Lecture 2: Probability
|
|
|
- Winfred Stanley
- 5 years ago
- Views:
Transcription
1 Lecture 2: Probability
2 Statistical Paradigms Statistical Estimator Method of Estimation Output Data Complexity Prior Info Classical Cost Function Analytical Solution Point Estimate Simple No Maximum Likelihood Probability Theory Numerical Optimization Point Estimate Intermediate No Bayesian Probability Theory Sampling Probability Distribution Complex Yes The unifying principal for this course is statistical estimation based on probability
3 Overview Basic probability Joint, marginal, conditional probability Bayes Rule Random variables Probability distribution Discrete Continuous Moments One could spend ½ a semester on this alone...
4 Example White-breasted fruit dove (Ptilinopus rivoli) Yellow-bibbed fruit dove (Ptilinopus solomonensis)
5 Events S Sc R 2 9 Rc 18 3 Pr(A) = probability that event A occurs Pr(R) =? Pr(Rc) =? Pr(S) =? Pr(Sc) =?
6 Events S Sc R 2 9 Rc 18 3 Pr(A) = probability that event A occurs Pr(R) = 11/32 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32
7 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 Joint Probability Pr(A,B) = probability that both A and B occur Pr(R,Sc) =? Pr(S,Rc) =? Pr(R,S) =? Pr(Rc,Sc) =?
8 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 Joint Probability Pr(A,B) = probability that both A and B occur Pr(R,Sc) = 9/32 Pr(S,Rc) = 18/32 Pr(R,S) = 2/32 Pr(Rc,Sc) = 3/32
9 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 Pr(A or B) = Pr(A) + Pr(B) Pr(A,B) = 1 Pr(neither) Pr(R or S) =?
10 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 Pr(A or B) = Pr(A) + Pr(B) Pr(A,B) = 1 Pr(neither) Pr(R or S) = 11/ /32 2/32 = 29/32 = 32/32 3/32 = 29/32
11 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 If Pr(A,B) = Pr(A) Pr(B) then A and B are independent Pr(R,S) = Pr(R) Pr(S)??
12 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 If Pr(A,B) = Pr(A) Pr(B) then A and B are independent = 2/32 = Pr(R,S) Pr(R) Pr(S) = 11/32 20/32 = 0.215
13 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 Conditional Probability Pr(A B) = Probability of A given B occurred Pr(A B) = Pr(A,B) / Pr(B) Pr(B A) = Pr(B,A) / Pr(A) Pr(R S) =?
14 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 Conditional Probability Pr(B A) = Pr(B,A) / Pr(A) Pr(A B) = Pr(A,B) / Pr(B) Pr(R S) = Pr(R,S) / Pr(S) = (2/32) / (20/32) = 2/20
15 Conditional Probability Pr(B A) = Pr(B,A) / Pr(A) Pr(A B) = Pr(A,B) / Pr(B) Pr(A,B) = Pr(B A) Pr(A) Pr(B,A) = Pr(A B) Pr(B)
16 Conditional Probability Pr(B A) = Pr(B,A) / Pr(A) Pr(A B) = Pr(A,B) / Pr(B) Pr(A,B) = Pr(B A) Pr(A) Pr(B,A) = Pr(A B) Pr(B) Pr(A B) Pr(B) = Pr(B A) Pr(A)
17 Conditional Probability Pr(B A) = Pr(B,A) / Pr(A) Pr(A B) = Pr(A,B) / Pr(B) Pr(A,B) = Pr(B A) Pr(A) Pr(B,A) = Pr(A B) Pr(B) Pr(A B) Pr(B) = Pr(B A) Pr(A) Pr(A B) = Pr(B A) Pr(A) / Pr(B) BAYES RULE
18 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 Bayes Rule Pr(A B) = Pr(B A) Pr(A) / Pr(B) Pr(R S) = Pr(S R) Pr(R) / Pr(S) = (2/11) (11/32) / (20/32) = 2/20
19 Events S Sc R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Pr(S) = 20/32 Pr(Sc) = 12/32 32 Marginal Probability Pr(B) = Σ Pr(B,A i ) Pr(B) = Σ Pr(B A i ) Pr(A i ) Pr(R) =?
20 Events S Sc Marginal R 2 9 Pr(R) = 11/32 Rc 18 3 Pr(Rc) = 21/32 Marginal Pr(S) = 20/32 Pr(Sc) = 12/32 32 Marginal Probability Pr(B) = Σ Pr(B A i ) Pr(A i ) Pr(R) = Pr(R S) Pr(S) + Pr(R Sc) Pr(Sc) = 2/20 20/32 + 9/12 12/32 = 2/32 + 9/32 = 11/32
21 BAYES RULE: alternate form Pr(A B) = Pr(B A) Pr(A) / Pr(B) Pr(A B) = Pr(B A) Pr(A) / (ΣPr(B A) Pr(A) ) Factoring probabilities: Pr(A,B,C) = Pr(A B,C)Pr(B,C) = Pr(A B,C)P(B C)Pr(C) Joint = Conditional x Marginal
22 Random Variables a variable that can take on more than one value, in which the values are determined by probabilities Pr(Z = z k ) = p k given: 0 p k 1 Random variables can be continuous or discrete
23 Discrete random variables k can only take on discrete values (typically integers) We can define two important and interrelated functions probability mass function (pmf): f(z) = Pr(Z = z k ) = p k where f(z) = 1 (if not met, is just a density fcn) Cumulative distribution function (cdf): F(z) = Pr(Z z k ) = f(z) summed up to k 0 F(z) 1 but can be infinite in z
24 Example For the set {1,2,3,4,5,6} f(z) = Pr(Z = z k ) = {1/6,1/6,1/6,1/6,1/6,1/6} F(z) = Pr(Z z k )= {1/6,2/6,3/6,4/6,5/6,6/6} For z < 1, f(z) = 0, F(z) = 0 For z > 6, f(z) = 0, F(z) = 1
25 Continuous Random Variables z is Real (though can still be bound) Cumulative distribution function (cdf): F(z) = Pr(Z z) where 0 F(z) 1 Pr(Z = z) is infinitely small Pr(z Z z + dz) = Pr(Z z+dz) Pr(Z z) = F(z+dz) F(z) Probability density function (pdf): f(z) = df/dz f(z) 0 but NOT bound by 1
26 Continuous Random Variables f is derivative of F F is integral of f z dz Pr z Z z dz = z f z ANY function that meets these rules (positive, integrate to 1) Monday we will be going over a number of standard number of distributions and discussing their interpretation/application.
27 Example: exponential distribution f z = exp z F z =1 exp z Where z 0 What are the values of F(z) and f(z): At z = 0? As z? What do F(z) and f(z) look like?
28 Exponential Exp x = exp x f(z) At z = 0 f(z) = λ F(z) = 0 As z f(z) = 0 F(z) = 1 F(z)
29 Moments of probability distributions E[x n ]= x n f x dx First moment = Expected value = mean Example: exponential E[x]= x f x dx= 0 = x exp x E[x]=1/ x exp x exp x
30 E[c] = c Properties of means E[x + c] = E[x] + c E[cx] = c E[x] E[x+y] = E[x] + E[y] (even if X is not independent of Y) E[E[x y]]= E[x] E[g(x)]!= g(e[x]) E(xy) = E[x]E[Y] iterated expectation Jensen's Inequality only if independent
31 Central Moments E[ x E[ x] n ]= x E[ x] n f x dx Second Central Moment = Variance = σ 2
32 Var ax =a 2 Var X Var X b =Var X Properties of variance Var X Y =Var X Var Y 2Cov X,Y Var ax by =a 2 Var X b 2 Var Y 2abCov X,Y Var X = Var X i 2 i j Cov X i, X j Var X =Var E[ X Y ] E[Var X Y ]
33 Distributions and Probability all the same properties apply to random variables Pr A, B joint distribution Pr A B = Pr A, B Pr B conditional distribution Pr A = Pr A B i Pr B i marginal distribution Pr A B = Pr B A Pr A Pr B Baye's Rule
34 Looking forward... Use probability distributions to: Quantify the match between models and data Represent uncertainty about model parameters Partition sources of process variability
Lecture 2: Probability
Lecture 2: Probability Statistical Paradigms Statistical Estimator Method of Estimation Output Data Complexity Prior Info Classical Cost Function Analytical Solution Point Estimate Simple No Maximum Likelihood
Lecture 25: Review. Statistics 104. April 23, Colin Rundel
Lecture 25: Review Statistics 104 Colin Rundel April 23, 2012 Joint CDF F (x, y) = P [X x, Y y] = P [(X, Y ) lies south-west of the point (x, y)] Y (x,y) X Statistics 104 (Colin Rundel) Lecture 25 April
Lecture 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
Chapter 4 Multiple Random Variables
Review for the previous lecture Definition: n-dimensional random vector, joint pmf (pdf), marginal pmf (pdf) Theorem: How to calculate marginal pmf (pdf) given joint pmf (pdf) Example: How to calculate
Multivariate Random Variable
Multivariate Random Variable Author: Author: Andrés Hincapié and Linyi Cao This Version: August 7, 2016 Multivariate Random Variable 3 Now we consider models with more than one r.v. These are called multivariate
Probability. 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
HW Solution 12 Due: Dec 2, 9:19 AM
ECS 315: Probability and Random Processes 2015/1 HW Solution 12 Due: Dec 2, 9:19 AM Lecturer: Prapun Suksompong, Ph.D. Problem 1. Let X E(3). (a) For each of the following function g(x). Indicate whether
Multiple Random Variables
Multiple Random Variables This Version: July 30, 2015 Multiple Random Variables 2 Now we consider models with more than one r.v. These are called multivariate models For instance: height and weight An
Probability Models. 4. What is the definition of the expectation of a discrete random variable?
1 Probability Models The list of questions below is provided in order to help you to prepare for the test and exam. It reflects only the theoretical part of the course. You should expect the questions
STT 441 Final Exam Fall 2013
STT 441 Final Exam Fall 2013 (12:45-2:45pm, Thursday, Dec. 12, 2013) NAME: ID: 1. No textbooks or class notes are allowed in this exam. 2. Be sure to show all of your work to receive credit. Credits are
Chapter 4 Multiple Random Variables
Review for the previous lecture Theorems and Examples: How to obtain the pmf (pdf) of U = g ( X Y 1 ) and V = g ( X Y) Chapter 4 Multiple Random Variables Chapter 43 Bivariate Transformations Continuous
ECE Lecture #9 Part 2 Overview
ECE 450 - Lecture #9 Part Overview Bivariate Moments Mean or Expected Value of Z = g(x, Y) Correlation and Covariance of RV s Functions of RV s: Z = g(x, Y); finding f Z (z) Method : First find F(z), by
2 (Statistics) Random variables
2 (Statistics) Random variables References: DeGroot and Schervish, chapters 3, 4 and 5; Stirzaker, chapters 4, 5 and 6 We will now study the main tools use for modeling experiments with unknown outcomes
BASICS OF PROBABILITY
October 10, 2018 BASICS OF PROBABILITY Randomness, sample space and probability Probability is concerned with random experiments. That is, an experiment, the outcome of which cannot be predicted with certainty,
Joint 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
Continuous Random Variables
1 / 24 Continuous Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 27, 2013 2 / 24 Continuous Random Variables
Stat 5101 Notes: Algorithms
Stat 5101 Notes: Algorithms Charles J. Geyer January 22, 2016 Contents 1 Calculating an Expectation or a Probability 3 1.1 From a PMF........................... 3 1.2 From a PDF...........................
Lecture 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
MAS113 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
This exam contains 6 questions. The questions are of equal weight. Print your name at the top of this page in the upper right hand corner.
GROUND RULES: This exam contains 6 questions. The questions are of equal weight. Print your name at the top of this page in the upper right hand corner. This exam is closed book and closed notes. Show
Lecture 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................................
ACE 562 Fall Lecture 2: Probability, Random Variables and Distributions. by Professor Scott H. Irwin
ACE 562 Fall 2005 Lecture 2: Probability, Random Variables and Distributions Required Readings: by Professor Scott H. Irwin Griffiths, Hill and Judge. Some Basic Ideas: Statistical Concepts for Economists,
Chapter 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
Functions of two random variables. Conditional pairs
Handout 10 Functions of two random variables. Conditional pairs "Science is a wonderful thing if one does not have to earn a living at it. One should earn one's living by work of which one is sure one
6.041/6.431 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
P (x). all other X j =x j. If X is a continuous random vector (see p.172), then the marginal distributions of X i are: f(x)dx 1 dx n
JOINT DENSITIES - RANDOM VECTORS - REVIEW Joint densities describe probability distributions of a random vector X: an n-dimensional vector of random variables, ie, X = (X 1,, X n ), where all X is are
Lecture 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:
Stat 5101 Notes: Algorithms (thru 2nd midterm)
Stat 5101 Notes: Algorithms (thru 2nd midterm) Charles J. Geyer October 18, 2012 Contents 1 Calculating an Expectation or a Probability 2 1.1 From a PMF........................... 2 1.2 From a PDF...........................
Algorithms 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
Math 151. Rumbos Fall Solutions to Review Problems for Exam 2. Pr(X = 1) = ) = Pr(X = 2) = Pr(X = 3) = p X. (k) =
Math 5. Rumbos Fall 07 Solutions to Review Problems for Exam. A bowl contains 5 chips of the same size and shape. Two chips are red and the other three are blue. Draw three chips from the bowl at random,
Random 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
Problem Y is an exponential random variable with parameter λ = 0.2. Given the event A = {Y < 2},
ECE32 Spring 25 HW Solutions April 6, 25 Solutions to HW 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 italics where
ECE Lecture #10 Overview
ECE 450 - Lecture #0 Overview Introduction to Random Vectors CDF, PDF Mean Vector, Covariance Matrix Jointly Gaussian RV s: vector form of pdf Introduction to Random (or Stochastic) Processes Definitions
4. Distributions of Functions of Random Variables
4. Distributions of Functions of Random Variables Setup: Consider as given the joint distribution of X 1,..., X n (i.e. consider as given f X1,...,X n and F X1,...,X n ) Consider k functions g 1 : R n
STAT/MATH 395 PROBABILITY II
STAT/MATH 395 PROBABILITY II Bivariate Distributions Néhémy Lim University of Washington Winter 2017 Outline Distributions of Two Random Variables Distributions of Two Discrete Random Variables Distributions
Introduction 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
Perhaps the simplest way of modeling two (discrete) random variables is by means of a joint PMF, defined as follows.
Chapter 5 Two Random Variables In a practical engineering problem, there is almost always causal relationship between different events. Some relationships are determined by physical laws, e.g., voltage
Summary of Random Variable Concepts March 17, 2000
Summar of Random Variable Concepts March 17, 2000 This is a list of important concepts we have covered, rather than a review that devives or eplains them. Tpes of random variables discrete A random variable
STAT 515 MIDTERM 2 EXAM November 14, 2018
STAT 55 MIDTERM 2 EXAM November 4, 28 NAME: Section Number: Instructor: In problems that require reasoning, algebraic calculation, or the use of your graphing calculator, it is not sufficient just to write
Review 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
ENGG2430A-Homework 2
ENGG3A-Homework Due on Feb 9th,. Independence vs correlation a For each of the following cases, compute the marginal pmfs from the joint pmfs. Explain whether the random variables X and Y are independent,
EEL 5544 Noise in Linear Systems Lecture 30. X (s) = E [ e sx] f X (x)e sx dx. Moments can be found from the Laplace transform as
![EEL 5544 Noise in Linear Systems Lecture 30. X (s) = E [ e sx] f X (x)e sx dx. Moments can be found from the Laplace transform as](/thumbs/85/92409620.jpg "EEL 5544 Noise in Linear Systems Lecture 30. X (s) = E [ e sx] f X (x)e sx dx. Moments can be found from the Laplace transform as")
L30-1 EEL 5544 Noise in Linear Systems Lecture 30 OTHER TRANSFORMS For a continuous, nonnegative RV X, the Laplace transform of X is X (s) = E [ e sx] = 0 f X (x)e sx dx. For a nonnegative RV, the Laplace
Bandits, Experts, and Games
Bandits, Experts, and Games CMSC 858G Fall 2016 University of Maryland Intro to Probability* Alex Slivkins Microsoft Research NYC * Many of the slides adopted from Ron Jin and Mohammad Hajiaghayi Outline
A Probability Review
A Probability Review Outline: A probability review Shorthand notation: RV stands for random variable EE 527, Detection and Estimation Theory, # 0b 1 A Probability Review Reading: Go over handouts 2 5 in
Probability. 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.........................................
f X, Y (x, y)dx (x), where f(x,y) is the joint pdf of X and Y. (x) dx
INDEPENDENCE, COVARIANCE AND CORRELATION Independence: Intuitive idea of "Y is independent of X": The distribution of Y doesn't depend on the value of X. In terms of the conditional pdf's: "f(y x doesn't
Introduction to Machine Learning
What does this mean? Outline Contents Introduction to Machine Learning Introduction to Probabilistic Methods Varun Chandola December 26, 2017 1 Introduction to Probability 1 2 Random Variables 3 3 Bayes
Recitation 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
ECE302 Exam 2 Version A April 21, You must show ALL of your work for full credit. Please leave fractions as fractions, but simplify them, etc.
ECE32 Exam 2 Version A April 21, 214 1 Name: Solution Score: /1 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully. Please check your answers
5 Operations on Multiple Random Variables
EE360 Random Signal analysis Chapter 5: Operations on Multiple Random Variables 5 Operations on Multiple Random Variables Expected value of a function of r.v. s Two r.v. s: ḡ = E[g(X, Y )] = g(x, y)f X,Y
MAS113 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
Random 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
Bivariate Distributions
STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 17 Néhémy Lim Bivariate Distributions 1 Distributions of Two Random Variables Definition 1.1. Let X and Y be two rrvs on probability space (Ω, A, P).
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES VARIABLE Studying the behavior of random variables, and more importantly functions of random variables is essential for both the
Math 151. Rumbos Spring Solutions to Review Problems for Exam 2
Math 5. Rumbos Spring 22 Solutions to Review Problems for Exam 2. Let X and Y be independent Normal(, ) random variables. Put Z = Y X. Compute the distribution functions (z) and (z). Solution: Since X,
ECSE B Solutions to Assignment 8 Fall 2008
ECSE 34-35B Solutions to Assignment 8 Fall 28 Problem 8.1 A manufacturing system is governed by a Poisson counting process {N t ; t < } with rate parameter λ >. If a counting event occurs at an instant
Chapter 4 continued. Chapter 4 sections
Chapter 4 sections Chapter 4 continued 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:
Preliminary Statistics. Lecture 3: Probability Models and Distributions
Preliminary Statistics Lecture 3: Probability Models and Distributions Rory Macqueen (rm43@soas.ac.uk), September 2015 Outline Revision of Lecture 2 Probability Density Functions Cumulative Distribution
Appendix 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
15-388/688 - Practical Data Science: Basic probability. J. Zico Kolter Carnegie Mellon University Spring 2018
15-388/688 - Practical Data Science: Basic probability J. Zico Kolter Carnegie Mellon University Spring 2018 1 Announcements Logistics of next few lectures Final project released, proposals/groups due
UC Berkeley Department of Electrical Engineering and Computer Sciences. EECS 126: Probability and Random Processes
UC Berkeley Department of Electrical Engineering and Computer Sciences EECS 6: Probability and Random Processes Problem Set 3 Spring 9 Self-Graded Scores Due: February 8, 9 Submit your self-graded scores
1 Probability Review. 1.1 Sample Spaces
1 Probability Review Probability is a critical tool for modern data analysis. It arises in dealing with uncertainty, in randomized algorithms, and in Bayesian analysis. To understand any of these concepts
Math 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential.
Math 8A Lecture 6 Friday May 7 th Epectation Recall the three main probability density functions so far () Uniform () Eponential (3) Power Law e, ( ), Math 8A Lecture 6 Friday May 7 th Epectation Eample
Statistics for Economists Lectures 6 & 7. Asrat Temesgen Stockholm University
Statistics for Economists Lectures 6 & 7 Asrat Temesgen Stockholm University 1 Chapter 4- Bivariate Distributions 41 Distributions of two random variables Definition 41-1: Let X and Y be two random variables
CS145: Probability & Computing Lecture 11: Derived Distributions, Functions of Random Variables
CS145: Probability & Computing Lecture 11: Derived Distributions, Functions of Random Variables Instructor: Erik Sudderth Brown University Computer Science March 5, 2015 Homework Submissions Electronic
Lecture 22: Variance and Covariance
EE5110 : Probability Foundations for Electrical Engineers July-November 2015 Lecture 22: Variance and Covariance Lecturer: Dr. Krishna Jagannathan Scribes: R.Ravi Kiran In this lecture we will introduce
UNIT Define joint distribution and joint probability density function for the two random variables X and Y.
UNIT 4 1. Define joint distribution and joint probability density function for the two random variables X and Y. Let and represent the probability distribution functions of two random variables X and Y
FINAL EXAM: Monday 8-10am
ECE 30: Probabilistic Methods in Electrical and Computer Engineering Fall 016 Instructor: Prof. A. R. Reibman FINAL EXAM: Monday 8-10am Fall 016, TTh 3-4:15pm (December 1, 016) This is a closed book exam.
Disjointness and Additivity
Midterm 2: Format Midterm 2 Review CS70 Summer 2016 - Lecture 6D David Dinh 28 July 2016 UC Berkeley 8 questions, 190 points, 110 minutes (same as MT1). Two pages (one double-sided sheet) of handwritten
Chapter 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
Expectation 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
conditional cdf, conditional pdf, total probability theorem?
6 Multiple Random Variables 6.0 INTRODUCTION scalar vs. random variable cdf, pdf transformation of a random variable conditional cdf, conditional pdf, total probability theorem expectation of a random
Machine Learning. Instructor: Pranjal Awasthi
Machine Learning Instructor: Pranjal Awasthi Course Info Requested an SPN and emailed me Wait for Carol Difrancesco to give them out. Not registered and need SPN Email me after class No promises It s a
Probability. Lecture Notes. Adolfo J. Rumbos
Probability Lecture Notes Adolfo J. Rumbos October 20, 204 2 Contents Introduction 5. An example from statistical inference................ 5 2 Probability Spaces 9 2. Sample Spaces and σ fields.....................
CHAPTER 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
LIST OF FORMULAS FOR STK1100 AND STK1110
LIST OF FORMULAS FOR STK1100 AND STK1110 (Version of 11. November 2015) 1. Probability Let A, B, A 1, A 2,..., B 1, B 2,... be events, that is, subsets of a sample space Ω. a) Axioms: A probability function
Review 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
Properties of Summation Operator
Econ 325 Section 003/004 Notes on Variance, Covariance, and Summation Operator By Hiro Kasahara Properties of Summation Operator For a sequence of the values {x 1, x 2,..., x n, we write the sum of x 1,
Bivariate 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
Part 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.
Midterm 2 Review. CS70 Summer Lecture 6D. David Dinh 28 July UC Berkeley
Midterm 2 Review CS70 Summer 2016 - Lecture 6D David Dinh 28 July 2016 UC Berkeley Midterm 2: Format 8 questions, 190 points, 110 minutes (same as MT1). Two pages (one double-sided sheet) of handwritten
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
Math 416 Lecture 2 DEFINITION. Here are the multivariate versions: X, Y, Z iff P(X = x, Y = y, Z =z) = p(x, y, z) of X, Y, Z iff for all sets A, B, C,
Math 416 Lecture 2 DEFINITION. Here are the multivariate versions: PMF case: p(x, y, z) is the joint Probability Mass Function of X, Y, Z iff P(X = x, Y = y, Z =z) = p(x, y, z) PDF case: f(x, y, z) is
Tutorial 1 : Probabilities
Lund University ETSN01 Advanced Telecommunication Tutorial 1 : Probabilities Author: Antonio Franco Emma Fitzgerald Tutor: Farnaz Moradi January 11, 2016 Contents I Before you start 3 II Exercises 3 1
Review: 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
REVIEW OF MAIN CONCEPTS AND FORMULAS A B = Ā B. Pr(A B C) = Pr(A) Pr(A B C) =Pr(A) Pr(B A) Pr(C A B)
REVIEW OF MAIN CONCEPTS AND FORMULAS Boolean algebra of events (subsets of a sample space) DeMorgan s formula: A B = Ā B A B = Ā B The notion of conditional probability, and of mutual independence of two
Probability Theory and Statistics. Peter Jochumzen
Probability Theory and Statistics Peter Jochumzen April 18, 2016 Contents 1 Probability Theory And Statistics 3 1.1 Experiment, Outcome and Event................................ 3 1.2 Probability............................................
More 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
Math 151. Rumbos Fall Solutions to Review Problems for Final Exam
Math 5. Rumbos Fall 23 Solutions to Review Problems for Final Exam. Three cards are in a bag. One card is red on both sides. Another card is white on both sides. The third card in red on one side and white
Math 426: Probability MWF 1pm, Gasson 310 Exam 3 SOLUTIONS
Name: ANSWE KEY Math 46: Probability MWF pm, Gasson Exam SOLUTIONS Problem Points Score 4 5 6 BONUS Total 6 Please write neatly. You may leave answers below unsimplified. Have fun and write your name above!
Closed book and notes. 60 minutes. Cover page and four pages of exam. No calculators.
IE 230 Seat # Closed book and notes. 60 minutes. Cover page and four pages of exam. No calculators. Score Exam #3a, Spring 2002 Schmeiser Closed book and notes. 60 minutes. 1. True or false. (for each,
Review (Probability & Linear Algebra)
Review (Probability & Linear Algebra) CE-725 : Statistical Pattern Recognition Sharif University of Technology Spring 2013 M. Soleymani Outline Axioms of probability theory Conditional probability, Joint
DO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO
QUESTION BOOKLET EE 26 Spring 2006 Final Exam Wednesday, May 7, 8am am DO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO You have 80 minutes to complete the final. The final consists of five
Deccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY. SECOND YEAR B.Sc. SEMESTER - III
Deccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY SECOND YEAR B.Sc. SEMESTER - III SYLLABUS FOR S. Y. B. Sc. STATISTICS Academic Year 07-8 S.Y. B.Sc. (Statistics)
ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables
Department o Electrical Engineering University o Arkansas ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Two discrete random variables
Problem Solving. Correlation and Covariance. Yi Lu. Problem Solving. Yi Lu ECE 313 2/51
Yi Lu Correlation and Covariance Yi Lu ECE 313 2/51 Definition Let X and Y be random variables with finite second moments. the correlation: E[XY ] Yi Lu ECE 313 3/51 Definition Let X and Y be random variables
Probability and Distributions
Probability and Distributions What is a statistical model? A statistical model is a set of assumptions by which the hypothetical population distribution of data is inferred. It is typically postulated
CMPSCI 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/
Random Variables and Expectations
Inside ECOOMICS Random Variables Introduction to Econometrics Random Variables and Expectations A random variable has an outcome that is determined by an experiment and takes on a numerical value. A procedure
Introduction to Bayesian Statistics
School of Computing & Communication, UTS January, 207 Random variables Pre-university: A number is just a fixed value. When we talk about probabilities: When X is a continuous random variable, it has a