Math/Stat 360-1: Probability and Statistics, Washington State University
|
|
- Gary Hart
- 7 years ago
- Views:
Transcription
1 Math/Stat 360-1: Probability and Statistics, Washington State University Haijun Li Department of Mathematics Washington State University Week 3 Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 1 / 31
2 Outline 1 Section 2.4: Conditional Probability 2 Section 2.5: Independence 3 Section 3.1: Random Variables 4 Section 3.2: Probability Distributions for Discrete Random Variables Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 2 / 31
3 Probabilistic Modeling Three Basic Ingredients: 1 Sample space Ω 2 Events E 3 Probability measure P(E) Motivation for Conditional Probability Measures It should be easier to estimate probabilities if more relevant information is given. The probability P(E) can be calculated by analyzing what could possibly happen under various possible scenarios. Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 3 / 31
4 Cosindering three possible scenarios... Figure: If B 1 occurs, then A occurs. If B 3 occurs, then A will not occur. If B 2 occurs, likelihood of A depends on P(A B 2 ). Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 4 / 31
5 Definition Let A and B be two events with P(B) > 0. The conditional probability of A given that B occurs is defined as P(A B) := P(A B), A, B Ω. P(B) Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 5 / 31
6 Example In a city, 60% of all households get Internet service from the local cable company, 80% get television service from that company, and 50% get both services from that company. Let A = {getting Internet service}, B = {getting TV service}. 1 What is the probability that a randomly selected household gets Internet service given that it gets TV service from that company? P(A B) = P(A B) P(B) = = What is the probability that a randomly selected household gets TV service given that it gets Internet service from that company? P(B A) = P(A B) P(A) = = Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 6 / 31
7 The Multiplication Rule Theorem 1 For any two events A 1, A 2 Ω, P(A 1 A 2 ) = P(A 1 A 2 )P(A 2 ) = P(A 2 A 1 )P(A 1 ). 2 For any three events A 1, A 2, A 3 Ω, P(A 1 A }{{} 2 A 3 ) = P(A 3 A 1 A 2 )P(A }{{} 1 A 2 ) = }{{} B B B P(A 3 A 1 A 2 )P(A 2 A 1 )P(A 1 ). 3 This can be extended to multiple events. Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 7 / 31
8 Example Four individuals have responded to a request by a blood bank for blood donations, and their blood types are unknown. Suppose only type O+ 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? Let A = {first type is not O+}, B = {second type is not O+} P(at least three individuals are typed) = P(A B) = P(B A)P(A) = = 0.5. Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 8 / 31
9 Example (cont d) Four individuals have responded to a request by a blood bank for blood donations, and their blood types are unknown. Suppose only type O+ is desired and only one of the four actually has this type. What is the probability that type O+ is typed on the third donor? Let C = {third type is O+}. P(O+ is typed on the third donor) = P(C A B)P(B A)P(A) = = Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 9 / 31
10 The Total Probability Law Theorem For any events A, B Ω, P(B) = P(A B) + P(A B) = P(B A)P(A) + P(B A )P(A ). Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 10 / 31
11 Remark Events {A 1, A 2,..., A k } constitute a partition of sample space Ω if they are mutually exclusive and k i=1 A i = Ω. For any event B, k k P(B) = P(A i B) = P(B A i )P(A i ). i=1 where P(B A i ), i = 1,..., k, are usually easier to calculate. i=1 Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 11 / 31
12 Example An individual has 3 different accounts. 70% of her messages come into account #1, whereas 20% come into account #2 and the remaining 10% into account #3. Of the messages into account #1, only 1% are spam, whereas the corresponding percentages for accounts #2 and #3 are 2% and 5%, respectively. What is the probability that a randomly selected message is spam? Let A i = {message is from account # i}, i = 1, 2, 3, B = {message is spam}. It follows from the total probability law that P(B) = P(B A 1 )P(A 1 ) + P(B A 2 )P(A 2 ) + P(B A 3 )P(A 3 ) = = Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 12 / 31
13 Bayes Rule Theorem Let {A 1, A 2,..., A k } be a partition of sample space Ω. For any events B Ω, P(A j B) = P(A j B) P(B) = P(B A j )P(A j ) k i=1 P(B A, j = 1,..., k. i)p(a i ) Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 13 / 31
14 Example An individual has 3 different accounts. 70% of her messages come into account #1, whereas 20% come into account #2 and the remaining 10% into account #3. Of the messages into account #1, only 1% are spam, whereas the corresponding percentages for accounts #2 and #3 are 2% and 5%, respectively. What is the probability that a randomly selected message is from account #1 given that it is spam? Let A i = {message is from account # i}, i = 1, 2, 3, B = {message is spam}. It follows from Bayes rule that = P(A 1 B) = P(B A 1 )P(A 1 ) P(B A 1 )P(A 1 ) + P(B A 2 )P(A 2 ) + P(B A 3 )P(A 3 ) = = Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 14 / 31
15 Example (Incidence of a rare disease) Only 1 in 1000 adults is afflicted with a rare disease for which a diagnostic test has been developed. The test is such that when an individual actually has the disease, a positive result will occur 99% of the time, whereas an individual without the disease will show a positive test result only 2% of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual has the disease? Let A 1 = individual has the disease, A 2 = individual does not have the disease, and B = positive test result. P(A 1 ) = 0.001, P(A 2 ) = 0.999, P(B A 1 ) = 0.99, P(B A 2 ) = Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 15 / 31
16 Example (cont d) P(A 1 B) = P(A 1 B) P(B) = = Figure: Path probabilities Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 16 / 31
17 Learning via Bayes Rule Let H be an event of interest, and E be an event representing the evidence. Figure: P(H) is updated to P(H E). Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 17 / 31
18 Independence Two events A and B are independent if P(A B) = P(A). The Product Form: Two events A and B are independent is equivalent to that P(A B) = P(A)P(B). Independence and mutually exclusive are different. Two mutually exclusive events are in fact highly dependent. If A and B are independent, then P(A B) = P(A)+P(B) P(A B) = P(A)+P(B) P(A)P(B). Definition Events A 1, A 2,..., A n are mutually independent if for any subset {i 1,..., i k } {1,..., n}, P(A i1 A ik ) = P(A i1 ) P(A ik ). Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 18 / 31
19 Example Consider a system consisting of two components #1 and #2. Assume that components work independently of one another and P(component works) = 0.9. Components #1 and #2 are connected in series, so that system works iff both #1 and #2 work. Calculate P(system works). P(system works) = P(#1)P(#2) = = Components #1 and #2 are connected in parallel, so that system works iff either #1 or #2 works. Calculate P(system works). P(system works) = P(#1) + P(#2) P(#1)P(#2) = = Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 19 / 31
20 Example (cont d) Consider a system consisting of two components #1 and #2. Assume that components work independently of one another and P(component works) = 0.9. Components #1 and #2 are connected in parallel. Given that the system fails, what is the probability that component #1 fails? P(component #1 fails system works) = = = P(#2 works) P(system works) Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 20 / 31
21 Example Consider the system of 4 components. Components 1 and 2 are connected in parallel; 3 and 4 are connected in series. If components work independently of one another and P(component works) = 0.9, calculate P(system works). P(1 or 2) = 0.99, P(3 and 4) = P(system works) = (0.99)(0.81) = Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 21 / 31
22 Random Variables Random variable (RV): A function defined on the sample space. Example: Toss a coin three times. Let N = number of heads in three tosses. N(TTH) = 1, N(HHH) = 3, N(HTH) = 2, N(TTT ) = 0. Example: Sample a product from an assembly line. Let T = lifelength of the item. Discrete random variable: Its values are limited to discrete points (i.e., finite or countably infinite) on the real line. Continuous random variable: It takes on continuous measurements. Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 22 / 31
23 Example Toss a fair coin three times. The sample space = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT }. Let N denote the number of heads in three tosses. P(N = 0) = 1/8, P(N = 1) = 3/8, P(N = 2) = 3/8, P(N = 3) = 1/8. Table: Probability Masses N = x P(N = x) 1/8 3/8 3/8 1/8 Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 23 / 31
24 Discrete Random Variables The distribution of a discrete random variable X is described by the probability mass function (PMF) p(x i ) = P(X = x i ), for all the possible values x i of X. Distribution of RV X: Likelihoods or relative frequencies of various values of X. Properties of PMF 1 0 p(x) 1. 2 all x s p(x) = 1. Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 24 / 31
25 Example Consider a group of five potential blood donors, a, b, c, d, and e, of whom only a and b have type O+ blood. Five blood samples, one from each individual, will be typed in random order until an O+ individual is identified. Let RV Y = the number of typings necessary to identify an O+ individual. Then the PMF of Y is Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 25 / 31
26 Example (cont d) Figure: The line graph for the PMF Figure: The histogram for the PMF Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 26 / 31
27 Cumulative Distribution Function = Cumulative Frequency Cumulative Distribution Function (CDF) of X F(x) = P(X x) = y:y x 1 F(x) is step-wise, non-decreasing. 2 0 F(x) 1. 3 F(x) 1 as x +. p(y). Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 27 / 31
28 PMF vs CDF P(a X b) = y:a y b P X(y) = F(b) F(a ). Figure: P X (x) = PMF, F X (x) = CDF Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 28 / 31
29 Example (Five Blood Samples) Let RV Y = the number of typings necessary to identify an O+ individual. The PMF of Y is given by The CDF of Y is given by F(x) = 0 if x < if 1 x < if 2 x < if 3 x < 4 1 if 4 x Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 29 / 31
30 Example (Geometric Distribution) Consider testing items coming off an assembly line one by one until a defective item (labeled F ) is found. Let X be the number of testing items necessary to find the first defective item. If P(F) = p, find the PMF and CDF of X. Let S denote a non-defective item, and so P(S) = 1 p. The PMF of X: p(k) = P(X = k) = (1 p) k 1 p, k 1. For the CDF, for any positive integer x 1, F(x) = P(X x) = k x p(k) = x (1 p) k 1 p k=1 = 1 (1 p)x p = 1 (1 p) x. p Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 30 / 31
31 Example (cont d): Geometric CDFs Haijun Li Math/Stat 360-1: Probability and Statistics, Washington State University Week 3 31 / 31
Random variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationCONTINGENCY (CROSS- TABULATION) TABLES
CONTINGENCY (CROSS- TABULATION) TABLES Presents counts of two or more variables A 1 A 2 Total B 1 a b a+b B 2 c d c+d Total a+c b+d n = a+b+c+d 1 Joint, Marginal, and Conditional Probability We study methods
More informationDefinition and Calculus of Probability
In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the
More informationBayes Theorem. Bayes Theorem- Example. Evaluation of Medical Screening Procedure. Evaluation of Medical Screening Procedure
Bayes Theorem P(C A) P(A) P(A C) = P(C A) P(A) + P(C B) P(B) P(E B) P(B) P(B E) = P(E B) P(B) + P(E A) P(A) P(D A) P(A) P(A D) = P(D A) P(A) + P(D B) P(B) Cost of procedure is $1,000,000 Data regarding
More informationData Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More information3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.
3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately
More informationPROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE
PROBABILITY 53 Chapter 3 PROBABILITY The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE 3. Introduction In earlier Classes, we have studied the probability as
More informationST 371 (IV): Discrete Random Variables
ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible
More informationV. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE
V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationAn Introduction to Basic Statistics and Probability
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random
More informationLesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationPeople have thought about, and defined, probability in different ways. important to note the consequences of the definition:
PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models
More informationChapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.
Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive
More informationSTATISTICS HIGHER SECONDARY - SECOND YEAR. Untouchability is a sin Untouchability is a crime Untouchability is inhuman
STATISTICS HIGHER SECONDARY - SECOND YEAR Untouchability is a sin Untouchability is a crime Untouchability is inhuman TAMILNADU TEXTBOOK CORPORATION College Road, Chennai- 600 006 i Government of Tamilnadu
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationStatistics in Geophysics: Introduction and Probability Theory
Statistics in Geophysics: Introduction and Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/32 What is Statistics? Introduction Statistics is the
More informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover
More informationSample Space and Probability
1 Sample Space and Probability Contents 1.1. Sets........................... p. 3 1.2. Probabilistic Models.................... p. 6 1.3. Conditional Probability................. p. 18 1.4. Total Probability
More informationMAS108 Probability I
1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Probability and Probability Distributions 1. Introduction 2. Probability 3. Basic rules of probability 4. Complementary events 5. Addition Law for
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2014 1 2 3 4 5 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Experiment, outcome, sample space, and
More informationHomework 8 Solutions
CSE 21 - Winter 2014 Homework Homework 8 Solutions 1 Of 330 male and 270 female employees at the Flagstaff Mall, 210 of the men and 180 of the women are on flex-time (flexible working hours). Given that
More informationChapter 4 - Practice Problems 1
Chapter 4 - Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula
More informationHomework 3 Solution, due July 16
Homework 3 Solution, due July 16 Problems from old actuarial exams are marked by a star. Problem 1*. Upon arrival at a hospital emergency room, patients are categorized according to their condition as
More informationProbability for Computer Scientists
Probability for Computer Scientists This material is provided for the educational use of students in CSE2400 at FIT. No further use or reproduction is permitted. Copyright G.A.Marin, 2008, All rights reserved.
More informationThe Calculus of Probability
The Calculus of Probability Let A and B be events in a sample space S. Partition rule: P(A) = P(A B) + P(A B ) Example: Roll a pair of fair dice P(Total of 10) = P(Total of 10 and double) + P(Total of
More informationChapter 9 Monté Carlo Simulation
MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?
More information5. Continuous Random Variables
5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be
More informationProbability & Probability Distributions
Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions
More informationChapter 5. Discrete Probability Distributions
Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable
More informationIntroduction to Probability
LECTURE NOTES Course 6.041-6.431 M.I.T. FALL 2000 Introduction to Probability Dimitri P. Bertsekas and John N. Tsitsiklis Professors of Electrical Engineering and Computer Science Massachusetts Institute
More informationIAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics
More informationLecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University
Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science Adam J. Lee adamlee@cs.pitt.edu 6111 Sennott Square Lecture #20: Bayes Theorem November 5, 2013 How can we incorporate prior knowledge? Sometimes we want to know
More informationRandom variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.
Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
More informationUnit 19: Probability Models
Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,
More informationBusiness Statistics 41000: Probability 1
Business Statistics 41000: Probability 1 Drew D. Creal University of Chicago, Booth School of Business Week 3: January 24 and 25, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More informationFor two disjoint subsets A and B of Ω, say that A and B are disjoint events. For disjoint events A and B we take an axiom P(A B) = P(A) + P(B)
Basic probability A probability space or event space is a set Ω together with a probability measure P on it. This means that to each subset A Ω we associate the probability P(A) = probability of A with
More informationMAS131: Introduction to Probability and Statistics Semester 1: Introduction to Probability Lecturer: Dr D J Wilkinson
MAS131: Introduction to Probability and Statistics Semester 1: Introduction to Probability Lecturer: Dr D J Wilkinson Statistics is concerned with making inferences about the way the world is, based upon
More informationChapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationProbability and statistics; Rehearsal for pattern recognition
Probability and statistics; Rehearsal for pattern recognition Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception
More informationChapter 13 & 14 - Probability PART
Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph
More informationProbability and statistical hypothesis testing. Holger Diessel holger.diessel@uni-jena.de
Probability and statistical hypothesis testing Holger Diessel holger.diessel@uni-jena.de Probability Two reasons why probability is important for the analysis of linguistic data: Joint and conditional
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationMath 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141
Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Law of total probability Bayes Theorem the Multiplication Rule, again Recall
More informationA Tutorial on Probability Theory
Paola Sebastiani Department of Mathematics and Statistics University of Massachusetts at Amherst Corresponding Author: Paola Sebastiani. Department of Mathematics and Statistics, University of Massachusetts,
More informationIntroduction to Probability
3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which
More informationChapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.
Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationQuestion: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationM2S1 Lecture Notes. G. A. Young http://www2.imperial.ac.uk/ ayoung
M2S1 Lecture Notes G. A. Young http://www2.imperial.ac.uk/ ayoung September 2011 ii Contents 1 DEFINITIONS, TERMINOLOGY, NOTATION 1 1.1 EVENTS AND THE SAMPLE SPACE......................... 1 1.1.1 OPERATIONS
More informationBayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
1 Learning Goals Bayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1. Be able to apply Bayes theorem to compute probabilities. 2. Be able to identify
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,
More informationProbability and Statistics Vocabulary List (Definitions for Middle School Teachers)
Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence
More information10-601. Machine Learning. http://www.cs.cmu.edu/afs/cs/academic/class/10601-f10/index.html
10-601 Machine Learning http://www.cs.cmu.edu/afs/cs/academic/class/10601-f10/index.html Course data All up-to-date info is on the course web page: http://www.cs.cmu.edu/afs/cs/academic/class/10601-f10/index.html
More informationGraphs. Exploratory data analysis. Graphs. Standard forms. A graph is a suitable way of representing data if:
Graphs Exploratory data analysis Dr. David Lucy d.lucy@lancaster.ac.uk Lancaster University A graph is a suitable way of representing data if: A line or area can represent the quantities in the data in
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More informationBNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I
BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential
More informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
More informationReliability Applications (Independence and Bayes Rule)
Reliability Applications (Independence and Bayes Rule ECE 313 Probability with Engineering Applications Lecture 5 Professor Ravi K. Iyer University of Illinois Today s Topics Review of Physical vs. Stochastic
More informationNotes on Probability. Peter J. Cameron
Notes on Probability Peter J. Cameron ii Preface Here are the course lecture notes for the course MAS108, Probability I, at Queen Mary, University of London, taken by most Mathematics students and some
More informationSection 6-5 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More informationBasic probability theory and n-gram models
theory and n-gram models INF4820 H2010 Institutt for Informatikk Universitetet i Oslo 28. september Outline 1 2 3 Outline 1 2 3 Gambling the beginning of probabilities Example Throwing a fair dice Example
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationPROBABILITY. Chapter. 0009T_c04_133-192.qxd 06/03/03 19:53 Page 133
0009T_c04_133-192.qxd 06/03/03 19:53 Page 133 Chapter 4 PROBABILITY Please stand up in front of the class and give your oral report on describing data using statistical methods. Does this request to speak
More informationSTAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE
STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE TROY BUTLER 1. Random variables and distributions We are often presented with descriptions of problems involving some level of uncertainty about
More informationSection 6.2 Definition of Probability
Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will
More informationChapter 4 - Practice Problems 2
Chapter - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) If you flip a coin three times, the
More informationPattern matching probabilities and paradoxes A new variation on Penney s coin game
Osaka Keidai Ronshu, Vol. 63 No. 4 November 2012 Pattern matching probabilities and paradoxes A new variation on Penney s coin game Yutaka Nishiyama Abstract This paper gives an outline of an interesting
More informationChapter 3. Probability
Chapter 3 Probability Every Day, each us makes decisions based on uncertainty. Should you buy an extended warranty for your new DVD player? It depends on the likelihood that it will fail during the warranty.
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationSTA 371G: Statistics and Modeling
STA 371G: Statistics and Modeling Decision Making Under Uncertainty: Probability, Betting Odds and Bayes Theorem Mingyuan Zhou McCombs School of Business The University of Texas at Austin http://mingyuanzhou.github.io/sta371g
More informationName Please Print MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Problems for Mid-Term 1, Fall 2012 (STA-120 Cal.Poly. Pomona) Name Please Print MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether
More informationBasic Probability Concepts
page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes
More informationSummer School in Statistics for Astronomers & Physicists, IV June 9-14, 2008
p. 1/4 Summer School in Statistics for Astronomers & Physicists, IV June 9-14, 2008 Laws of Probability, Bayes theorem, and the Central Limit Theorem June 10, 8:45-10:15 am Mosuk Chow Department of Statistics
More informationLecture 4: BK inequality 27th August and 6th September, 2007
CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationConcepts of Probability
Concepts of Probability Trial question: we are given a die. How can we determine the probability that any given throw results in a six? Try doing many tosses: Plot cumulative proportion of sixes Also look
More informationIntroduction to Probability 2nd Edition Problem Solutions
Introduction to Probability nd Edition Problem Solutions (last updated: 9/8/5) c Dimitri P. Bertsekas and John N. Tsitsiklis Massachusetts Institute of Technology WWW site for book information and orders
More informationECE302 Spring 2006 HW3 Solutions February 2, 2006 1
ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
More informationProbability Distributions
CHAPTER 6 Probability Distributions Calculator Note 6A: Computing Expected Value, Variance, and Standard Deviation from a Probability Distribution Table Using Lists to Compute Expected Value, Variance,
More informationLecture 9: Introduction to Pattern Analysis
Lecture 9: Introduction to Pattern Analysis g Features, patterns and classifiers g Components of a PR system g An example g Probability definitions g Bayes Theorem g Gaussian densities Features, patterns
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected
More informationHomework Assignment #2: Answer Key
Homework Assignment #2: Answer Key Chapter 4: #3 Assuming that the current interest rate is 3 percent, compute the value of a five-year, 5 percent coupon bond with a face value of $,000. What happens if
More informationECE302 Spring 2006 HW1 Solutions January 16, 2006 1
ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 Solutions to HW1 Note: These solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics
More informationDecision Making Under Uncertainty. Professor Peter Cramton Economics 300
Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate
More informationINSTRUCTOR S SOLUTION MANUAL. for
INSTRUCTOR S SOLUTION MANUAL KEYING YE AND SHARON MYERS for PROBABILITY & STATISTICS FOR ENGINEERS & SCIENTISTS EIGHTH EDITION WALPOLE, MYERS, MYERS, YE Contents Introduction to Statistics and Data Analysis
More informationLecture 7: Continuous Random Variables
Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider
More informationSTATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS
STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS 1. If two events (both with probability greater than 0) are mutually exclusive, then: A. They also must be independent. B. They also could
More informationProbabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I
Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with
More informationLecture 6: Discrete & Continuous Probability and Random Variables
Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September
More informationRandom Variables. Chapter 2. Random Variables 1
Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets
More informationConditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence
More information