Lecture 1: Review of Probability
|
|
- Erica Briggs
- 5 years ago
- Views:
Transcription
1 EAS31136/B9036: Statistics in Earth & Atmospheric Sciences Lecture 1: Review of Probability Instructor: Prof. Johnny Luo
2 Dates Topic Reading (Based on the 2 nd Edition of Wilks book) Other Activity Aug 31 Introduction; Review of probability Wilks, Chap 2 Pre-test Sep 7 Matlab tutorial (optional) Sep 14 Review of probability; Probability Distribution 1 Wilks, Chap 2, 3 Sep 21 Probability Distribution 2 Wilks, Chap 3, 4 Sep 28 Hypothesis testing Wilks, Chap 5 Oct 5 Linear regression I Wilks Chap 6; von Storch 8-9 Oct 12 Linear regression II Wilks Chap 6; von Storch 8-9 Oct 19 Time series analysis I Wilks 8; von Storch Oct 26 Midterm; discussion of final project Nov 2 Time series analysis II Wilks 8; von Storch Nov 9 Principal Component Analysis & Empirical orthogonal functions I Wilks 11; von Storch 13 Project 1-page abstract due Nov 16 Principal Component Analysis & Empirical orthogonal functions II Wilks 11; von Storch 13 Project progress report due Nov 30 Cluster analysis Wilks 14 Dec 7 Final project presentation
3 Outlines 1. Definition of terms 2. Axioms of Probability & different views 3. Some properties of probability
4 Probability deals with uncertainties Ø When facing uncertainties, we need a way to describe it. We can go with qualitative descriptors such as rain likely, unlikely or possible. Ø Probability is a quantitative way of expressing uncertainty, e.g., 40% chance of rain. Ø (Dictionary) Probability: the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible. Ø Probability builds upon an abstract mathematical system.
5 A Few Terms Ø Events: A set of possible (uncertain) outcomes (e.g., flipping coin: you won t know for sure which face will come up). Ø Sample Space (or Event Space): the set of all possible events. Usually use capital letter S to represent it. Ø Mutually Exclusive and Collectively Exhaustive (MECE) events; ME: no more than one of the events can occur; CE: at least one of the events will occur.
6 These are null space Venn Diagram
7 4. There are only three companies that make electric cars: companies A, B, and C. Market shares of companies, A, B, C are 10%, 25%, 65%, respectively. If you randomly pick an electric car, what s the probability it was made by A, by B, or by C? P(A) = P(B) = P(c) = 10% 25% 65%
8 5. Now you know the defect rate of new electric cars is 30%, 20%, and 10% for companies A, B, C, respectively. If you randomly pick an electric car and it is defected, what s the probability it was made by A, by B, or by C? 10% x 30% = % x 20% = % x 10% = % 25% 65%
9 5. Now you know the defect rate of new electric cars is 30%, 20%, and 10% for companies A, B, C, respectively. If you randomly pick an electric car and it is defected, what s the probability it was made by A, by B, or by C? 10% x 30% = % x 20% = % x 10% =0.065 Total = = P(A) = 0.03/0.145 = 21% P(B) = 0.05/0.145 = 34% P(C) = 0.065/0.145 = 45%
10 Outlines 1. Definition of terms 2. Axioms of Probability & different views 3. Some properties of probability
11 Axioms of Probability (Dictionary) Axiom: A self-evident truth that requires no proof; a universally accepted principle; (mathematics) a proposition that is assumed without proof for the sake of studying the consequences that follow from it. For an event E in a sample space S 1.0 P(E) 1 2. P( S) = 1 3. P( E1 È E2) = P( E1) + P( E2) where E1 and E 2 mutually exclusive
12 The axioms are like the US Constitution. They are not very informative about how to estimate and interpret probability. There are two dominant views of the meaning of probability: the Frequency view and the Bayesian view. Frequency view: The true probability of of event {E} exists and can be estimated through a long series of trials. Bayesian view: There is no such a thing as true probability; we just estimate it based on whatever information we have in hand.
13 Monty Hall Problem What s probability of winning for each door? You pick one and then the host will open up one with a goat You are now offered an opportunity to switch your choice. Will switch change the probability for the two remaining doors? What are the winning probabilities for door 1 and door 2 now?
14 Monty Hall Problem What about 100 doors? You pick one. Then, I open 98: all goats! Will you now switch to the remaining one? You are now offered an opportunity to stay with your choice, or switch. Will switch give you better chance?
15 Monty Hall Problem Frequency view: The true prob of of event {E} exists and can be estimated through a long series of trials. You are now offered an opportunity to stay with your choice, or switch. Will switch give you better chance? Bayesian view: There is no such a thing as true prob; we just estimate it based on whatever info we have in hand.
Lecture 4: Statistical Hypothesis Testing
EAS31136/B9036: Statistics in Earth & Atmospheric Sciences Lecture 4: Statistical Hypothesis Testing Instructor: Prof. Johnny Luo www.sci.ccny.cuny.edu/~luo Dates Topic Reading (Based on the 2 nd Edition
More informationLecture 5: Linear Regression
EAS31136/B9036: Statistics in Earth & Atmospheric Sciences Lecture 5: Linear Regression Instructor: Prof. Johnny Luo www.sci.ccny.cuny.edu/~luo Dates Topic Reading (Based on the 2 nd Edition of Wilks book)
More informationLecture 2: Probability Distributions
EAS31136/B9036: Statistics in Earth & Atmospheric Sciences Lecture 2: Probability Distributions Instructor: Prof. Johnny Luo www.sci.ccny.cuny.edu/~luo Dates Topic Reading (Based on the 2 nd Edition of
More informationConsider an experiment that may have different outcomes. We are interested to know what is the probability of a particular set of outcomes.
CMSC 310 Artificial Intelligence Probabilistic Reasoning and Bayesian Belief Networks Probabilities, Random Variables, Probability Distribution, Conditional Probability, Joint Distributions, Bayes Theorem
More informationLecture 2. Conditional Probability
Math 408 - Mathematical Statistics Lecture 2. Conditional Probability January 18, 2013 Konstantin Zuev (USC) Math 408, Lecture 2 January 18, 2013 1 / 9 Agenda Motivation and Definition Properties of Conditional
More information2.4 Conditional Probability
2.4 Conditional Probability The probabilities assigned to various events depend on what is known about the experimental situation when the assignment is made. Example: Suppose a pair of dice is tossed.
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14 Introduction One of the key properties of coin flips is independence: if you flip a fair coin ten times and get ten
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 10
EECS 70 Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 10 Introduction to Basic Discrete Probability In the last note we considered the probabilistic experiment where we flipped
More informationConditional probability
CHAPTER 4 Conditional probability 4.1. Introduction Suppose there are 200 men, of which 100 are smokers, and 100 women, of which 20 are smokers. What is the probability that a person chosen at random will
More informationLecture 3. January 7, () Lecture 3 January 7, / 35
Lecture 3 January 7, 2013 () Lecture 3 January 7, 2013 1 / 35 Outline This week s lecture: Fast review of last week s lecture: Conditional probability. Partition, Partition theorem. Bayes theorem and its
More informationMonty Hall Puzzle. Draw a tree diagram of possible choices (a possibility tree ) One for each strategy switch or no-switch
Monty Hall Puzzle Example: You are asked to select one of the three doors to open. There is a large prize behind one of the doors and if you select that door, you win the prize. After you select a door,
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 informationEE126: Probability and Random Processes
EE126: Probability and Random Processes Lecture 1: Probability Models Abhay Parekh UC Berkeley January 18, 2011 1 Logistics 2 Introduction 3 Model 4 Examples What is this course about? Most real-world
More informationBasic Probabilistic Reasoning SEG
Basic Probabilistic Reasoning SEG 7450 1 Introduction Reasoning under uncertainty using probability theory Dealing with uncertainty is one of the main advantages of an expert system over a simple decision
More information4. Probability of an event A for equally likely outcomes:
University of California, Los Angeles Department of Statistics Statistics 110A Instructor: Nicolas Christou Probability Probability: A measure of the chance that something will occur. 1. Random experiment:
More informationElementary Discrete Probability
Elementary Discrete Probability MATH 472 Financial Mathematics J Robert Buchanan 2018 Objectives In this lesson we will learn: the terminology of elementary probability, elementary rules of probability,
More informationSTAT509: Probability
University of South Carolina August 20, 2014 The Engineering Method and Statistical Thinking The general steps of engineering method are: 1. Develop a clear and concise description of the problem. 2. Identify
More informationUniversity of Technology, Building and Construction Engineering Department (Undergraduate study) PROBABILITY THEORY
ENGIEERING STATISTICS (Lectures) University of Technology, Building and Construction Engineering Department (Undergraduate study) PROBABILITY THEORY Dr. Maan S. Hassan Lecturer: Azhar H. Mahdi Probability
More informationFundamentals of Probability CE 311S
Fundamentals of Probability CE 311S OUTLINE Review Elementary set theory Probability fundamentals: outcomes, sample spaces, events Outline ELEMENTARY SET THEORY Basic probability concepts can be cast in
More informationStatistics 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 informationWith Question/Answer Animations. Chapter 7
With Question/Answer Animations Chapter 7 Chapter Summary Introduction to Discrete Probability Probability Theory Bayes Theorem Section 7.1 Section Summary Finite Probability Probabilities of Complements
More informationMATH MW Elementary Probability Course Notes Part I: Models and Counting
MATH 2030 3.00MW Elementary Probability Course Notes Part I: Models and Counting Tom Salisbury salt@yorku.ca York University Winter 2010 Introduction [Jan 5] Probability: the mathematics used for Statistics
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 informationLecture 04: Conditional Probability. Lisa Yan July 2, 2018
Lecture 04: Conditional Probability Lisa Yan July 2, 2018 Announcements Problem Set #1 due on Friday Gradescope submission portal up Use Piazza No class or OH on Wednesday July 4 th 2 Summary from last
More informationProbabilistic models
Probabilistic models Kolmogorov (Andrei Nikolaevich, 1903 1987) put forward an axiomatic system for probability theory. Foundations of the Calculus of Probabilities, published in 1933, immediately became
More informationWhere are we in CS 440?
Where are we in CS 440? Now leaving: sequential deterministic reasoning Entering: probabilistic reasoning and machine learning robability: Review of main concepts Chapter 3 Making decisions under uncertainty
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 informationORF 245 Fundamentals of Statistics Chapter 5 Probability
ORF 245 Fundamentals of Statistics Chapter 5 Probability Robert Vanderbei Oct 2015 Slides last edited on October 14, 2015 http://www.princeton.edu/ rvdb Sample Spaces (aka Populations) and Events When
More informationUncertainty. Russell & Norvig Chapter 13.
Uncertainty Russell & Norvig Chapter 13 http://toonut.com/wp-content/uploads/2011/12/69wp.jpg Uncertainty Let A t be the action of leaving for the airport t minutes before your flight Will A t get you
More informationLecture 4 Bayes Theorem
Lecture 4 Bayes Theorem Thais Paiva STA 111 - Summer 2013 Term II July 5, 2013 Lecture Plan 1 Probability Review 2 Bayes Theorem 3 More worked problems Why Study Probability? A probability model describes
More informationCogs 14B: Introduction to Statistical Analysis
Cogs 14B: Introduction to Statistical Analysis Statistical Tools: Description vs. Prediction/Inference Description Averages Variability Correlation Prediction (Inference) Regression Confidence intervals/
More informationSome Basic Concepts of Probability and Information Theory: Pt. 1
Some Basic Concepts of Probability and Information Theory: Pt. 1 PHYS 476Q - Southern Illinois University January 18, 2018 PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and
More informationCS 361: Probability & Statistics
February 12, 2018 CS 361: Probability & Statistics Random Variables Monty hall problem Recall the setup, there are 3 doors, behind two of them are indistinguishable goats, behind one is a car. You pick
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 informationIntroduction to Probability Theory, Algebra, and Set Theory
Summer School on Mathematical Philosophy for Female Students Introduction to Probability Theory, Algebra, and Set Theory Catrin Campbell-Moore and Sebastian Lutz July 28, 2014 Question 1. Draw Venn diagrams
More informationAxioms of Probability
Sample Space (denoted by S) The set of all possible outcomes of a random experiment is called the Sample Space of the experiment, and is denoted by S. Example 1.10 If the experiment consists of tossing
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 informationCS626 Data Analysis and Simulation
CS626 Data Analysis and Simulation Instructor: Peter Kemper R 104A, phone 221-3462, email:kemper@cs.wm.edu Today: Probability Primer Quick Reference: Sheldon Ross: Introduction to Probability Models 9th
More informationProbabilistic models
Kolmogorov (Andrei Nikolaevich, 1903 1987) put forward an axiomatic system for probability theory. Foundations of the Calculus of Probabilities, published in 1933, immediately became the definitive formulation
More informationLecture 4 Bayes Theorem
1 / 24 Lecture 4 Bayes Theorem September 09, 2010 2 / 24 Lesson Plan 1. Bayes Theorem 2. Simpson s Paradox 3. More worked problems 3 / 24 Why Study Probability? A probability model describes the random
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 informationECE 340 Probabilistic Methods in Engineering M/W 3-4:15. Lecture 2: Random Experiments. Prof. Vince Calhoun
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Lecture 2: Random Experiments Prof. Vince Calhoun Reading This class: Section 2.1-2.2 Next class: Section 2.3-2.4 Homework: Assignment 1: From the
More informationSets and Set notation. Algebra 2 Unit 8 Notes
Sets and Set notation Section 11-2 Probability Experimental Probability experimental probability of an event: Theoretical Probability number of time the event occurs P(event) = number of trials Sample
More informationn How to represent uncertainty in knowledge? n Which action to choose under uncertainty? q Assume the car does not have a flat tire
Uncertainty Uncertainty Russell & Norvig Chapter 13 Let A t be the action of leaving for the airport t minutes before your flight Will A t get you there on time? A purely logical approach either 1. risks
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 informationCS395T Computational Statistics with Application to Bioinformatics
CS395T Computational Statistics with Application to Bioinformatics Prof. William H. Press Spring Term, 2010 The University of Texas at Austin Unit 1. Probability and Inference The University of Texas at
More informationStatistics 251: Statistical Methods
Statistics 251: Statistical Methods Probability Module 3 2018 file:///volumes/users/r/renaes/documents/classes/lectures/251301/renae/markdown/master%20versions/module3.html#1 1/33 Terminology probability:
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 informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions
Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions 1999 Prentice-Hall, Inc. Chap. 4-1 Chapter Topics Basic Probability Concepts: Sample
More informationORF 245 Fundamentals of Statistics Chapter 1 Probability
ORF 245 Fundamentals of Statistics Chapter 1 Probability Robert Vanderbei Sept 2014 Slides last edited on September 19, 2014 http://www.princeton.edu/ rvdb Course Info Prereqs: Textbook: Three semesters
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 informationTopic 4 Probability. Terminology. Sample Space and Event
Topic 4 Probability The Sample Space is the collection of all possible outcomes Experimental outcome An outcome from a sample space with one characteristic Event May involve two or more outcomes simultaneously
More informationLecture Overview. Introduction to Artificial Intelligence COMP 3501 / COMP Lecture 11: Uncertainty. Uncertainty.
Lecture Overview COMP 3501 / COMP 4704-4 Lecture 11: Uncertainty Return HW 1/Midterm Short HW 2 discussion Uncertainty / Probability Prof. JGH 318 Uncertainty Previous approaches dealt with relatively
More informationStatistical Inference
Statistical Inference Lecture 1: Probability Theory MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 11, 2018 Outline Introduction Set Theory Basics of Probability Theory
More informationCMPSCI 240: Reasoning about Uncertainty
CMPSCI 240: Reasoning about Uncertainty Lecture 2: Sets and Events Andrew McGregor University of Massachusetts Last Compiled: January 27, 2017 Outline 1 Recap 2 Experiments and Events 3 Probabilistic Models
More informationCIVL Why are we studying probability and statistics? Learning Objectives. Basic Laws and Axioms of Probability
CIVL 3103 Basic Laws and Axioms of Probability Why are we studying probability and statistics? How can we quantify risks of decisions based on samples from a population? How should samples be selected
More informationSTAT 430/510 Probability
STAT 430/510 Probability Hui Nie Lecture 3 May 28th, 2009 Review We have discussed counting techniques in Chapter 1. Introduce the concept of the probability of an event. Compute probabilities in certain
More informationIntroduction to Informatics
Introduction to Informatics Lecture 19: Probability Readings until now Lecture notes Posted online http://informatics.indiana.edu/rocha/i101 The Nature of Information Technology Modeling the World @ infoport
More informationCOS 424: Interacting with Data. Lecturer: Dave Blei Lecture #2 Scribe: Ellen Kim February 7, 2008
COS 424: Interacting with Data Lecturer: Dave Blei Lecture #2 Scribe: Ellen Kim February 7, 2008 1 Probability Review 1.1 Random Variables A random variable can be used to model any probabilistic outcome.
More informationDiscrete Random Variables
Discrete Random Variables An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Introduction The markets can be thought of as a complex interaction of a large number of random
More informationthe time it takes until a radioactive substance undergoes a decay
1 Probabilities 1.1 Experiments with randomness Wewillusethetermexperimentinaverygeneralwaytorefertosomeprocess that produces a random outcome. Examples: (Ask class for some first) Here are some discrete
More informationConditional Probability P( )
Conditional Probability P( ) 1 conditional probability where P(F) > 0 Conditional probability of E given F: probability that E occurs given that F has occurred. Conditioning on F Written as P(E F) Means
More informationMachine Learning. CS Spring 2015 a Bayesian Learning (I) Uncertainty
Machine Learning CS6375 --- Spring 2015 a Bayesian Learning (I) 1 Uncertainty Most real-world problems deal with uncertain information Diagnosis: Likely disease given observed symptoms Equipment repair:
More informationWhere are we in CS 440?
Where are we in CS 440? Now leaving: sequential deterministic reasoning Entering: probabilistic reasoning and machine learning robability: Review of main concepts Chapter 3 Motivation: lanning under uncertainty
More informationBayes Rule for probability
Bayes Rule for probability P A B P A P B A PAP B A P AP B A An generalization of Bayes Rule Let A, A 2,, A k denote a set of events such that S A A2 Ak and Ai Aj for all i and j. Then P A i B P Ai P B
More informationEnM Probability and Random Processes
Historical Note: EnM 503 - Probability and Random Processes Probability has its roots in games of chance, which have been played since prehistoric time. Games and equipment have been found in Egyptian
More informationInformation Science 2
Information Science 2 Probability Theory: An Overview Week 12 College of Information Science and Engineering Ritsumeikan University Agenda Terms and concepts from Week 11 Basic concepts of probability
More informationProbability Pr(A) 0, for any event A. 2. Pr(S) = 1, for the sample space S. 3. If A and B are mutually exclusive, Pr(A or B) = Pr(A) + Pr(B).
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
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 informationBasic Probability and Statistics
Basic Probability and Statistics Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [based on slides from Jerry Zhu, Mark Craven] slide 1 Reasoning with Uncertainty
More informationIntro to Stats Lecture 11
Outliers and influential points Intro to Stats Lecture 11 Collect data this week! Midterm is coming! Terms X outliers: observations outlying the overall pattern of the X- variable Y outliers: observations
More informationOrigins of Probability Theory
1 16.584: INTRODUCTION Theory and Tools of Probability required to analyze and design systems subject to uncertain outcomes/unpredictability/randomness. Such systems more generally referred to as Experiments.
More informationProbability Distributions. Conditional Probability
Probability Distributions. Conditional Probability Russell Impagliazzo and Miles Jones Thanks to Janine Tiefenbruck http://cseweb.ucsd.edu/classes/sp16/cse21-bd/ May 16, 2016 In probability, we want to
More informationSample Space: Specify all possible outcomes from an experiment. Event: Specify a particular outcome or combination of outcomes.
Chapter 2 Introduction to Probability 2.1 Probability Model Probability concerns about the chance of observing certain outcome resulting from an experiment. However, since chance is an abstraction of something
More informationChapter 2 PROBABILITY SAMPLE SPACE
Chapter 2 PROBABILITY Key words: Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation,
More informationProbability Review Lecturer: Ji Liu Thank Jerry Zhu for sharing his slides
Probability Review Lecturer: Ji Liu Thank Jerry Zhu for sharing his slides slide 1 Inference with Bayes rule: Example In a bag there are two envelopes one has a red ball (worth $100) and a black ball one
More informationINDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR. NPTEL National Programme on Technology Enhanced Learning. Probability Methods in Civil Engineering
INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR NPTEL National Programme on Technology Enhanced Learning Probability Methods in Civil Engineering Prof. Rajib Maity Department of Civil Engineering IIT Kharagpur
More informationAST 418/518 Instrumentation and Statistics
AST 418/518 Instrumentation and Statistics Class Website: http://ircamera.as.arizona.edu/astr_518 Class Texts: Practical Statistics for Astronomers, J.V. Wall, and C.R. Jenkins Measuring the Universe,
More informationProbability Distributions. Conditional Probability.
Probability Distributions. Conditional Probability. CSE21 Winter 2017, Day 21 (B00), Day 14 (A00) March 6, 2017 http://vlsicad.ucsd.edu/courses/cse21-w17 Probability Rosen p. 446, 453 Sample space, S:
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 informationFormalizing Probability. Choosing the Sample Space. Probability Measures
Formalizing Probability Choosing the Sample Space What do we assign probability to? Intuitively, we assign them to possible events (things that might happen, outcomes of an experiment) Formally, we take
More informationUncertainty. Michael Peters December 27, 2013
Uncertainty Michael Peters December 27, 20 Lotteries In many problems in economics, people are forced to make decisions without knowing exactly what the consequences will be. For example, when you buy
More informationJi-Sun Kang and Eugenia Kalnay
Review of Probability Wilks, Chapter Ji-Sun Kang and Eugenia Kalnay AOSC630 Spring, 008, updated in 07 Definition Event: set, class, or group of possible uncertain outcomes A compound event can be decomposed
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 informationLING 473: Day 5. START THE RECORDING Bayes Theorem. University of Washington Linguistics 473: Computational Linguistics Fundamentals
LING 473: Day 5 START THE RECORDING 1 Announcements I will not be physically here August 8 & 10 Lectures will be made available right before I go to sleep in Oslo So, something like 2:30-3:00pm here. I
More informationAn Introduction to Bayesian Reasoning
Tilburg Center for Logic and Philosophy of Science (TiLPS) Tilburg University, The Netherlands EPS Seminar, TiLPS, 9 October 2013 Overview of the Tutorial This tutorial aims at giving you an idea of why
More informationFinal Examination. Adrian Georgi Josh Karen Lee Min Nikos Tina. There are 12 problems totaling 150 points. Total time is 170 minutes.
Massachusetts Institute of Technology 6.042J/18.062J, Fall 02: Mathematics for Computer Science Prof. Albert Meyer and Dr. Radhika Nagpal Final Examination Your name: Circle the name of your Tutorial Instructor:
More informationStatistical Methods for Astronomy
Statistical Methods for Astronomy Probability (Lecture 1) Statistics (Lecture 2) Why do we need statistics? Useful Statistics Definitions Error Analysis Probability distributions Error Propagation Binomial
More informationProbability (Devore Chapter Two)
Probability (Devore Chapter Two) 1016-345-01: Probability and Statistics for Engineers Fall 2012 Contents 0 Administrata 2 0.1 Outline....................................... 3 1 Axiomatic Probability 3
More informationLecture 4. Selected material from: Ch. 6 Probability
Lecture 4 Selected material from: Ch. 6 Probability Example: Music preferences F M Suppose you want to know what types of CD s males and females are more likely to buy. The CD s are classified as Classical,
More informationChapter 1 Principles of Probability
Chapter Principles of Probability Combining independent probabilities You have applied to three medical schools: University of California at San Francisco (UCSF), Duluth School of Mines (DSM), and Harvard
More informationLecture 2: Probability
Lecture 2: Probability MSU-STT-351-Sum-17B (P. Vellaisamy: MSU-STT-351-Sum-17B) Probability & Statistics for Engineers 1 / 39 Chance Experiment We discuss in this lecture 1 Random Experiments 2 Sample
More informationMATH STUDENT BOOK. 12th Grade Unit 9
MATH STUDENT BOOK 12th Grade Unit 9 Unit 9 COUNTING PRINCIPLES MATH 1209 COUNTING PRINCIPLES INTRODUCTION 1. PROBABILITY DEFINITIONS, SAMPLE SPACES, AND PROBABILITY ADDITION OF PROBABILITIES 11 MULTIPLICATION
More informationChap 1: Experiments, Models, and Probabilities. Random Processes. Chap 1 : Experiments, Models, and Probabilities
EE8103 Random Processes Chap 1 : Experiments, Models, and Probabilities Introduction Real world word exhibits randomness Today s temperature; Flip a coin, head or tail? WalkAt to a bus station, how long
More informationLower bound for sorting/probability Distributions
Lower bound for sorting/probability Distributions CSE21 Winter 2017, Day 20 (B00), Day 14 (A00) March 3, 2017 http://vlsicad.ucsd.edu/courses/cse21-w17 Another application of counting lower bounds Sorting
More informationChapter 2 Probability
Basic Axioms and properties Chapter 2 Probability 1. 0 Pr(A) 1 2. Pr(S) = 1 3. A B Pr(A) Pr(B) 4. Pr(φ ) = 0 5. If A and B are disjoint i.e. A B φ P( A B) = P(A) + P(B). 6. For any two events A and B,
More informationProbabilistic Reasoning
Course 16 :198 :520 : Introduction To Artificial Intelligence Lecture 7 Probabilistic Reasoning Abdeslam Boularias Monday, September 28, 2015 1 / 17 Outline We show how to reason and act under uncertainty.
More informationP (A B) P ((B C) A) P (B A) = P (B A) + P (C A) P (A) = P (B A) + P (C A) = Q(A) + Q(B).
Lectures 7-8 jacques@ucsdedu 41 Conditional Probability Let (Ω, F, P ) be a probability space Suppose that we have prior information which leads us to conclude that an event A F occurs Based on this information,
More informationCSC Discrete Math I, Spring Discrete Probability
CSC 125 - Discrete Math I, Spring 2017 Discrete Probability Probability of an Event Pierre-Simon Laplace s classical theory of probability: Definition of terms: An experiment is a procedure that yields
More informationMATHEMATICS 191, FALL 2004 MATHEMATICAL PROBABILITY Outline #3 (Conditional Probability)
Last modified: October 5, 2004 References: PRP, sections 1.4, 1.5 and 1.7 EP, Chapter 2 MATHEMATICS, FALL 2004 MATHEMATICAL PROBABILITY Outline #3 (Conditional Probability) 1. Define conditional probability
More information