Probability Review. Chao Lan
|
|
- Gregory Johns
- 5 years ago
- Views:
Transcription
1 Probability Review Chao Lan
2 Let s start with a single random variable
3 Random Experiment A random experiment has three elements 1. sample space Ω: set of all possible outcomes e.g.,ω={1,2,3,4,5,6} 2. event space E: collection of subsets of Ω e.g., E={{1},...,{1,2},...,{3,4,6},...,{1,2,3,4,5,6}}=2 Ω 3. probability measure P: a special function P : E R e.g., P({1})=1/6, P({3,4,6})=3/6
4 Axioms of Probability Probability measure P needs to satisfy 1. P(e) 0, for any event e E e.g.,p({3,4,6})=3/ P(Ω)=1 e.g., P({1,2,3,4,5,6})=1 3. if e1,e2 E are disjoint, then P(e1 e2)=p(e1)+p(e2) e.g., P({1} {2,4,6})=P({1})+P({2,4,6})
5 Exercise Tomorrow s Temperature Tomorrow s Weather Sample Space Ω =? Sample Space Ω =? Event Space E =? Event Space E =? Probability Measure P? Probability Measure P?
6 Exercise A random experiment is flipping a coin twice (getting H or T in each trial) 1. sample space Ω =? 2. event space E =? 3. probability measure P? A random experiment can be a single trial or a sequence of trails.
7 Random Variable A random variable X is a function X : Ω R. Q: are the following random variables? 1. X({1}) = 1, X({3}) = 3, 2. X({1}) = 19, X({3}) = 45.7, 3. X({1}) = 1, X({3}) = 1,
8 Random Variable A random variable X is a function X : E R. Q: are the following random variables? 1. X({1}) = 1, X({3}) = 3, 2. X({1}) = 19, X({3}) = 45.7, 3. X({1}) = 1, X({3}) = 1, Q: why bother?
9 Random Variable Random variable makes presentation easier! Example 1 If X({1})=19 and X({3})=45.7, instead of writing P({1}), P({3}), we can write P(X=19), P(X=45.7). Example 2 If X counts odds over three trials, e.g., X({1, 2, 6})=1, X({1,2,3})=2, instead of writing P(1 odd) or P(2 odds), we can write P(X=1) or P(X=2).
10 Discrete Variable vs Continuous Variable X is discrete if it maps E to a discrete domain (our focus) - e.g., X({3})=1, X({1,2,3})=2 - specify P(X=1), P(X=2), etc X is continuous if it maps E to a continuous domain - e.g., X(angle) = specify P(0 X 30), etc For continuous random variable X, P(X=a)=0.
11 Properties of Probability 1. if e1 e2, then P(e1) P(e2) e.g., since {2} {2,4,6}, we have P({2}) P({2,4,6}) 2. P(e1 e2) min{p(e1), P(e2)} e.g., P({2,3} {2,4,6}) min{p({2,3}),p({2,4,6})} 3. P(e1 e2) P(e1) + P(e2) [union bound] e.g., P({2,3} {2,4,6}) P({2,3}) + P({2,4,6})
12 Properties of Probability 4. P(Ω \ e) = 1 - P(e) e.g., P({1,2,3,4,5,6}\{1,2,3}) = 1 - P({1,2,3}) 5. if e1,e2 are disjoint, e1 e2=ω, then P(e1)+P(e2)=1 [law of total probability] e.g., P({1,2,3})+P({4,5,6}) = 1
13 What is the distribution of a random variable?
14 Cumulative Distribution Function (CDF) CDF of X is a function CDFX: R [0,1] specifying a special probability CDFX(a) = P(X a) Example If X counts odds, P(X 3) is the chance of getting three trials with no more than 3 odds, i.e., P(X 3) = P({e:X(e) 3}) If X is temperature, P(X 27) is the chance of getting tomorrow s temperature of no higher than 27F i.e., P(X 27) = P({e:X(e) 27})
15 Properties of CDF CDF of X is a function CDFX: R [0,1] specifying a special probability CDFX(a) = P(X a) 1. 0 CDF(a) 1 2. lima - CDF(a) = 0 3. lima + CDF(a) = 1 4. if a b, then CDF(a) CDF(b)
16 Probability Density Function (PDF) PDF of a continuous random variable X is the derivative of its CDF p(x) = CDF(x) / x
17 Example: Gaussian Distribution If X follows a Gaussian distribution (or, normal distribution), then
18 Different PDFs
19 Properties of PDF PDF of a continuous random variable X is the derivative of its CDF p(x) = CDF(x) / x 1. p(x) 0 2. x (-,+ ) p(x) dx = 1 3. x S p(x) dx = P(X S)
20 Probability Mass Function (PMF) PMF of a discrete random variable X is a table of probabilities for all events.
21 Different PMFs
22 Properties of PMF PMF of a discrete random variable X is a table of probabilities for all events p(x) 0 2. Σx (-,+ ) p(x) = 1 3. Σx S p(x) = P(X S)
23 Based on distribution, we can define expectation and variance of a variable...
24 Expectation Expectation of a random variable is its weighted average value. E[X] = Σx D x*p(x) X p(x) E[X] = Σx D x*p(x) = 1* * * = 1.75 or E[X] = x D x*p(x) dx
25 Expectation of Standard Normal Distribution Expectation of a normally distributed random variable ~N(0,σ2) is zero. One can similarly prove expectation of variable from N(m,σ2) is m.
26 Properties of Expectation Expectation of a random variable is its weighted average value. E[X] = Σx D x*p(x) 1. E[a] = a, for any constant a 2. E[a*X] = a*e[x] 3. E[X+Y] = E[X] + E[Y] 4. E[1X=k] = P(X=k) or E[X] = x D x*p(x) dx
27 Variance Variance of a random variable is its average deviation from its expectation. Var[X] = E [ (X-E[X])2 ] X p(x) E[X] Var[X] = Σx D x*p(x) = Σx D (X-E[X])2*p(x) = 1* * *0.25 = (1-1.75)2*0.5 + (2-1.75)2* (3-1.75)2* = 1.75 = Note expectation E[X] is a constant.
28 Variance of Normally Distributed Variable
29 Variance of Other Distribution Poisson Distribution Rayleigh Distribution
30 Properties of Variance Variance of a random variable is its average deviation from its expectation. Var[X] = E [ (X-E[X])2 ] 1. Var[X] = E[X2] - E[X]2 2. Var[a] = 0 for constant a 3. Var[aX] = a2 Var[X]
31 Common Distributions
32 Now we look at two (discrete) random variables... The following slides are largely built on the lecture slides of probability by Klein and Abbeel.
33 Recap: Distribution of Single Variable Q: what is the probability that tomorrow is hot and sun, i.e. P(T=hot, W=sun)?
34 Joint Distribution of Two Variables Joint distribution is a table of probabilities of all joint events.
35 Marginal Distribution Sometimes we are interested in the distribution of one variable. We can obtain it from the joint distribution by marginalizing it. marginal distribution
36 Exercise
37 Conditional Distribution Sometimes we are interested in the distribution of one variable X while fixing the other Y. This is conditional distribution and defined as P(X Y) = P(X,Y) / P(Y).
38 Conditional Distribution Sometimes we are interested in the distribution of one variable X while fixing the other Y. This is conditional distribution and defined as P(X Y) = P(X,Y) / P(Y).
39 Exercise
40 Remark Variable X can have different distributions when conditioned on different Y.
41 Product Rule Infer joint probability from marginal probability and conditional probability.
42 Exercise
43 Chain Rule Split a joint probability of multiple variables into conditional probabilities of each.
44 Chain Rule Split a joint probability of multiple variables into conditional probabilities of each. Q: how to derive it?
45 Bayes s Rule Bayes s rule connects two conditional probabilities. - P(A) is prior of A, P(A B) is posterior of A - p(a=a B=b) = p(b=b A=a)*p(A=a) / p(b=b) [discrete] - p(a b) = p(b a)*p(a) / p(b) [continuous] Q: how to derive Bayes s Rule? Why Bayes s rule?
46 Example We randomly pick up a coin and flip it to get head (H) or tail (T). We know a priori that - 40% coins are fair (F), and 60% are unfair (U) - 80% chance we get a head (from an arbitrary coin) If we get head, what is the chance the flipped coin is fair? P(F H)?
47 Exercise
48 Independence Two variables X and Y are independent if P(X, Y) = P(X) P(Y) - or, equivalently P(X Y) = P(X) - p(x=x,y=y) = p(x=x)*p(y=y) [discrete] - p(x,y) = p(x)*p(y) [continuous]
49 Expectation Expectation of two random variables X,Y is E[g(X,Y)] = Σ(x,y) D g(x,y)*p(x,y) or E[g(X,Y)] = (x,y) D g(x,y)*p(x) dxdy
50 Covariance Covariance of two variables X, Y is Cov(X,Y) = E[(X-E[X])(Y - E[Y])] - if X=Y, then Cov(X,Y) = Var(X) = Var(Y) - Cov(X,Y) = E[XY] - E[X] E[Y] - Var[X+Y] = Var[X] + Var[Y] + Cov[X,Y] - if X, Y are independent, then Cov[X,Y] = 0 (uncorrelated)
51 Property of Covariance
Probability Review. Yutian Li. January 18, Stanford University. Yutian Li (Stanford University) Probability Review January 18, / 27
Probability Review Yutian Li Stanford University January 18, 2018 Yutian Li (Stanford University) Probability Review January 18, 2018 1 / 27 Outline 1 Elements of probability 2 Random variables 3 Multiple
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 informationRecitation 2: Probability
Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions
More informationJoint Distribution of Two or More Random Variables
Joint Distribution of Two or More Random Variables Sometimes more than one measurement in the form of random variable is taken on each member of the sample space. In cases like this there will be a few
More informationAppendix A : Introduction to Probability and stochastic processes
A-1 Mathematical methods in communication July 5th, 2009 Appendix A : Introduction to Probability and stochastic processes Lecturer: Haim Permuter Scribe: Shai Shapira and Uri Livnat The probability of
More informationLecture 2: Repetition of probability theory and statistics
Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:
More informationMore on Distribution Function
More on Distribution Function The distribution of a random variable X can be determined directly from its cumulative distribution function F X. Theorem: Let X be any random variable, with cumulative distribution
More informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Tobias Neckel, Ionuț-Gabriel Farcaș Lehrstuhl Informatik V Summer Semester 2017 Lecture 2: Repetition of probability theory and statistics Example: coin flip Example
More informationRandom Variables. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 13 Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 8, 2013 2 / 13 Random Variable Definition A real-valued
More informationReview of Probability. CS1538: Introduction to Simulations
Review of Probability CS1538: Introduction to Simulations Probability and Statistics in Simulation Why do we need probability and statistics in simulation? Needed to validate the simulation model Needed
More informationProbability. Paul Schrimpf. January 23, UBC Economics 326. Probability. Paul Schrimpf. Definitions. Properties. Random variables.
Probability UBC Economics 326 January 23, 2018 1 2 3 Wooldridge (2013) appendix B Stock and Watson (2009) chapter 2 Linton (2017) chapters 1-5 Abbring (2001) sections 2.1-2.3 Diez, Barr, and Cetinkaya-Rundel
More informationRandom variables (discrete)
Random variables (discrete) Saad Mneimneh 1 Introducing random variables A random variable is a mapping from the sample space to the real line. We usually denote the random variable by X, and a value that
More informationChapter 2. Some Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables
Chapter 2 Some Basic Probability Concepts 2.1 Experiments, Outcomes and Random Variables A random variable is a variable whose value is unknown until it is observed. The value of a random variable results
More informationProbability. Paul Schrimpf. January 23, Definitions 2. 2 Properties 3
Probability Paul Schrimpf January 23, 2018 Contents 1 Definitions 2 2 Properties 3 3 Random variables 4 3.1 Discrete........................................... 4 3.2 Continuous.........................................
More informationBayesian statistics, simulation and software
Module 1: Course intro and probability brush-up Department of Mathematical Sciences Aalborg University 1/22 Bayesian Statistics, Simulations and Software Course outline Course consists of 12 half-days
More informationE X A M. Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours. Number of pages incl.
E X A M Course code: Course name: Number of pages incl. front page: 6 MA430-G Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours Resources allowed: Notes: Pocket calculator,
More informationSTAT 430/510 Probability Lecture 7: Random Variable and Expectation
STAT 430/510 Probability Lecture 7: Random Variable and Expectation Pengyuan (Penelope) Wang June 2, 2011 Review Properties of Probability Conditional Probability The Law of Total Probability Bayes Formula
More informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationFundamental Tools - Probability Theory II
Fundamental Tools - Probability Theory II MSc Financial Mathematics The University of Warwick September 29, 2015 MSc Financial Mathematics Fundamental Tools - Probability Theory II 1 / 22 Measurable random
More informationDeep Learning for Computer Vision
Deep Learning for Computer Vision Lecture 3: Probability, Bayes Theorem, and Bayes Classification Peter Belhumeur Computer Science Columbia University Probability Should you play this game? Game: A fair
More informationHomework 4 Solution, due July 23
Homework 4 Solution, due July 23 Random Variables Problem 1. Let X be the random number on a die: from 1 to. (i) What is the distribution of X? (ii) Calculate EX. (iii) Calculate EX 2. (iv) Calculate Var
More informationRandom Variables. Random variables. A numerically valued map X of an outcome ω from a sample space Ω to the real line R
In probabilistic models, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. As a function or a map, it maps from an element (or an outcome) of a sample
More informationSTAT2201. Analysis of Engineering & Scientific Data. Unit 3
STAT2201 Analysis of Engineering & Scientific Data Unit 3 Slava Vaisman The University of Queensland School of Mathematics and Physics What we learned in Unit 2 (1) We defined a sample space of a random
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 informationProbability Theory for Machine Learning. Chris Cremer September 2015
Probability Theory for Machine Learning Chris Cremer September 2015 Outline Motivation Probability Definitions and Rules Probability Distributions MLE for Gaussian Parameter Estimation MLE and Least Squares
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
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 informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationSummary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016
8. For any two events E and F, P (E) = P (E F ) + P (E F c ). Summary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016 Sample space. A sample space consists of a underlying
More informationMore than one variable
Chapter More than one variable.1 Bivariate discrete distributions Suppose that the r.v. s X and Y are discrete and take on the values x j and y j, j 1, respectively. Then the joint p.d.f. of X and Y, to
More informationReview: mostly probability and some statistics
Review: mostly probability and some statistics C2 1 Content robability (should know already) Axioms and properties Conditional probability and independence Law of Total probability and Bayes theorem Random
More informationFundamentals. CS 281A: Statistical Learning Theory. Yangqing Jia. August, Based on tutorial slides by Lester Mackey and Ariel Kleiner
Fundamentals CS 281A: Statistical Learning Theory Yangqing Jia Based on tutorial slides by Lester Mackey and Ariel Kleiner August, 2011 Outline 1 Probability 2 Statistics 3 Linear Algebra 4 Optimization
More information01 Probability Theory and Statistics Review
NAVARCH/EECS 568, ROB 530 - Winter 2018 01 Probability Theory and Statistics Review Maani Ghaffari January 08, 2018 Last Time: Bayes Filters Given: Stream of observations z 1:t and action data u 1:t Sensor/measurement
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 information6.041/6.431 Fall 2010 Quiz 2 Solutions
6.04/6.43: Probabilistic Systems Analysis (Fall 200) 6.04/6.43 Fall 200 Quiz 2 Solutions Problem. (80 points) In this problem: (i) X is a (continuous) uniform random variable on [0, 4]. (ii) Y is an exponential
More informationLecture 1: Probability Fundamentals
Lecture 1: Probability Fundamentals IB Paper 7: Probability and Statistics Carl Edward Rasmussen Department of Engineering, University of Cambridge January 22nd, 2008 Rasmussen (CUED) Lecture 1: Probability
More information1 Random Variable: Topics
Note: Handouts DO NOT replace the book. In most cases, they only provide a guideline on topics and an intuitive feel. 1 Random Variable: Topics Chap 2, 2.1-2.4 and Chap 3, 3.1-3.3 What is a random variable?
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 14: Continuous random variables Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin www.cs.cmu.edu/
More informationLecture 16. Lectures 1-15 Review
18.440: Lecture 16 Lectures 1-15 Review Scott Sheffield MIT 1 Outline Counting tricks and basic principles of probability Discrete random variables 2 Outline Counting tricks and basic principles of probability
More informationLecture 3. Discrete Random Variables
Math 408 - Mathematical Statistics Lecture 3. Discrete Random Variables January 23, 2013 Konstantin Zuev (USC) Math 408, Lecture 3 January 23, 2013 1 / 14 Agenda Random Variable: Motivation and Definition
More informationLECTURE 1. 1 Introduction. 1.1 Sample spaces and events
LECTURE 1 1 Introduction The first part of our adventure is a highly selective review of probability theory, focusing especially on things that are most useful in statistics. 1.1 Sample spaces and events
More informationQuick Tour of Basic Probability Theory and Linear Algebra
Quick Tour of and Linear Algebra Quick Tour of and Linear Algebra CS224w: Social and Information Network Analysis Fall 2011 Quick Tour of and Linear Algebra Quick Tour of and Linear Algebra Outline Definitions
More informationReview of Probabilities and Basic Statistics
Alex Smola Barnabas Poczos TA: Ina Fiterau 4 th year PhD student MLD Review of Probabilities and Basic Statistics 10-701 Recitations 1/25/2013 Recitation 1: Statistics Intro 1 Overview Introduction to
More information3 Multiple Discrete Random Variables
3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f
More informationDISCRETE RANDOM VARIABLES: PMF s & CDF s [DEVORE 3.2]
DISCRETE RANDOM VARIABLES: PMF s & CDF s [DEVORE 3.2] PROBABILITY MASS FUNCTION (PMF) DEFINITION): Let X be a discrete random variable. Then, its pmf, denoted as p X(k), is defined as follows: p X(k) :=
More informationBivariate distributions
Bivariate distributions 3 th October 017 lecture based on Hogg Tanis Zimmerman: Probability and Statistical Inference (9th ed.) Bivariate Distributions of the Discrete Type The Correlation Coefficient
More informationIntroduction to Machine Learning
Introduction to Machine Learning Introduction to Probabilistic Methods Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB
More informationLecture 10: Probability distributions TUESDAY, FEBRUARY 19, 2019
Lecture 10: Probability distributions DANIEL WELLER TUESDAY, FEBRUARY 19, 2019 Agenda What is probability? (again) Describing probabilities (distributions) Understanding probabilities (expectation) Partial
More informationa zoo of (discrete) random variables
a zoo of (discrete) random variables 42 uniform random variable Takes each possible value, say {1..n} with equal probability. Say random variable uniform on S Recall envelopes problem on homework... Randomization
More informationSingle Maths B: Introduction to Probability
Single Maths B: Introduction to Probability Overview Lecturer Email Office Homework Webpage Dr Jonathan Cumming j.a.cumming@durham.ac.uk CM233 None! http://maths.dur.ac.uk/stats/people/jac/singleb/ 1 Introduction
More informationChapter 2. Probability
2-1 Chapter 2 Probability 2-2 Section 2.1: Basic Ideas Definition: An experiment is a process that results in an outcome that cannot be predicted in advance with certainty. Examples: rolling a die tossing
More informationLecture 11. Probability Theory: an Overveiw
Math 408 - Mathematical Statistics Lecture 11. Probability Theory: an Overveiw February 11, 2013 Konstantin Zuev (USC) Math 408, Lecture 11 February 11, 2013 1 / 24 The starting point in developing the
More informationChapter 2. Continuous random variables
Chapter 2 Continuous random variables Outline Review of probability: events and probability Random variable Probability and Cumulative distribution function Review of discrete random variable Introduction
More informationMAS113 Introduction to Probability and Statistics. Proofs of theorems
MAS113 Introduction to Probability and Statistics Proofs of theorems Theorem 1 De Morgan s Laws) See MAS110 Theorem 2 M1 By definition, B and A \ B are disjoint, and their union is A So, because m is a
More informationCMPSCI 240: Reasoning Under Uncertainty
CMPSCI 240: Reasoning Under Uncertainty Lecture 8 Prof. Hanna Wallach wallach@cs.umass.edu February 16, 2012 Reminders Check the course website: http://www.cs.umass.edu/ ~wallach/courses/s12/cmpsci240/
More information1 Presessional Probability
1 Presessional Probability Probability theory is essential for the development of mathematical models in finance, because of the randomness nature of price fluctuations in the markets. This presessional
More informationProbability. Computer Science Tripos, Part IA. R.J. Gibbens. Computer Laboratory University of Cambridge. Easter Term 2008/9
Probability Computer Science Tripos, Part IA R.J. Gibbens Computer Laboratory University of Cambridge Easter Term 2008/9 Last revision: 2009-05-06/r-36 1 Outline Elementary probability theory (2 lectures)
More informationLecture 10: Bayes' Theorem, Expected Value and Variance Lecturer: Lale Özkahya
BBM 205 Discrete Mathematics Hacettepe University http://web.cs.hacettepe.edu.tr/ bbm205 Lecture 10: Bayes' Theorem, Expected Value and Variance Lecturer: Lale Özkahya Resources: Kenneth Rosen, Discrete
More informationPart IA Probability. Definitions. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Definitions Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationData Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2017 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
More informationBrief Review of Probability
Brief Review of Probability Nuno Vasconcelos (Ken Kreutz-Delgado) ECE Department, UCSD Probability Probability theory is a mathematical language to deal with processes or experiments that are non-deterministic
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 informationIntroduction to Stochastic Processes
Stat251/551 (Spring 2017) Stochastic Processes Lecture: 1 Introduction to Stochastic Processes Lecturer: Sahand Negahban Scribe: Sahand Negahban 1 Organization Issues We will use canvas as the course webpage.
More informationReview of probability
Review of probability Computer Sciences 760 Spring 2014 http://pages.cs.wisc.edu/~dpage/cs760/ Goals for the lecture you should understand the following concepts definition of probability random variables
More informationIntroduction to Statistical Inference Self-study
Introduction to Statistical Inference Self-study Contents Definition, sample space The fundamental object in probability is a nonempty sample space Ω. An event is a subset A Ω. Definition, σ-algebra A
More informationThe Binomial distribution. Probability theory 2. Example. The Binomial distribution
Probability theory Tron Anders Moger September th 7 The Binomial distribution Bernoulli distribution: One experiment X i with two possible outcomes, probability of success P. If the experiment is repeated
More informationReview of Probability Mark Craven and David Page Computer Sciences 760.
Review of Probability Mark Craven and David Page Computer Sciences 760 www.biostat.wisc.edu/~craven/cs760/ Goals for the lecture you should understand the following concepts definition of probability random
More informationfor valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I
Code: 15A04304 R15 B.Tech II Year I Semester (R15) Regular Examinations November/December 016 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks:
More informationMath Review Sheet, Fall 2008
1 Descriptive Statistics Math 3070-5 Review Sheet, Fall 2008 First we need to know about the relationship among Population Samples Objects The distribution of the population can be given in one of the
More informationLecture 10. Variance and standard deviation
18.440: Lecture 10 Variance and standard deviation Scott Sheffield MIT 1 Outline Defining variance Examples Properties Decomposition trick 2 Outline Defining variance Examples Properties Decomposition
More informationChapter 4. Chapter 4 sections
Chapter 4 sections 4.1 Expectation 4.2 Properties of Expectations 4.3 Variance 4.4 Moments 4.5 The Mean and the Median 4.6 Covariance and Correlation 4.7 Conditional Expectation SKIP: 4.8 Utility Expectation
More information1.1 Review of Probability Theory
1.1 Review of Probability Theory Angela Peace Biomathemtics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology. CRC Press,
More informationMAS113 Introduction to Probability and Statistics. Proofs of theorems
MAS113 Introduction to Probability and Statistics Proofs of theorems Theorem 1 De Morgan s Laws) See MAS110 Theorem 2 M1 By definition, B and A \ B are disjoint, and their union is A So, because m is a
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 informationLecture Notes 5 Convergence and Limit Theorems. Convergence with Probability 1. Convergence in Mean Square. Convergence in Probability, WLLN
Lecture Notes 5 Convergence and Limit Theorems Motivation Convergence with Probability Convergence in Mean Square Convergence in Probability, WLLN Convergence in Distribution, CLT EE 278: Convergence and
More informationBasics on Probability. Jingrui He 09/11/2007
Basics on Probability Jingrui He 09/11/2007 Coin Flips You flip a coin Head with probability 0.5 You flip 100 coins How many heads would you expect Coin Flips cont. You flip a coin Head with probability
More informationReview of Probability Theory
Review of Probability Theory Arian Maleki and Tom Do Stanford University Probability theory is the study of uncertainty Through this class, we will be relying on concepts from probability theory for deriving
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 informationEE514A Information Theory I Fall 2013
EE514A Information Theory I Fall 2013 K. Mohan, Prof. J. Bilmes University of Washington, Seattle Department of Electrical Engineering Fall Quarter, 2013 http://j.ee.washington.edu/~bilmes/classes/ee514a_fall_2013/
More informationSTA 111: Probability & Statistical Inference
STA 111: Probability & Statistical Inference Lecture Four Expectation and Continuous Random Variables Instructor: Olanrewaju Michael Akande Department of Statistical Science, Duke University Instructor:
More informationLecture 2. October 21, Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University.
Lecture 2 Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University October 21, 2007 1 2 3 4 5 6 Define probability calculus Basic axioms of probability Define
More informationBasic Probability. Introduction
Basic Probability Introduction The world is an uncertain place. Making predictions about something as seemingly mundane as tomorrow s weather, for example, is actually quite a difficult task. Even with
More informationECE531: Principles of Detection and Estimation Course Introduction
ECE531: Principles of Detection and Estimation Course Introduction D. Richard Brown III WPI 22-January-2009 WPI D. Richard Brown III 22-January-2009 1 / 37 Lecture 1 Major Topics 1. Web page. 2. Syllabus
More informationRandom Variables. Cumulative Distribution Function (CDF) Amappingthattransformstheeventstotherealline.
Random Variables Amappingthattransformstheeventstotherealline. Example 1. Toss a fair coin. Define a random variable X where X is 1 if head appears and X is if tail appears. P (X =)=1/2 P (X =1)=1/2 Example
More informationExercises with solutions (Set D)
Exercises with solutions Set D. A fair die is rolled at the same time as a fair coin is tossed. Let A be the number on the upper surface of the die and let B describe the outcome of the coin toss, where
More informationB4 Estimation and Inference
B4 Estimation and Inference 6 Lectures Hilary Term 27 2 Tutorial Sheets A. Zisserman Overview Lectures 1 & 2: Introduction sensors, and basics of probability density functions for representing sensor error
More informationA random variable is a quantity whose value is determined by the outcome of an experiment.
Random Variables A random variable is a quantity whose value is determined by the outcome of an experiment. Before the experiment is carried out, all we know is the range of possible values. Birthday example
More information1 Probability theory. 2 Random variables and probability theory.
Probability theory Here we summarize some of the probability theory we need. If this is totally unfamiliar to you, you should look at one of the sources given in the readings. In essence, for the major
More informationL2: Review of probability and statistics
Probability L2: Review of probability and statistics Definition of probability Axioms and properties Conditional probability Bayes theorem Random variables Definition of a random variable Cumulative distribution
More informationLecture 2: Review of Probability
Lecture 2: Review of Probability Zheng Tian Contents 1 Random Variables and Probability Distributions 2 1.1 Defining probabilities and random variables..................... 2 1.2 Probability distributions................................
More informationLecture 3: Basic Statistical Tools. Bruce Walsh lecture notes Tucson Winter Institute 7-9 Jan 2013
Lecture 3: Basic Statistical Tools Bruce Walsh lecture notes Tucson Winter Institute 7-9 Jan 2013 1 Basic probability Events are possible outcomes from some random process e.g., a genotype is AA, a phenotype
More informationDiscrete Probability Refresher
ECE 1502 Information Theory Discrete Probability Refresher F. R. Kschischang Dept. of Electrical and Computer Engineering University of Toronto January 13, 1999 revised January 11, 2006 Probability theory
More informationExpectation of Random Variables
1 / 19 Expectation of Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 13, 2015 2 / 19 Expectation of Discrete
More information1 INFO Sep 05
Events A 1,...A n are said to be mutually independent if for all subsets S {1,..., n}, p( i S A i ) = p(a i ). (For example, flip a coin N times, then the events {A i = i th flip is heads} are mutually
More informationStochastic Models of Manufacturing Systems
Stochastic Models of Manufacturing Systems Ivo Adan Organization 2/47 7 lectures (lecture of May 12 is canceled) Studyguide available (with notes, slides, assignments, references), see http://www.win.tue.nl/
More informationCHAPTER 4 MATHEMATICAL EXPECTATION. 4.1 Mean of a Random Variable
CHAPTER 4 MATHEMATICAL EXPECTATION 4.1 Mean of a Random Variable The expected value, or mathematical expectation E(X) of a random variable X is the long-run average value of X that would emerge after a
More informationSTAT 430/510: Lecture 10
STAT 430/510: Lecture 10 James Piette June 9, 2010 Updates HW2 is due today! Pick up your HW1 s up in stat dept. There is a box located right when you enter that is labeled "Stat 430 HW1". It ll be out
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 17: Continuous random variables: conditional PDF Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin
More information1: PROBABILITY REVIEW
1: PROBABILITY REVIEW Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) Slides 1: Probability Review 1 / 56 Outline We will review the following
More informationOutline PMF, CDF and PDF Mean, Variance and Percentiles Some Common Distributions. Week 5 Random Variables and Their Distributions
Week 5 Random Variables and Their Distributions Week 5 Objectives This week we give more general definitions of mean value, variance and percentiles, and introduce the first probability models for discrete
More information