Appendix A. Math Reviews 03Jan2007. A.1 From Simple to Complex. Objectives. 1. Review tools that are needed for studying models for CLDVs.
|
|
- Anis Stokes
- 6 years ago
- Views:
Transcription
1 Appendix A Math Reviews 03Jan007 Objectives. Review tools that are needed for studying models for CLDVs.. Get you used to the notation that will be used. Readings. Read this appendix before class.. Pay special attention to the results marked with a *. 3. Review any other algebra text as needed. A. From Simple to Complex I With a simple equation: x = y I Or a complex equation: y = b 0 + b x + b x + + u I The same rules apply. Don t confuse messy and complex with hard and incomprehensible! i
2 ii APPENDIX A. MATH REVIEWS 03JAN007 A. Basic Rules Distributive law a (b + c) = (a b) + (a c) (A.) 4 ( + 3) = (4 ) + (4 3) ( ) ( 0 + x + x ) = ( ) (A.) = = ( 0 + x + x ) ( 0 + x + x ) = [ 0 + x + x ] [ 0 + x + x ] Multiplying by a b = a b = k k a b = ka kb (A.3) A.3 Solving Equations 3 = 3 = = = 8 = 3 Let p be the probability of an event, and = p p the odds (Note: = ex ). You should be able to work this derivation from to p and from p to without looking. = p p 9 = :9 : ( p) = p 9 ( :9) = :9 p = p 9 9 (:9) = :9 = p + p 9 = 9 (:9) + :9 Therefore, p = = p ( + ) 9 = :9 ( + 9) + = p = :9 + = ex + e x :
3 A.4. EXPONENTS AND RADICALS iii A.4 Exponents and Radicals Zero exponent a 0 = (A.4) 3 0 = :788 0 = = e 0 Integer exponent a k = a (k) a; where (k) means repeat k times (A.5) 3 = = 8 e 3 = :788 :788 :788 = 0:086 Negative integer exponent a k = 3 = a (k) a = a k = 8 (A.6) Base e e = : : : : is a useful base. Notation is: e x or exp(x). e 0 = e = : 78 e = e e = 7:389 e 3 = e e e = 0:086 * Product of powers: multiplying as the sum of powers * Quotient of powers a M a N = [a (M + N) a] = a M+N (A.7) 3 4 = ( ) ( ) = 3+4 = 7 e 3 e 4 = (e e e) (e e e e) = e 3+4 = e 7 (A.8) a M [a (M) a] = an [a (N) a] = am N (A.9) e 5 = e e e e e e 3 e e e = e 5 3 = e
4 iv APPENDIX A. MATH REVIEWS 03JAN007 Power of powers (a M ) N = a MN (A.0) e 5 A.5 ** Natural Logarithms = (e e) (e e) (e e) (e e) (e e) = e 0 = e 5 Natural logarithms and exponentials are used extensively in statistics. A key reason is that they turn multiplication into addition. Here s why:. Every positive real number m can be written as m = e p. Example: Let m = 3. Find p such that e p = 3: (a) e = 7:389 : : : and e 3 = 0:086 : : : ) < p < 3. (b) e :5 = :8 : : : and e :6 = 3:464 : : : ) :5 < p < :6. (c) And so on until e :565::: = 3 3. De nition of the Log (a) If m = e p, then p = ln m: The log of m is p: (b) Or, ln m = p which is equivalent to e p = m (c) Which looks like:
5 A.5. ** NATURAL LOGARITHMS v 4. * Log of Products (a) Let m = e p () ln m = p n = e q () ln n = q (b) Then, multiply m times n: m n = e p e q = e (p+q) (c) Taking the log of both sides: ln (m n) = ln e (p+q) = p + q = ln m + ln n (d) For example: 3 = 6 ln ( 3) = ln + ln 3 = = 0: 6935::: + :0986::: = :798::: = ln 6 5. * Log of Quotients m ln n ln 3 The logit: = ln m ln n = : = ln 3 ln = : 0986 : 6935 p ln p = 6. Inverse operations (a) ln(k) is that power of e that equals k: k = e ln k (b) ln(e k ) is that power of e that equals e k, namely k: ln e k = k
6 vi APPENDIX A. MATH REVIEWS 03JAN007 (c) and e ln k = k 7. Log of Power 8. Example from Regression ln m n = n ln m ln 3 = ln 9 = : 97 = ln 3 = (: 0986) (a) Assume that (b) Taking logs: y = x x " ln y = ln x x " = ln + ln x + ln x + ln " = ln + ln x + ln x + " = + x + x + " A.6 Vector Algebra. Consider the regression equation for observation i: y i = 0 + x i + x i + " i Vector multiplication allows us to write this more simply.. For example, let 0 = 0 and x = x x, then 0 x = x 0 A = 0 + x + x 3. More generally, consider K and x K, then by de nition: x = 0 + KX i x i = 0 + x + x + 3 x 3 + i=
7 A.7. PROBABILITY DISTRIBUTIONS vii A.7 Probability Distributions I Let X be a random variable with discrete outcomes x. The frequency of those outcomes is the probability distribution: f(x) = Pr(X = x) Bernoulli Distribution For example, let y indicate the outcome of a fair coin. Then, y = 0 or, and Pr (y = 0) = :4 and Pr (y = ) = :6 I For all probability distributions:. All probabilities are between zero and one: 0 f(x). Probabilities sum to one: P x f(x i) =
8 viii APPENDIX A. MATH REVIEWS 03JAN007 I For a continuous random variable, f(x) is called a probability density function or pdf.. f(x) = 0. Why? Pick any two numbers. Can you nd a number in between them?. Pr(a x b) = 3. Z f(t)dt = Normal Distribution Z b a f(x)dx 0. The pdf mean and standard deviation, x N(; ) ; is:! f(x j ; ) = p (x ) exp (A.). This de nes the classic bell curve: 3. If = 0 and = : (x) = f(x j 0; ) = p x exp (A.)
9 A.7. PROBABILITY DISTRIBUTIONS ix A.7. Cumulative Distribution Function (cdf) I The cdf is the probability of a value up to or equal to a speci c value. For discrete random variables: F (x) = P Xx f(x) = Pr (X x) For a continuous variable: F (x) = I For the cdf:. 0 F (x).. If x > y, then F (x) F (y). 3. F ( ) = 0 and F () =. Z x f(t)dt = Pr (X x) A.7. * Computing the Area Within a Distribution I Consider the distribution f (x), where F (x) = Pr (X x): Pr (a X b) = Pr (X b) Pr (X a) = F (b) F (a)
10 x APPENDIX A. MATH REVIEWS 03JAN007 A.7.3 * Expectation I The mean of N sample values of X is: For example: = x = P N i= X i N = I The expectation is de ned in terms of the population: For discrete variables: E(X) = X x f(x)x = X x Pr(X = x)x For continuous variables: Z E(X) = f(x)x dx x * Example of Expectation of Binary Variable If X has values 0 and with probabilities 4 and 3 4, then E(X) = = = Pr (x = ) = [Value Pr (Value )] + [Value Pr (Value )]
11 A.7. PROBABILITY DISTRIBUTIONS xi * Expectation of Sums I If X and Y are random variables, and a, b and c are constants, then E(a + bx + cy ) = a + be(x) + ce(y ) (A.3) I Example: Let Then KX y i = + k x ki + " i k=! KX E(y i ) = E + k x ki + " i = E () + E k=! KX k x ki + E (" i ) k= KX = + k E(x ik ) k= Conditional Expectations I Conditioning means holding some things constant while something else changes. I Example: Let $ be income. E($) tells us the mean $; but is not useful for telling us how other variables a ect $. Let S be the sex of the respondent. We might compute: E($ j S = female) = Expected $ for females This allows us to see how the expectation varies by the level of other variables. I Example: If y = x + ", then E(y j x) = E(x + ") = E(x) + E(") = x
12 xii APPENDIX A. MATH REVIEWS 03JAN007 A.7.4 The Variance I The variance is de ned as P N s i= = (x i x) N I Variance for a Population: Let f(x) = Pr(X = x). If x is discrete: Var(X) = X x [x E(x)] f(x) (A.4) If x is continuous: Z Var(X) = [x E(x)] f(x)dx (A.5) x Example of Variance of Binary Variable If X has values 0 and with probabilities 4 and 3 4, then E(X) = 3 4, and! 3 Var(X) = = = E (X) [ E (X)] * Variance of a Linear Transformation! I Let X be a random variable, and a and b be constants. Then, = = 64 = 3 6 Var(a + bx) = b Var(X) (A.6) * Variance of a Sum I Let X and Y be two random variables with constants a and b: Var(aX + by ) = a Var(X) + b Var(Y ) + abcov(x; Y ) (A.7) I Let Y = P K i= X i. If the X s are uncorrelated, then KX Var(Y ) = Var( X i ) = i= KX Var(X i ) i= (A.8)
13 A.8. **RESCALING VARIABLES xiii A.8 **Rescaling Variables Often we want to use addition and multiplication to change a variable with mean and variance into a variable with mean 0 and variance. This is called rescaling.. Consider X where E (x) = and Var (x) =. By subtracting the mean, the expectation becomes zero: E (x ) = E (x) E () = = 0 3. But the variance is unchanged: Var (x ) = Var (x) = 4. Dividing by : x E = E x = E (x) = = 5. Subtracting and dividing by does not change the mean: x E = E (x ) = 0 = 0 (A.9) 6. But, the variance becomes one: x Var = Var (x ) = Var (x) = (A.0)
14 xiv APPENDIX A. MATH REVIEWS 03JAN007 Stata: Standardizing Variables. use science, clear. sum pub9 Variable Obs Mean Std. Dev. Min Max pub gen p9_mn = pub9 - r(mean). sum p9_mn Variable Obs Mean Std. Dev. Min Max p9_mn e gen p9_sd = p9_mn/ gen p9_sd = (pub )/ egen p9_sdez = std(pub9). sum p9_sd p9_sd p9_sdez Variable Obs Mean Std. Dev. Min Max p9_sd e p9_sd 308.8e p9_sdez e
15 A.9. DISTRIBUTIONS xv A.9 Distributions A.9. Bernoulli I X has a Bernoulli distribution if it has two possible outcomes: Pr(X = ) = p and Pr(X = 0) = p I Then: I That is: f(x j p) = p x ( p) x = Pr(X = x j p) I It can be shown that f(0 j p) = p 0 ( p) = p and f( j p) = p ( p) 0 = p I Note how the variance is related to the mean: E(X) = p and Var (X) = p ( p) (A.) E(X) = p Var (X) = p ( p)
16 xvi APPENDIX A. MATH REVIEWS 03JAN007 A.9. Normal I The pdf for a normal distribution with mean and standard deviation is! f(x j ; ) = p (x ) exp (A.) I If x is distributed normally with mean and standard deviation : x N ; I The cdf is de ned as F (x j ; ) = Z x f(t j ; )dt
17 A.9. DISTRIBUTIONS xvii Standardized Normal I If x N(0; ), we de ne: x pdf: (x) = p exp cdf: (x) = R x (t)dt I You can move from an unstandardized to a standardized normal distribution. Let x N(0; ) Then, f(x j 0; ) = = x p exp = x p exp x = (A.3) Area Under the Curve I If x N(0; ), then Pr (a x b) = (b) (a) (A.4) Linear Transformation of a Normal I If I Then Sums of Normals x N ; a + bx N a + b; b (A.5) I Let Cor(x ; x ) =, where x N ; and x N ; I Then x + x N([ + ] ; + + ) I When = 0: x + x N([ + ] ; + ) (A.6)
18 xviii A.9.3 I Let i Chi-square (i = to df) be independent, standard normal variates. APPENDIX A. MATH REVIEWS 03JAN007 I De ne: Xdf dfx i= i df I The chi-square distribution is de ned as the sum of independent squared normal variables. I The mean and variance: E Xdf = df and Var X df = df Adding Chi-squares Let x df x and y df y If x and y are independent: x + y df x +df y Shape With df, the distribution is highly skewed. As df!, the chi-square becomes distributed normally. Question from Intro to Statistics X = X all cells (obs exp) exp Consider the chi-square test in contingency tables: X df with df = (#rows )(#columns ) Why would this be distributed as chi-square? Why those degrees of freedom? A.9.4 F-distribution I Let X and X be independent chi-square variables with degrees of freedom r and r. I The F-distribution is de ned as: F r ;r X =r X =r
19 A.9. DISTRIBUTIONS xix A.9.5 t-distribution I Consider z and x df, where z and x are independent. I Then the t-distribution with df degrees of freedom is de ned as: t df z p x=df A.9.6 Relationships among normal, t, chi-square and F. z = t. z = X = F ; = t 3. t df = F ;df 4. X df df = F r;
20 xx APPENDIX A. MATH REVIEWS 03JAN007 A.0 Calculus I The two central ideas in calculus are the derivative and the integral. I Derivative: The derivative is the slope of a curve y = f (x): dy dx = f 0 (x) (A.7) I The second derivative indicates how quickly the slope of the curve is changing: d y dx = d dy dx = f 00 (x) (A.8) dx I If the curve is de ned as y = f(x; z), we write the partial derivative with respect to x z) Imagine half of a hard boiled egg setting on a table; slice it from the top to the table. The partial derivative is the slope on the resulting curve. I Integral: The integral is the area under a curve. I For example, if a curve is de ned as y = f(x), the area under the curve from point a to point b is computed with the integral: Z a b f (t) dt
21 A.. MATRIX ALGEBRA xxi A. Matrix Algebra A.. Matrix Basic De nitions is an array of numbers, arranged in rows and columns: a a A = a 3 a a a 3 A.. Transposing a Matrix I The transpose is indicated by the prime or superscript T. For example: A 0 or A T : If A =, then A 0 = I Transposing the Transpose I If A is a symmetric matrix, then A 0 = A A..3 I Addition: A 00 = A Addition and Subtraction 4 5 I Transposes of added matrices: + A + B = fa rc + b rc g 3 7 = (A.30) I Subtraction of matrices: (A + B) 0 = A 0 + B 0 (A.3) = = + = A B = fa rc b rc g = A..4 Scalar Multiplication A = fa rc g = f a rc g =
22 xxii A..5 Vector Matrix Multiplication is a matrix with one dimension equal to one. APPENDIX A. MATH REVIEWS 03JAN007 I A column vector is an R matrix: 3 c = 4 5 or c 0 = 3 3 I A row vector is a C matrix: r = 3 4 * Vector Multiplication Consider K and x K, then by de nition: x = 3X i x i = x + x + 3 x 3 i= I For example, let 0 = 0 and x = x x, then 0 0 x = x A = 0 + x + x Matrix Multiplication For A RK and B KC, the matrix product C RC = AB equals: ( X K fc rc g = i= a ri b ic ) I Note that element c rc is the vector multiplication of row r from A and column c from B. I Example: a b c d e f g h i I Example from Regression: 3 5 = a + d + 3g b + e + 3h c + f + 3i 4a + 5d + 6g 4b + 5e + 6h 4c + 5f + 6i 6 4 y. y N y = X + " 3 x x = 4... x N x N = x + x + ". 0 + x N + x N + " N ". " N 3 7 5
23 A.. MATRIX ALGEBRA xxiii A..6 Inverse I An identity matrix is a square matrix with s on the diagonal, and 0 s elsewhere. I If A is square, then A is the inverse of A if and only if AA = I (A.3) = 0 0. If A exists, it is unique.. If A does not exist, A is called singular. A..7 Rank I Rank is the size of the largest submatrix that can be inverted. I A matrix is of full rank if the rank is equal to the minimum of the number of rows and columns. I Problems occur in estimation when a matrix is encountered that is not of full rank. I When this occurs, messages such as the following are generated: Matrix is not of full rank. Singular matrix encountered. Matrix cannot be inverted. An inverse does not exist.
REFRESHER. William Stallings
BASIC MATH REFRESHER William Stallings Trigonometric Identities...2 Logarithms and Exponentials...4 Log Scales...5 Vectors, Matrices, and Determinants...7 Arithmetic...7 Determinants...8 Inverse of a Matrix...9
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 informationPreliminary Statistics Lecture 3: Probability Models and Distributions (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 3: Probability Models and Distributions (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics,
More information1. 4 2y 1 2 = x = x 1 2 x + 1 = x x + 1 = x = 6. w = 2. 5 x
.... VII x + x + = x x x 8 x x = x + a = a + x x = x + x x Solve the absolute value equations.. z = 8. x + 7 =. x =. x =. y = 7 + y VIII Solve the exponential equations.. 0 x = 000. 0 x+ = 00. x+ = 8.
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 informationQuestion Points Score Total: 76
Math 447 Test 2 March 17, Spring 216 No books, no notes, only SOA-approved calculators. true/false or fill-in-the-blank question. You must show work, unless the question is a Name: Question Points Score
More information2 (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
More informationGATE Engineering Mathematics SAMPLE STUDY MATERIAL. Postal Correspondence Course GATE. Engineering. Mathematics GATE ENGINEERING MATHEMATICS
SAMPLE STUDY MATERIAL Postal Correspondence Course GATE Engineering Mathematics GATE ENGINEERING MATHEMATICS ENGINEERING MATHEMATICS GATE Syllabus CIVIL ENGINEERING CE CHEMICAL ENGINEERING CH MECHANICAL
More informationTABLE OF CONTENTS INTRODUCTION, APPROXIMATION & ERRORS 1. Chapter Introduction to numerical methods 1 Multiple-choice test 7 Problem set 9
TABLE OF CONTENTS INTRODUCTION, APPROXIMATION & ERRORS 1 Chapter 01.01 Introduction to numerical methods 1 Multiple-choice test 7 Problem set 9 Chapter 01.02 Measuring errors 11 True error 11 Relative
More informationMATH Solutions to Probability Exercises
MATH 5 9 MATH 5 9 Problem. Suppose we flip a fair coin once and observe either T for tails or H for heads. Let X denote the random variable that equals when we observe tails and equals when we observe
More informationMathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS
Mathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS (i) Sets and their Representations: Finite and Infinite sets; Empty set; Equal sets; Subsets; Power set; Universal set; Venn Diagrams;
More informationProbability 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
More informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More informationContents of the Supplemental Information
Contents of the Supplemental Information Fig. S1. Description of sampling regions for the micromilled septa 47407A and 47407B. Table S2. Results of isotope-dillution ICP-MS determined Mg/Ca and Sr/Ca ratios
More informationContinuous Random Variables
MATH 38 Continuous Random Variables Dr. Neal, WKU Throughout, let Ω be a sample space with a defined probability measure P. Definition. A continuous random variable is a real-valued function X defined
More informationLecture Notes Part 2: Matrix Algebra
17.874 Lecture Notes Part 2: Matrix Algebra 2. Matrix Algebra 2.1. Introduction: Design Matrices and Data Matrices Matrices are arrays of numbers. We encounter them in statistics in at least three di erent
More informationECON Fundamentals of Probability
ECON 351 - Fundamentals of Probability Maggie Jones 1 / 32 Random Variables A random variable is one that takes on numerical values, i.e. numerical summary of a random outcome e.g., prices, total GDP,
More informationBasics of Calculus and Algebra
Monika Department of Economics ISCTE-IUL September 2012 Basics of linear algebra Real valued Functions Differential Calculus Integral Calculus Optimization Introduction I A matrix is a rectangular array
More information[POLS 8500] Review of Linear Algebra, Probability and Information Theory
[POLS 8500] Review of Linear Algebra, Probability and Information Theory Professor Jason Anastasopoulos ljanastas@uga.edu January 12, 2017 For today... Basic linear algebra. Basic probability. Programming
More informationLinear Algebra Review
Linear Algebra Review Yang Feng http://www.stat.columbia.edu/~yangfeng Yang Feng (Columbia University) Linear Algebra Review 1 / 45 Definition of Matrix Rectangular array of elements arranged in rows and
More informationxvi xxiii xxvi Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7
About the Author v Preface to the Instructor xvi WileyPLUS xxii Acknowledgments xxiii Preface to the Student xxvi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real
More informationPolitical Science 6000: Beginnings and Mini Math Boot Camp
Political Science 6000: Beginnings and Mini Math Boot Camp January 20, 2010 First things first Syllabus This is the most important course you will take. 1. You need to understand these concepts in order
More informationA FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Matrix Arithmetic Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)
More informationRandom 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
More informationWeek 2: Review of probability and statistics
Week 2: Review of probability and statistics Marcelo Coca Perraillon University of Colorado Anschutz Medical Campus Health Services Research Methods I HSMP 7607 2017 c 2017 PERRAILLON ALL RIGHTS RESERVED
More informationMR. YATES. Vocabulary. Quadratic Cubic Monomial Binomial Trinomial Term Leading Term Leading Coefficient
ALGEBRA II WITH TRIGONOMETRY COURSE OUTLINE SPRING 2009. MR. YATES Vocabulary Unit 1: Polynomials Scientific Notation Exponent Base Polynomial Degree (of a polynomial) Constant Linear Quadratic Cubic Monomial
More informationMath RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5
Math 201-203-RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 What is the Antiderivative? In a derivative problem, a function f(x) is given and you find the derivative f (x) using
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Revision: Probability and Linear Algebra Week 1, Lecture 2
MA 575 Linear Models: Cedric E Ginestet, Boston University Revision: Probability and Linear Algebra Week 1, Lecture 2 1 Revision: Probability Theory 11 Random Variables A real-valued random variable is
More informationContinuous random variables
Continuous random variables Can take on an uncountably infinite number of values Any value within an interval over which the variable is definied has some probability of occuring This is different from
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 informationSTATISTICS; An Introductory Analysis. 2nd hidition TARO YAMANE NEW YORK UNIVERSITY A HARPER INTERNATIONAL EDITION
2nd hidition TARO YAMANE NEW YORK UNIVERSITY STATISTICS; An Introductory Analysis A HARPER INTERNATIONAL EDITION jointly published by HARPER & ROW, NEW YORK, EVANSTON & LONDON AND JOHN WEATHERHILL, INC.,
More information1 Review of Probability and Distributions
Random variables. A numerically valued function X of an outcome ω from a sample space Ω X : Ω R : ω X(ω) is called a random variable (r.v.), and usually determined by an experiment. We conventionally denote
More information1 Proof techniques. CS 224W Linear Algebra, Probability, and Proof Techniques
1 Proof techniques Here we will learn to prove universal mathematical statements, like the square of any odd number is odd. It s easy enough to show that this is true in specific cases for example, 3 2
More informationProbability 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 information3. Probability and Statistics
FE661 - Statistical Methods for Financial Engineering 3. Probability and Statistics Jitkomut Songsiri definitions, probability measures conditional expectations correlation and covariance some important
More informationArkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan
2.4 Random Variables Arkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan By definition, a random variable X is a function with domain the sample space and range a subset of the
More informationI) Simplifying fractions: x x. 1) 1 1 y x. 1 1 x 1. 4 x. 13x. x y xy. x 2. Factoring: 10) 13) 12) III) Solving: x 9 Prime (using only) 11)
AP Calculus Summer Packet Answer Key Reminders:. This is not an assignment.. This will not be collected.. You WILL be assessed on these skills at various times throughout the course.. You are epected to
More informationREAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS
REAL LINEAR ALGEBRA: PROBLEMS WITH SOLUTIONS The problems listed below are intended as review problems to do before the final They are organied in groups according to sections in my notes, but it is not
More informationEC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)
1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For
More informationELECTROMAGNETIC WAVES
Physics 4D ELECTROMAGNETIC WAVE Hans P. Paar 26 January 2006 i Chapter 1 Vector Calculus 1.1 Introduction Vector calculus is a branch of mathematics that allows differentiation and integration of (scalar)
More informationMath 215 HW #11 Solutions
Math 215 HW #11 Solutions 1 Problem 556 Find the lengths and the inner product of 2 x and y [ 2 + ] Answer: First, x 2 x H x [2 + ] 2 (4 + 16) + 16 36, so x 6 Likewise, so y 6 Finally, x, y x H y [2 +
More informationREVIEW 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
More informationSkew-symmetric tensor decomposition
[Arrondo,, Macias Marques, Mourrain] University of Trento, Italy September 28, 2018 Warsaw Symmetric-rank C[x 0,..., x n ] d F = r λ i L d i i=1 S d C n+1 F = r i=1 λ i v d i this corresponds to find the
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 informationAlbertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.
Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2015 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the
More informationDETAILED CONTENTS PART I INTRODUCTION AND DESCRIPTIVE STATISTICS. 1. Introduction to Statistics
DETAILED CONTENTS About the Author Preface to the Instructor To the Student How to Use SPSS With This Book PART I INTRODUCTION AND DESCRIPTIVE STATISTICS 1. Introduction to Statistics 1.1 Descriptive and
More informationBASICS 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,
More informationNo books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.
Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points
More informationMath 0031, Final Exam Study Guide December 7, 2015
Math 0031, Final Exam Study Guide December 7, 2015 Chapter 1. Equations of a line: (a) Standard Form: A y + B x = C. (b) Point-slope Form: y y 0 = m (x x 0 ), where m is the slope and (x 0, y 0 ) is a
More informationDistributions of Functions of Random Variables. 5.1 Functions of One Random Variable
Distributions of Functions of Random Variables 5.1 Functions of One Random Variable 5.2 Transformations of Two Random Variables 5.3 Several Random Variables 5.4 The Moment-Generating Function Technique
More informationMath 105 Course Outline
Math 105 Course Outline Week 9 Overview This week we give a very brief introduction to random variables and probability theory. Most observable phenomena have at least some element of randomness associated
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Final Revision CLASS XII CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION.
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Final Revision CLASS XII 2016 17 CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.),
More informationMatrices. Chapter What is a Matrix? We review the basic matrix operations. An array of numbers a a 1n A = a m1...
Chapter Matrices We review the basic matrix operations What is a Matrix? An array of numbers a a n A = a m a mn with m rows and n columns is a m n matrix Element a ij in located in position (i, j The elements
More informationEconometrics Lecture 1 Introduction and Review on Statistics
Econometrics Lecture 1 Introduction and Review on Statistics Chau, Tak Wai Shanghai University of Finance and Economics Spring 2014 1 / 69 Introduction This course is about Econometrics. Metrics means
More informationMath 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,
More informationGuidelines for Solving Probability Problems
Guidelines for Solving Probability Problems CS 1538: Introduction to Simulation 1 Steps for Problem Solving Suggested steps for approaching a problem: 1. Identify the distribution What distribution does
More informationADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET
NAME ADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET Part I. Order of Operations (PEMDAS) Parenthesis and other grouping symbols. Exponential expressions. Multiplication & Division. Addition & Subtraction. Tutorial:
More informationA Introduction to Matrix Algebra and the Multivariate Normal Distribution
A Introduction to Matrix Algebra and the Multivariate Normal Distribution PRE 905: Multivariate Analysis Spring 2014 Lecture 6 PRE 905: Lecture 7 Matrix Algebra and the MVN Distribution Today s Class An
More informationUpdated: January 16, 2016 Calculus II 7.4. Math 230. Calculus II. Brian Veitch Fall 2015 Northern Illinois University
Math 30 Calculus II Brian Veitch Fall 015 Northern Illinois University Integration of Rational Functions by Partial Fractions From algebra, we learned how to find common denominators so we can do something
More information401 Review. 6. Power analysis for one/two-sample hypothesis tests and for correlation analysis.
401 Review Major topics of the course 1. Univariate analysis 2. Bivariate analysis 3. Simple linear regression 4. Linear algebra 5. Multiple regression analysis Major analysis methods 1. Graphical analysis
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 informationContinuous Distributions
Chapter 3 Continuous Distributions 3.1 Continuous-Type Data In Chapter 2, we discuss random variables whose space S contains a countable number of outcomes (i.e. of discrete type). In Chapter 3, we study
More informationSimplifying Radical Expressions
Simplifying Radical Expressions Product Property of Radicals For any real numbers a and b, and any integer n, n>1, 1. If n is even, then When a and b are both nonnegative. n ab n a n b 2. If n is odd,
More information6.1 Matrices. Definition: A Matrix A is a rectangular array of the form. A 11 A 12 A 1n A 21. A 2n. A m1 A m2 A mn A 22.
61 Matrices Definition: A Matrix A is a rectangular array of the form A 11 A 12 A 1n A 21 A 22 A 2n A m1 A m2 A mn The size of A is m n, where m is the number of rows and n is the number of columns The
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 informationProbability theory for Networks (Part 1) CS 249B: Science of Networks Week 02: Monday, 02/04/08 Daniel Bilar Wellesley College Spring 2008
Probability theory for Networks (Part 1) CS 249B: Science of Networks Week 02: Monday, 02/04/08 Daniel Bilar Wellesley College Spring 2008 1 Review We saw some basic metrics that helped us characterize
More informationMath 313 Chapter 1 Review
Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations
More informationMAT063 and MAT065 FINAL EXAM REVIEW FORM 1R x
Page NEW YORK CITY COLLEGE OF TECHNOLOGY of the City University of New York R DEPARTMENT OF MATHEMATICS Revised Spring 0 W. Colucci, D. DeSantis, and P. Deraney. Updated Fall 0 S. Singh MAT06 and MAT06
More informationReview Let A, B, and C be matrices of the same size, and let r and s be scalars. Then
1 Sec 21 Matrix Operations Review Let A, B, and C be matrices of the same size, and let r and s be scalars Then (i) A + B = B + A (iv) r(a + B) = ra + rb (ii) (A + B) + C = A + (B + C) (v) (r + s)a = ra
More informationExponential and. Logarithmic Functions. Exponential Functions. Logarithmic Functions
Chapter Five Exponential and Logarithmic Functions Exponential Functions Logarithmic Functions Properties of Logarithms Exponential Equations Exponential Situations Logarithmic Equations Exponential Functions
More informationMATH Mathematics for Agriculture II
MATH 10240 Mathematics for Agriculture II Academic year 2018 2019 UCD School of Mathematics and Statistics Contents Chapter 1. Linear Algebra 1 1. Introduction to Matrices 1 2. Matrix Multiplication 3
More informationp. 6-1 Continuous Random Variables p. 6-2
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability (>). Often, there is interest in random variables
More informationIntroduction to Linear Algebra. Tyrone L. Vincent
Introduction to Linear Algebra Tyrone L. Vincent Engineering Division, Colorado School of Mines, Golden, CO E-mail address: tvincent@mines.edu URL: http://egweb.mines.edu/~tvincent Contents Chapter. Revew
More informationReview of Basic Probability Theory
Review of Basic Probability Theory James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) 1 / 35 Review of Basic Probability Theory
More information0.1 Rational Canonical Forms
We have already seen that it is useful and simpler to study linear systems using matrices. But matrices are themselves cumbersome, as they are stuffed with many entries, and it turns out that it s best
More information3 Continuous Random Variables
Jinguo Lian Math437 Notes January 15, 016 3 Continuous Random Variables Remember that discrete random variables can take only a countable number of possible values. On the other hand, a continuous random
More informationStatistical Pattern Recognition
Statistical Pattern Recognition A Brief Mathematical Review Hamid R. Rabiee Jafar Muhammadi, Ali Jalali, Alireza Ghasemi Spring 2012 http://ce.sharif.edu/courses/90-91/2/ce725-1/ Agenda Probability theory
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 informationCLASS 12 ALGEBRA OF MATRICES
CLASS 12 ALGEBRA OF MATRICES Deepak Sir 9811291604 SHRI SAI MASTERS TUITION CENTER CLASS 12 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements
More informationSeventeen generic formulas that may generate prime-producing quadratic polynomials
Seventeen generic formulas that may generate prime-producing quadratic polynomials Marius Coman Bucuresti, Romania email: mariuscoman13@gmail.com Abstract. In one of my previous papers I listed forty-two
More informationLinear Equations and Matrix
1/60 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear
More informationALGEBRA I FORM I. Textbook: Algebra, Second Edition;Prentice Hall,2002
ALGEBRA I FORM I Textbook: Algebra, Second Edition;Prentice Hall,00 Prerequisites: Students are expected to have a knowledge of Pre Algebra and proficiency of basic math skills including: positive and
More informationLinear Algebra Review (Course Notes for Math 308H - Spring 2016)
Linear Algebra Review (Course Notes for Math 308H - Spring 2016) Dr. Michael S. Pilant February 12, 2016 1 Background: We begin with one of the most fundamental notions in R 2, distance. Letting (x 1,
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 information3 (Maths) Linear Algebra
3 (Maths) Linear Algebra References: Simon and Blume, chapters 6 to 11, 16 and 23; Pemberton and Rau, chapters 11 to 13 and 25; Sundaram, sections 1.3 and 1.5. The methods and concepts of linear algebra
More informationax 2 + bx + c = 0 where
Chapter P Prerequisites Section P.1 Real Numbers Real numbers The set of numbers formed by joining the set of rational numbers and the set of irrational numbers. Real number line A line used to graphically
More informationSouthington High School 720 Pleasant Street Southington, CT 06489
BLUE KNIGHTS Southington High School 720 Pleasant Street Southington, CT 06489 Phone: (860) 628-3229 Fax: (860) 628-3397 Home Page: www.southingtonschools.org Principal Brian Stranieri Assistant Principals
More information!"#$%&'(&)*$%&+",#$$-$%&+./#-+ (&)*$%&+%"-$+0!#1%&
!"#$%&'(&)*$%&",#$$-$%&./#- (&)*$%&%"-$0!#1%&23 44444444444444444444444444444444444444444444444444444444444444444444 &53.67689:5;978?58"@A9;8=B!=89C7DE,6=8FG=CD=CF(76F9C7D!)#!/($"%*$H!I"%"&1/%/.!"JK$&3
More informationDr. Junchao Xia Center of Biophysics and Computational Biology. Fall /13/2016 1/33
BIO5312 Biostatistics Lecture 03: Discrete and Continuous Probability Distributions Dr. Junchao Xia Center of Biophysics and Computational Biology Fall 2016 9/13/2016 1/33 Introduction In this lecture,
More informationA 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
More informationMath 121 Calculus 1 Fall 2009 Outcomes List for Final Exam
Math 121 Calculus 1 Fall 2009 Outcomes List for Final Exam This outcomes list summarizes what skills and knowledge you should have reviewed and/or acquired during this entire quarter in Math 121, and what
More informationMULTIVARIATE DISTRIBUTIONS
Chapter 9 MULTIVARIATE DISTRIBUTIONS John Wishart (1898-1956) British statistician. Wishart was an assistant to Pearson at University College and to Fisher at Rothamsted. In 1928 he derived the distribution
More informationJoint Probability Distributions and Random Samples (Devore Chapter Five)
Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01: Probability and Statistics for Engineers Spring 2013 Contents 1 Joint Probability Distributions 2 1.1 Two Discrete
More informationPractice Questions for Final
Math 39 Practice Questions for Final June. 8th 4 Name : 8. Continuous Probability Models You should know Continuous Random Variables Discrete Probability Distributions Expected Value of Discrete Random
More informationPROBABILITY DISTRIBUTIONS: DISCRETE AND CONTINUOUS
PROBABILITY DISTRIBUTIONS: DISCRETE AND CONTINUOUS Univariate Probability Distributions. Let S be a sample space with a probability measure P defined over it, and let x be a real scalar-valued set function
More informationCollege Algebra. Third Edition. Concepts Through Functions. Michael Sullivan. Michael Sullivan, III. Chicago State University. Joliet Junior College
College Algebra Concepts Through Functions Third Edition Michael Sullivan Chicago State University Michael Sullivan, III Joliet Junior College PEARSON Boston Columbus Indianapolis New York San Francisco
More informationDiscrete Random Variables
Discrete Random Variables We have a probability space (S, Pr). A random variable is a function X : S V (X ) for some set V (X ). In this discussion, we must have V (X ) is the real numbers X induces a
More informationIAM 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
More informationABELIAN FUNCTIONS Abel's theorem and the allied theory of theta functions
ABELIAN FUNCTIONS Abel's theorem and the allied theory of theta functions H. F Baker St John's College, Cambridge CAMBRIDGE UNIVERSITY PRESS CHAPTER I. THE SUBJECT OF INVESTIGATION. I Fundamental algebraic
More informationCS 246 Review of Proof Techniques and Probability 01/14/19
Note: This document has been adapted from a similar review session for CS224W (Autumn 2018). It was originally compiled by Jessica Su, with minor edits by Jayadev Bhaskaran. 1 Proof techniques Here we
More information