prentice-Hall, Inc. Chap. 4-36
37 Hypergeometric Distribution ( )( ) P ( X ) X n - X ( N ) P(X) = probability that X successes given n, N, and A n = sample size N = population size A = number of successes in population X = number of successes in sample (X = 0, 1, 2,..., n) A N - A 1999 Prentice-Hall, Inc. Chap n 3 Light bulbs were selected from 10. Of the 10 there were 4 defective. What is the probability that 2 of the 3 selected are defective? 4 6 P(2) = ( 2 )( 1 ) ( ) 10 3 =.30
38 Hypergeometric Characteristics Mean m E ( X ) n A N Standard Deviation Finite Population Correction s na ( N N 2 A ) N n N Prentice-Hall, Inc. Chap. 4-38
39 Covariance s XY N X E( X ) Y E(Y ) P( X Y ) i 1 i i i i X = discrete random variable X X i = ith outcome of X P(X i Y i ) = probability of occurrence of the ith outcome of Y Y = discrete random variable Y Y i = ith outcome of Y i = 1, 2,, N 1999 Prentice-Hall, Inc. Chap. 4-39
40 Computing the Mean for Investment Returns Return per $1,000 for two types of investments Investment P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession -$100 -$200.5 Stable Economy Expanding Economy E(X) = m X = (-100)(.2) + (100)(.5) + (250)(.3) = $105 E(Y) = m Y = (-200)(.2) + (50)(.5) + (350)(.3) = $ Prentice-Hall, Inc. Chap. 4-40
41 Computing the Variance for Investment Returns Investment P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession -$100 -$200.5 Stable Economy Expanding Economy s X Var(X) = = (.2)( )2 + (.5)( ) 2 + (.3)( ) 2 2 s Y = 14,725, s X = Var(Y) = = (.2)( )2 + (.5)(50-90) 2 + (.3)( )2 = 37,900, s Y = Prentice-Hall, Inc. Chap. 4-41
42 Computing the Covariance for Investment Returns Investment P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession -$100 -$200.5 Stable Economy Expanding Economy s XY = (.2)( )( ) + (.5)( )(50-90) + (.3)( )(350-90) = 23,300 The Covariance of 23,000 indicates that the two investments are strongly related and will vary together in the same direction Prentice-Hall, Inc. Chap. 4-42
43 Chapter Summary Discussed Basic Probability Concepts: Sample Spaces and Events, Simple Probability, and Joint Probability Defined Conditional Probability Discussed Bayes Theorem Addressed the Probability of a Discrete Random Variable Discussed Binomial, Poisson, and Hypergeometric Distributions Addressed Covariance and its Applications in Finance 1999 Prentice-Hall, Inc. Chap. 4-43
Statistics for Managers Using Microsoft Excel (3 rd Edition)
Statistics for Managers Using Microsoft Excel (3 rd Edition) Chapter 4 Basic Probability and Discrete Probability Distributions 2002 Prentice-Hall, Inc. Chap 4-1 Chapter Topics Basic probability concepts
More informationDistribusi Binomial, Poisson, dan Hipergeometrik
Distribusi Binomial, Poisson, dan Hipergeometrik CHAPTER TOPICS The Probability of a Discrete Random Variable Covariance and Its Applications in Finance Binomial Distribution Poisson Distribution Hypergeometric
More informationWhat is Probability? Probability. Sample Spaces and Events. Simple Event
What is Probability? Probability Peter Lo Probability is the numerical measure of likelihood that the event will occur. Simple Event Joint Event Compound Event Lies between 0 & 1 Sum of events is 1 1.5
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Lecture 2 Probability Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 3-1 3.1 Definition Random Experiment a process leading to an uncertain
More informationReview Basic Probability Concept
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.9008 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 3 Probability Contents 1. Events, Sample Spaces, and Probability 2. Unions and Intersections 3. Complementary Events 4. The Additive Rule and Mutually Exclusive
More informationIntroduction to probability
Introduction to probability 4.1 The Basics of Probability Probability The chance that a particular event will occur The probability value will be in the range 0 to 1 Experiment A process that produces
More informationI - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability
What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)
More informationStatistics for Business and Economics
Statistics for Business and Economics Basic Probability Learning Objectives In this lecture(s), you learn: Basic probability concepts Conditional probability To use Bayes Theorem to revise probabilities
More informationIntroduction to Probability
Introduction to Probability Content Experiments, Counting Rules, and Assigning Probabilities Events and Their Probability Some Basic Relationships of Probability Conditional Probability Bayes Theorem 2
More informationReview of Statistics
Review of Statistics Topics Descriptive Statistics Mean, Variance Probability Union event, joint event Random Variables Discrete and Continuous Distributions, Moments Two Random Variables Covariance and
More informationChapter 4 - Introduction to Probability
Chapter 4 - Introduction to Probability Probability is a numerical measure of the likelihood that an event will occur. Probability values are always assigned on a scale from 0 to 1. A probability near
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER / Probability
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER 2 2017/2018 DR. ANTHONY BROWN 5.1. Introduction to Probability. 5. Probability You are probably familiar with the elementary
More informationProbability Year 9. Terminology
Probability Year 9 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationProbability the chance that an uncertain event will occur (always between 0 and 1)
Quantitative Methods 2013 1 Probability as a Numerical Measure of the Likelihood of Occurrence Probability the chance that an uncertain event will occur (always between 0 and 1) Increasing Likelihood of
More informationProbability Year 10. Terminology
Probability Year 10 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More informationRandom Variable. Discrete Random Variable. Continuous Random Variable. Discrete Random Variable. Discrete Probability Distribution
Random Variable Theoretical Probability Distribution Random Variable Discrete Probability Distributions A variable that assumes a numerical description for the outcome of a random eperiment (by chance).
More informationAn event described by a single characteristic e.g., A day in January from all days in 2012
Events Each possible outcome of a variable is an event. Simple event An event described by a single characteristic e.g., A day in January from all days in 2012 Joint event An event described by two or
More informationLecture 3 Probability Basics
Lecture 3 Probability Basics Thais Paiva STA 111 - Summer 2013 Term II July 3, 2013 Lecture Plan 1 Definitions of probability 2 Rules of probability 3 Conditional probability What is Probability? Probability
More informationBinomial and Poisson Probability Distributions
Binomial and Poisson Probability Distributions Esra Akdeniz March 3, 2016 Bernoulli Random Variable Any random variable whose only possible values are 0 or 1 is called a Bernoulli random variable. What
More informationProbability Distribution
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.98 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More informationTOPIC 12 PROBABILITY SCHEMATIC DIAGRAM
TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of Importance References NCERT Book Vol. II Probability (i) Conditional Probability *** Article 1.2 and 1.2.1 Solved Examples 1 to 6 Q. Nos
More informationLecture Lecture 5
Lecture 4 --- Lecture 5 A. Basic Concepts (4.1-4.2) 1. Experiment: A process of observing a phenomenon that has variation in its outcome. Examples: (E1). Rolling a die, (E2). Drawing a card form a shuffled
More informationProbability Dr. Manjula Gunarathna 1
Probability Dr. Manjula Gunarathna Probability Dr. Manjula Gunarathna 1 Introduction Probability theory was originated from gambling theory Probability Dr. Manjula Gunarathna 2 History of Probability Galileo
More informationFault-Tolerant Computer System Design ECE 60872/CS 590. Topic 2: Discrete Distributions
Fault-Tolerant Computer System Design ECE 60872/CS 590 Topic 2: Discrete Distributions Saurabh Bagchi ECE/CS Purdue University Outline Basic probability Conditional probability Independence of events Series-parallel
More informationUnit 7 Probability M2 13.1,2,4, 5,6
+ Unit 7 Probability M2 13.1,2,4, 5,6 7.1 Probability n Obj.: I will be able to determine the experimental and theoretical probabilities of an event, or its complement, occurring. n Vocabulary o Outcome
More informationDept. of Linguistics, Indiana University Fall 2015
L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 34 To start out the course, we need to know something about statistics and This is only an introduction; for a fuller understanding, you would
More informationBinomial random variable
Binomial random variable Toss a coin with prob p of Heads n times X: # Heads in n tosses X is a Binomial random variable with parameter n,p. X is Bin(n, p) An X that counts the number of successes in many
More informationM378K In-Class Assignment #1
The following problems are a review of M6K. M7K In-Class Assignment # Problem.. Complete the definition of mutual exclusivity of events below: Events A, B Ω are said to be mutually exclusive if A B =.
More informationMutually Exclusive Events
172 CHAPTER 3 PROBABILITY TOPICS c. QS, 7D, 6D, KS Mutually Exclusive Events A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes
More information7.1 What is it and why should we care?
Chapter 7 Probability In this section, we go over some simple concepts from probability theory. We integrate these with ideas from formal language theory in the next chapter. 7.1 What is it and why should
More informationDiscrete probability distributions
Discrete probability s BSAD 30 Dave Novak Fall 08 Source: Anderson et al., 05 Quantitative Methods for Business th edition some slides are directly from J. Loucks 03 Cengage Learning Covered so far Chapter
More informationSTA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6. With SOLUTIONS
STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru Venkateswara Rao, Ph.D. STA 2023 Fall 2016 Venkat Mu ALL THE CONTENT IN THESE SOLUTIONS PRESENTED IN BLUE AND BLACK
More informationPermutation. Permutation. Permutation. Permutation. Permutation
Conditional Probability Consider the possible arrangements of the letters A, B, and C. The possible arrangements are: ABC, ACB, BAC, BCA, CAB, CBA. If the order of the arrangement is important then we
More informationCHAPTER - 16 PROBABILITY Random Experiment : If an experiment has more than one possible out come and it is not possible to predict the outcome in advance then experiment is called random experiment. Sample
More informationTopic 3: Introduction to Probability
Topic 3: Introduction to Probability 1 Contents 1. Introduction 2. Simple Definitions 3. Types of Probability 4. Theorems of Probability 5. Probabilities under conditions of statistically independent events
More informationChapter 2: Probability Part 1
Engineering Probability & Statistics (AGE 1150) Chapter 2: Probability Part 1 Dr. O. Phillips Agboola Sample Space (S) Experiment: is some procedure (or process) that we do and it results in an outcome.
More information5. Conditional Distributions
1 of 12 7/16/2009 5:36 AM Virtual Laboratories > 3. Distributions > 1 2 3 4 5 6 7 8 5. Conditional Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an
More informationChapter 6: Probability The Study of Randomness
Chapter 6: Probability The Study of Randomness 6.1 The Idea of Probability 6.2 Probability Models 6.3 General Probability Rules 1 Simple Question: If tossing a coin, what is the probability of the coin
More informationDiscrete Distributions
Discrete Distributions Applications of the Binomial Distribution A manufacturing plant labels items as either defective or acceptable A firm bidding for contracts will either get a contract or not A marketing
More informationSTA 2023 EXAM-2 Practice Problems. Ven Mudunuru. From Chapters 4, 5, & Partly 6. With SOLUTIONS
STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru, Venkateswara Rao STA 2023 Spring 2016 1 1. A committee of 5 persons is to be formed from 6 men and 4 women. What
More informationTopic 2: Probability & Distributions. Road Map Probability & Distributions. ECO220Y5Y: Quantitative Methods in Economics. Dr.
Topic 2: Probability & Distributions ECO220Y5Y: Quantitative Methods in Economics Dr. Nick Zammit University of Toronto Department of Economics Room KN3272 n.zammit utoronto.ca November 21, 2017 Dr. Nick
More informationSTAT Chapter 3: Probability
Basic Definitions STAT 515 --- Chapter 3: Probability Experiment: A process which leads to a single outcome (called a sample point) that cannot be predicted with certainty. Sample Space (of an experiment):
More informationIf S = {O 1, O 2,, O n }, where O i is the i th elementary outcome, and p i is the probability of the i th elementary outcome, then
1.1 Probabilities Def n: A random experiment is a process that, when performed, results in one and only one of many observations (or outcomes). The sample space S is the set of all elementary outcomes
More informationChap 4 Probability p227 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a long series of
Chap 4 Probability p227 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a long series of repetitions. (p229) That is, probability is a long-term
More informationG.PULLAIAH COLLEGE OF ENGINEERING & TECHNOLOGY DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING PROBABILITY THEORY & STOCHASTIC PROCESSES
G.PULLAIAH COLLEGE OF ENGINEERING & TECHNOLOGY DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING PROBABILITY THEORY & STOCHASTIC PROCESSES LECTURE NOTES ON PTSP (15A04304) B.TECH ECE II YEAR I SEMESTER
More informationIntroduction to Probability and Stocastic Processes - Part I
Introduction to Probability and Stocastic Processes - Part I Lecture 1 Henrik Vie Christensen vie@control.auc.dk Department of Control Engineering Institute of Electronic Systems Aalborg University Denmark
More informationProbability 5-4 The Multiplication Rules and Conditional Probability
Outline Lecture 8 5-1 Introduction 5-2 Sample Spaces and 5-3 The Addition Rules for 5-4 The Multiplication Rules and Conditional 5-11 Introduction 5-11 Introduction as a general concept can be defined
More informationChapter 2: Discrete Distributions. 2.1 Random Variables of the Discrete Type
Chapter 2: Discrete Distributions 2.1 Random Variables of the Discrete Type 2.2 Mathematical Expectation 2.3 Special Mathematical Expectations 2.4 Binomial Distribution 2.5 Negative Binomial Distribution
More informationChapter 1 (Basic Probability)
Chapter 1 (Basic Probability) What is probability? Consider the following experiments: 1. Count the number of arrival requests to a web server in a day. 2. Determine the execution time of a program. 3.
More informationRelative Risks (RR) and Odds Ratios (OR) 20
BSTT523: Pagano & Gavreau, Chapter 6 1 Chapter 6: Probability slide: Definitions (6.1 in P&G) 2 Experiments; trials; probabilities Event operations 4 Intersection; Union; Complement Venn diagrams Conditional
More informationPreliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix
More informationMath 218 Supplemental Instruction Spring 2008 Final Review Part A
Spring 2008 Final Review Part A SI leaders: Mario Panak, Jackie Hu, Christina Tasooji Chapters 3, 4, and 5 Topics Covered: General probability (probability laws, conditional, joint probabilities, independence)
More informationProbability Theory and Applications
Probability Theory and Applications Videos of the topics covered in this manual are available at the following links: Lesson 4 Probability I http://faculty.citadel.edu/silver/ba205/online course/lesson
More informationProbability- describes the pattern of chance outcomes
Chapter 6 Probability the study of randomness Probability- describes the pattern of chance outcomes Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long
More informationCSE 312 Final Review: Section AA
CSE 312 TAs December 8, 2011 General Information General Information Comprehensive Midterm General Information Comprehensive Midterm Heavily weighted toward material after the midterm Pre-Midterm Material
More informationStatistics for Engineers
Statistics for Engineers Antony Lewis http://cosmologist.info/teaching/stat/ Starter question Have you previously done any statistics? 1. Yes 2. No 54% 46% 1 2 BOOKS Chatfield C, 1989. Statistics for
More informationBasic Concepts of Probability
Probability Probability theory is the branch of math that deals with unpredictable or random events Probability is used to describe how likely a particular outcome is in a random event the probability
More informationTable of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z).
Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z). For example P(X.04) =.8508. For z < 0 subtract the value from,
More informationConditional Probability. CS231 Dianna Xu
Conditional Probability CS231 Dianna Xu 1 Boy or Girl? A couple has two children, one of them is a girl. What is the probability that the other one is also a girl? Assuming 50/50 chances of conceiving
More informationThe probability of an event is viewed as a numerical measure of the chance that the event will occur.
Chapter 5 This chapter introduces probability to quantify randomness. Section 5.1: How Can Probability Quantify Randomness? The probability of an event is viewed as a numerical measure of the chance that
More informationCME 106: Review Probability theory
: Probability theory Sven Schmit April 3, 2015 1 Overview In the first half of the course, we covered topics from probability theory. The difference between statistics and probability theory is the following:
More informationReview of Basic Probability Theory
Review of Basic Probability Theory James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) 1 / 35 Review of Basic Probability Theory
More informationProbability and random variables
Probability and random variables Events A simple event is the outcome of an experiment. For example, the experiment of tossing a coin twice has four possible outcomes: HH, HT, TH, TT. A compound event
More informationChapter 1: Revie of Calculus and Probability
Chapter 1: Revie of Calculus and Probability Refer to Text Book: Operations Research: Applications and Algorithms By Wayne L. Winston,Ch. 12 Operations Research: An Introduction By Hamdi Taha, Ch. 12 OR441-Dr.Khalid
More informationDiscrete Random Variables
Discrete Random Variables An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan Introduction The markets can be thought of as a complex interaction of a large number of random processes,
More informationTopic 2 Probability. Basic probability Conditional probability and independence Bayes rule Basic reliability
Topic 2 Probability Basic probability Conditional probability and independence Bayes rule Basic reliability Random process: a process whose outcome can not be predicted with certainty Examples: rolling
More informationSet/deck of playing cards. Spades Hearts Diamonds Clubs
TC Mathematics S2 Coins Die dice Tale Head Set/deck of playing cards Spades Hearts Diamonds Clubs TC Mathematics S2 PROBABILITIES : intuitive? Experiment tossing a coin Event it s a head Probability 1/2
More informationSTAT 414: Introduction to Probability Theory
STAT 414: Introduction to Probability Theory Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical Exercises
More informationProbability Theory and Random Variables
Probability Theory and Random Variables One of the most noticeable aspects of many computer science related phenomena is the lack of certainty. When a job is submitted to a batch oriented computer system,
More informationWeek 2. Section Texas A& M University. Department of Mathematics Texas A& M University, College Station 22 January-24 January 2019
Week 2 Section 1.2-1.4 Texas A& M University Department of Mathematics Texas A& M University, College Station 22 January-24 January 2019 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 1
More information02 Background Minimum background on probability. Random process
0 Background 0.03 Minimum background on probability Random processes Probability Conditional probability Bayes theorem Random variables Sampling and estimation Variance, covariance and correlation Probability
More information4 Lecture 4 Notes: Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio
4 Lecture 4 Notes: Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio Wrong is right. Thelonious Monk 4.1 Three Definitions of
More informationRelationship between probability set function and random variable - 2 -
2.0 Random Variables A rat is selected at random from a cage and its sex is determined. The set of possible outcomes is female and male. Thus outcome space is S = {female, male} = {F, M}. If we let X be
More informationProbability Theory Review
Cogsci 118A: Natural Computation I Lecture 2 (01/07/10) Lecturer: Angela Yu Probability Theory Review Scribe: Joseph Schilz Lecture Summary 1. Set theory: terms and operators In this section, we provide
More information2. AXIOMATIC PROBABILITY
IA Probability Lent Term 2. AXIOMATIC PROBABILITY 2. The axioms The formulation for classical probability in which all outcomes or points in the sample space are equally likely is too restrictive to develop
More informationb. ( ) ( ) ( ) ( ) ( ) 5. Independence: Two events (A & B) are independent if one of the conditions listed below is satisfied; ( ) ( ) ( )
1. Set a. b. 2. Definitions a. Random Experiment: An experiment that can result in different outcomes, even though it is performed under the same conditions and in the same manner. b. Sample Space: This
More informationIntroduction to Probability. Experiments. Sample Space. Event. Basic Requirements for Assigning Probabilities. Experiments
Introduction to Probability Experiments These are processes that generate welldefined outcomes Experiments Counting Rules Combinations Permutations Assigning Probabilities Experiment Experimental Outcomes
More informationProbability Pearson Education, Inc. Slide
Probability The study of probability is concerned with random phenomena. Even though we cannot be certain whether a given result will occur, we often can obtain a good measure of its likelihood, or probability.
More informationMath 1313 Experiments, Events and Sample Spaces
Math 1313 Experiments, Events and Sample Spaces At the end of this recording, you should be able to define and use the basic terminology used in defining experiments. Terminology The next main topic in
More informationPresentation on Theo e ry r y o f P r P o r bab a il i i l t i y
Presentation on Theory of Probability Meaning of Probability: Chance of occurrence of any event In practical life we come across situation where the result are uncertain Theory of probability was originated
More informationPROBABILITY CHAPTER LEARNING OBJECTIVES UNIT OVERVIEW
CHAPTER 16 PROBABILITY LEARNING OBJECTIVES Concept of probability is used in accounting and finance to understand the likelihood of occurrence or non-occurrence of a variable. It helps in developing financial
More informationProbability Theory. Introduction to Probability Theory. Principles of Counting Examples. Principles of Counting. Probability spaces.
Probability Theory To start out the course, we need to know something about statistics and probability Introduction to Probability Theory L645 Advanced NLP Autumn 2009 This is only an introduction; for
More informationCounting principles, including permutations and combinations.
1 Counting principles, including permutations and combinations. The binomial theorem: expansion of a + b n, n ε N. THE PRODUCT RULE If there are m different ways of performing an operation and for each
More informationSome Special Discrete Distributions
Mathematics Department De La Salle University Manila February 6, 2017 Some Discrete Distributions Often, the observations generated by different statistical experiments have the same general type of behaviour.
More informationan event with one outcome is called a simple event.
Ch5Probability Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations
More informationStatistics for Economists. Lectures 3 & 4
Statistics for Economists Lectures 3 & 4 Asrat Temesgen Stockholm University 1 CHAPTER 2- Discrete Distributions 2.1. Random variables of the Discrete Type Definition 2.1.1: Given a random experiment with
More informationChapter 2: The Random Variable
Chapter : The Random Variable The outcome of a random eperiment need not be a number, for eample tossing a coin or selecting a color ball from a bo. However we are usually interested not in the outcome
More information9/6/2016. Section 5.1 Probability. Equally Likely Model. The Division Rule: P(A)=#(A)/#(S) Some Popular Randomizers.
Chapter 5: Probability and Discrete Probability Distribution Learn. Probability Binomial Distribution Poisson Distribution Some Popular Randomizers Rolling dice Spinning a wheel Flipping a coin Drawing
More informationDiscrete Probability Distribution
Shapes of binomial distributions Discrete Probability Distribution Week 11 For this activity you will use a web applet. Go to http://socr.stat.ucla.edu/htmls/socr_eperiments.html and choose Binomial coin
More informationChapter. Probability
Chapter 3 Probability Section 3.1 Basic Concepts of Probability Section 3.1 Objectives Identify the sample space of a probability experiment Identify simple events Use the Fundamental Counting Principle
More informationConditional Probability
Conditional Probability Idea have performed a chance experiment but don t know the outcome (ω), but have some partial information (event A) about ω. Question: given this partial information what s the
More informationReview for Exam Spring 2018
Review for Exam 1 18.05 Spring 2018 Extra office hours Tuesday: David 3 5 in 2-355 Watch web site for more Friday, Saturday, Sunday March 9 11: no office hours March 2, 2018 2 / 23 Exam 1 Designed to be
More informationSTAT 418: Probability and Stochastic Processes
STAT 418: Probability and Stochastic Processes Spring 2016; Homework Assignments Latest updated on April 29, 2016 HW1 (Due on Jan. 21) Chapter 1 Problems 1, 8, 9, 10, 11, 18, 19, 26, 28, 30 Theoretical
More information4. Suppose that we roll two die and let X be equal to the maximum of the two rolls. Find P (X {1, 3, 5}) and draw the PMF for X.
Math 10B with Professor Stankova Worksheet, Midterm #2; Wednesday, 3/21/2018 GSI name: Roy Zhao 1 Problems 1.1 Bayes Theorem 1. Suppose a test is 99% accurate and 1% of people have a disease. What is the
More informationProbability & Random Variables
& Random Variables Probability Probability theory is the branch of math that deals with random events, processes, and variables What does randomness mean to you? How would you define probability in your
More informationPart (A): Review of Probability [Statistics I revision]
Part (A): Review of Probability [Statistics I revision] 1 Definition of Probability 1.1 Experiment An experiment is any procedure whose outcome is uncertain ffl toss a coin ffl throw a die ffl buy a lottery
More informationProbability (10A) Young Won Lim 6/12/17
Probability (10A) Copyright (c) 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationMotivation. Stat Camp for the MBA Program. Probability. Experiments and Outcomes. Daniel Solow 5/10/2017
Stat Camp for the MBA Program Daniel Solow Lecture 2 Probability Motivation You often need to make decisions under uncertainty, that is, facing an unknown future. Examples: How many computers should I
More information