# We have discussed the notion of probabilistic dependence above and indicated that dependence is

Save this PDF as:

Size: px
Start display at page:

Download "We have discussed the notion of probabilistic dependence above and indicated that dependence is"

## Transcription

1 1 CHAPTER 7 Online Supplement Covariance and Correlation for Measuring Dependence We have discussed the notion of probabilistic dependence above and indicated that dependence is defined in terms of conditional distributions. In some cases, though, the use of conditional distributions can be difficult, and another approach to measuring dependence is worthwhile. Covariance is a quantity that is closely related to the idea of variance. Covariance and its close relative correlation can be used to measure certain kinds of dependence. The covariance between two uncertain quantities X and Y is calculated mathematically by: Cov(X, Y) = [x 1 E(X)][y 1 E(Y)]P(X = x 1 and Y = y 1 ) + + [x n E(X)][y n E(Y)]P(X = x n and Y = y n ) n m = [x i E(X)] y j E(Y) P X = x i and Y = y j i=1 j=1 = E X E(X) Y E(Y) Although this is a complicated formula, with a little interpretation we can get some insight into it. First, it really is similar to the formula for variance. There we calculated an average of squared deviations of X from its expected value E(X). Here, instead of squaring the deviations, we multiply X s deviation times Y s deviation. This is sometimes called a cross product because it is a product of deviations for two different quantities (X and Y). Although it may not be evident on the surface, the covariance can be either positive or negative. Suppose that large values of X tend to occur with large values of Y, and small with small. On the other hand, the probability that high values of X and low values of Y (and vice versa) occur together is low. Now consider what happens with the cross products in the formula. When both X and Y are high, they will both be above their corresponding expected values, making both deviations and their cross product positive. When both quantities are low, they will both be below their expected values. Both deviations will be negative, but the cross product will again be positive. When one is high and one is low (that is, one above

2 2 its expected value and one below), the cross product will be negative. If large X s tend to go with large Y s, and small with small, then the positive cross products will get more weight (higher probabilities) than negative cross products in the formula, and the overall calculation will yield a positive covariance. On the other hand, if large X s tend to occur with small Y s, and vice versa, then the negative cross products will get more weight, resulting in a negative covariance. Thus, a positive covariance reflects quantities that tend to move in the same direction, and this is called a positive or direct relationship. Likewise, a negative covariance indicates that the quantities tend to move in opposite directions, which is called a negative or indirect relationship. A simple example will help clarify the idea of covariance and its calculation. Suppose an investor is considering purchasing shares in American Rivets Corporation (ARC). The investor already has shares of Sundance Solar Power (SSP). One of the things the investor would like to accomplish is to stabilize the rate of return of the portfolio; ideally, when the return on one stock goes down, the other would go up. On the other hand, a positive relationship would be bad because the returns would tend to go up and down together, making the overall return on the portfolio vary considerably. A simple model of the returns on the two stocks involves only two possible outcomes for each one. ARC could have returns of 10% or -5%. SSP, on the other hand, could have returns of 12% or -8%. The probabilities of the possible outcomes are P(ARC = 10% and SSP = 12%) = 0.35 P(ARC = 10% and SSP = 8%) = 0.10 P(ARC = 5% and SSP = 12%) = 0.15 P(ARC = 5% and SSP = 8%) = 0.40 You can see that the two stocks tend to move in the same direction. There is a 75% chance that they are both high or both low. Likewise there is only a 25% chance that one is high and the other low. Thus, we expect the covariance to be positive.

3 3 Calculating the covariance requires first calculating the expected values for each stock. Considering ARC first, we can use the law of total probability: P(ARC = 10%) = P(ARC = 10% and SSP = 12%) + P(ARC = 10% and SSP = 8%) = = 0.45 Thus, P(ARC = 5%) = 1 P(ARC = 10%) = 0.55, and we can calculate E(ARC) = 0.45(10%) ( 5%) = 1.75% Likewise, we can calculate E(SSP): P(SSP = 12%) = P(ARC = 10% and SSP = 12%) + P(ARC = 5% and SSP = 12%) = = 0.50 P(SSP = 8%) = 1 P(SSP = 12%) = 0.50 Therefore: E(SSP) = 0.50(12%) ( 8%) = 2%

4 4 Now we can calculate the covariance between ARC and SSP: Cov(ARC, SSP) = [10% 1.75%][12% 2%]P(ARC = 10% and SSP = 12%) + [10% 1.75%][ 8% 2%]P(ARC = 10% and SSP = 8%) + [ 5% 1.75%][12% 2%]P(ARC = 5% and SSP = 12%) + [ 5% 1.75%][ 8% 2%]P(ARC = 5% and SSP = 8%) = 8.25% 10% % ( 10%) ( 6.75%) 10% ( 6.75%) ( 10%) 0.40 = 37.5(% squared) As expected, the covariance is positive. The problem, however, is that the magnitude of the covariance is not very meaningful because it depends on the range of variation in the two quantities. Also, as with the variance, the covariance carries units that are not meaningful. In the case of the two stock returns, the units are percentage squared. But suppose we wanted to calculate the covariance between hemline height and stock market return; the calculation would involve multiplying inches times percentage points of return, and so the units would be in percentage inches. What in the world is a percentage inch? To solve these two problems, we often transform the covariance to get a standardized measure of dependence. This standardized measure is called the correlation coefficient, and the Greek symbol ρ (rho) is used to represent it. To calculate ρ, divide the covariance of X and Y by the standard deviations of these two uncertain quantities: ρ XY = Cov(X, Y) σ X σ Y

5 5 The correlation ρ XY or simply ρ has a number of useful properties. First, it ranges between +1 (perfect positive dependence) and 1 (perfect negative dependence). A correlation of zero suggests no relationship, although certain kinds of dependence are possible even though the correlation is zero. Complete Exercise 7S.1 and calculate the correlation to see an example. Also, ρ has no units. In the hemline stock market example, we divide the covariance, which is in percentage inches, by the standard deviation of return, which is in percentage, and the standard deviation of hemline height (in inches), and the units cancel each other out. As a result, the correlation is a unitless measure. An implication is that the correlation is useful for comparing the strength of the relationship in one case with the strength of the relationship in another that involves different variables altogether. To continue the example, we calculate the correlation between the returns for stocks ARC and SSP. To do this calculation, we must first calculate the standard deviation for each of the two individual stocks. These are σ ARC = 7.46% and σ SSP = 10%. Thus, ρ ARC,SSP = Cov(ARC, SSP) σ ARC σ SSP = = This correlation of gives the investor an indication of the extent to which the returns of the two stocks are related to each other. By comparing the correlations of different pairs of stocks, the investor can try to locate those with lower (or even negative) correlations in order to accomplish the objective of stabilizing the return of the overall portfolio. A portfolio of two assets can have a lower risk (variation) than the individual risks of either asset. When two assets are negatively correlated, increases in one asset tend to occur with decreases in the other, thereby stabilizing the returns. This lowers the risk of the portfolio without sacrificing return because the expected return is still the weighted sum of the two assets returns. But what happens when the portfolio has more than two assets? Then the number of dependent relations grows quickly. With two assets, there is only one correlation to monitor, with three assets there are three correlations, and with 30 assets there are over 400 correlations! Clearly, we need a formula that keeps track of the interplay among all the correlations and precisely relates the risk of a portfolio to the risk of each asset.

6 6 As in the text, we will denote Portfolio P s return by R P. To derive the formula for the variance, Var(R P ), we will use the expected-value operator. We have already noted that the variance can be written as the expected value of squared differences: Var(X) = E[(X E(X)) 2 ]. In a similar way, the covariance is an expected value, this time the expected value of the cross product: Cov(X, Y) = E[(X E(X))(Y E(Y))]. Using the properties of expected value, we have: Var(R P ) = E[(R P E(R P )) 2 ] = E (w AB R AB + w CD R CD + w EF R EF ) E(w AB R AB + w CD R CD + w EF R EF ) 2 = E[(w AB R AB E(w AB R AB ) + w CD R CD E(w CD R CD ) + w EF R EF E(w EF R EF )) 2 ] = E w AB R AB E(w AB R AB ) 2 + E[(w CD R CD E(w CD R CD )) 2 ] + E[(w EF R EF E(w EF R EF )) 2 ] + E 2 (w AB R AB E(w AB R AB ))(w CD R CD E(w CD R CD )) 2 + E[2((w AB R AB E(w AB R AB ))(w EF R EF E(w EF R EF ))) 2 ] + E 2 (w CD R CD E(w CD R CD ))(w EF R EF E(w EF R EF )) 2 = Var(w AB R AB ) + Var(w CD R CD ) + Var(w EF R EF ) + 2Cov(w AB R AB, w CD R CD ) + 2Cov(w AB R AB, w EF R EF ) + 2Cov(w CD R CD, w EF R EF ) = w 2 AB Var(R AB ) + w 2 CD Var(R CD ) + w 2 EF Var(R EF ) + 2w AB w CD Cov(R AB, R CD ) + 2w AB w EF Cov(R AB, R EF ) + 2w CD w EF Cov(R CD, R EF ). Thus, the variance of a portfolio is the squared weights times the variance of each asset plus twice the product of the weights times the covariance for each pair of variables. It is this complicated formula that makes managing portfolios both difficult and interesting. To better understand this formula, let s simplify by considering the two asset portfolio: Var(R P ) = w 2 AB Var(R AB ) + w 2 CD Var(R CD ) + 2w AB w CD Cov(R AB, R CD ).

7 7 Problem 7.7 asks you to show that Var(R AB ) = , Var(R CD ) = , and Var(R EF ) = Thus, the variance of an equally weighted portfolio is: Var(R P ) = (0.0019) (0.0003) Cov(R AB, R CD ) = Cov(R AB, R CD ) = ρ AB,CD. Now, we can compare the portfolio s risk to that of the individual assets. The standard deviation of AB is 4.3% and the standard deviation of CD is 1.8%. If the correlation between AB and CD were 0.75, then the standard deviation of the portfolio is 2.89%, which is less than AB s standard deviation but larger than CD s. If, however, the stocks were negatively correlated, say -0.75, then the portfolio s risk drops to 1.62%, below both assets, demonstrating the power of portfolio diversification. One final warning is in order before leaving the ideas of covariance and correlation. Although these measures of dependence are widely used, they only provide insight into a certain kind of dependence. That is, as long as the relationship is such that an increase in one variable suggests an increase (or a decrease) in the other, then the covariance and correlation will reflect this relationship. If the relationship is more complex, however, such a relationship may not be adequately reflected in the covariance and correlation. For example as X increases up to a certain point, Y might be expected to increase. If, after that point, as X continues to increase, Y is expected to decrease, then we call this a nonmonotonic relationship. Covariance and correlations should only be used for monotonic relationships as they poorly reflect nonmonotonicity. Stochastic Dominance and Multiple Attributes Now we are prepared to follow up the discussion of stochastic dominance with multiple attributes in Chapter 4. In the case of multiple attributes, one must consider the joint distribution for all of the attributes together. To develop this, we need to introduce some notation.

8 8 Figure 7S.1 CDF s for three investment alternatives. Investment B stochastically dominates Investment A. First, let F(X) denote the CDF for a variable X. That is, F(X) = P(X x) For example, in the case of yearly profit for the duplex, F(\$6000) = P(Yearly Profit \$6000). Now we can write an algebraic formula for stochastic dominance in terms of the F s. Considering the investments in Figure 7S.1, B dominates A because F B (x) F A (x) for all values of x on the horizontal axis. This condition asserts that the CDF for B must lie to the right of the CDF for A. Again, we warn the reader that it is easy to reverse dominance. Note that B dominates A when B s CDF F B (x) is always less than or equal to A s CDF F A (x). When there are more attributes, the CDF must encompass all of the attributes. For example, recall the summer-job example from Chapter 4, in which we discussed uncertainty about both salary and summer fun. We would have to look at the CDF for both uncertain quantities. The CDF would be denoted by F x s, x f = P Salary x s and I x f. An alternative B (a specific job like the in-town job) dominates alternative A if F B x s, x f F A x s, x f for all values of x s and x f and is strictly less for some x s and x f values. If we could draw the picture of the graph in three-dimensional space, you could see that this means that the CDF for B must be entirely below the CDF for A and shifted toward larger values for both Salary and Fun. In Chapter 4, we made the claim that if the uncertain quantities are independent, then stochastic dominance on each of the individual attributes implies overall stochastic dominance. To see this, consider

9 9 the summer-job example further. Stochastic dominance requires that F B x s, x f F A x s, x f for all values of x s and x f, which is the same as P B Salary x s and Fun x f P A Salary x s and Fun x f for all values of x s and x f. If Salary and Fun are independent for both alternatives A and B, we now know that the joint probabilities in this condition can be rewritten as the product of the individual (or marginal) probabilities: P B (Salary x s )P B Fun x f P a (Salary x s )P A Fun x f Now, suppose that B dominates A individually on each attribute. This means that P B (Salary x s ) P A (Salary x s ) and P B Fun x f P A Fun x f If this is true, it is certainly the case that the overall stochastic-dominance condition is met, because the product P B (Salary x s )P B Fun x f must be less than or equal to P A (Salary x s )P A Fun x f. A final word of caution is in order here. The reasoning above only goes in one direction. That is, if the attributes are independent and if the individual stochastic-dominance conditions are met, then the overall stochastic-dominance condition is also met. That is, we have identified sufficient conditions for overall stochastic dominance. However, it is possible for overall stochastic dominance to exist even though the uncertain quantities are not independent or do not display stochastic dominance in the individual attributes. In other words, in some cases, you might have to go back to the definition of overall stochastic dominance [F B (x 1,, x n ) F A (x 1,, x n ) for all x 1,, x n ] in order to determine whether B dominates A. Covariance and Correlation: The Continuous Case Covariance and correlation also have counterparts when the uncertain quantities are continuous. As with expected value and variance, the definition of covariance uses an integral sign instead of a summation: *start box

10 10 x + y + Cov(X, Y) = [x E(X)][y E(Y)]f(x, y)dydx x y As before, the correlation ρ XY is calculated by dividing Cov(X, Y) by σ X σ Y. The double integral in the formula above replaces the double summation in the previous formula for the covariance of two discrete uncertain quantities. The term f(x, y) refers to the joint density function for uncertain quantities X and Y. This joint density function is a natural extension of the density function for a single variable; it can be interpreted as a function that indicates the relative likelihood of different (x, y) pairs occurring, and the probability that X and Y fall into any given region can be calculated from f(x, y). Problems 7S.1 Consider the following probabilities: P(X = 2) = 0.3 P(X = 4) = 0.7 P(Y = 10 X = 2) = 0.9 P(Y = 20 X = 2) = 0.1 P(Y = 10 X = 4) = 0.25 P(Y = 20 X = 4) = 0.75 Calculate the covariance and correlation between X and Y. 7S.2 Consider the following joint probability distribution for uncertain quantities X and Y: P(X = 2 and Y = 2) = 0.2 P(X = 1 and Y = 1) = 0.2 P(X = 0 and Y = 0) = 0.2 P(X = 1 and Y = 1) = 0.2 P(X = 2 and Y = 2) = 0.2 Calculate the covariance and correlation between X and Y.

### Covariance and Correlation

Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such

### Joint 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 Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### Covariance and Correlation. Consider the joint probability distribution f XY (x, y).

Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Section 5-2 Covariance and Correlation Consider the joint probability distribution f XY (x, y). Is there a relationship between X and Y? If so, what kind?

### Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single

### 4. Joint Distributions of Two Random Variables

4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint

### Bivariate Distributions

Chapter 4 Bivariate Distributions 4.1 Distributions of Two Random Variables In many practical cases it is desirable to take more than one measurement of a random observation: (brief examples) 1. What is

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### ST 371 (VIII): Theory of Joint Distributions

ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or

### Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS

Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS 5-1 a. Stand-alone risk is only a part of total risk and pertains to the risk an investor takes by holding only one asset. Risk is

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

### Joint Distributions. Tieming Ji. Fall 2012

Joint Distributions Tieming Ji Fall 2012 1 / 33 X : univariate random variable. (X, Y ): bivariate random variable. In this chapter, we are going to study the distributions of bivariate random variables

### Definition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).

Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The

### Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture - 2 Simple Linear Regression

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 2 Simple Linear Regression Hi, this is my second lecture in module one and on simple

### Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

### Chapter 5. Random variables

Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

### Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall

### Random Variables, Expectation, Distributions

Random Variables, Expectation, Distributions CS 5960/6960: Nonparametric Methods Tom Fletcher January 21, 2009 Review Random Variables Definition A random variable is a function defined on a probability

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation

Statistical Foundations: and Central Tendency and and Lecture 4 September 5, 2006 Psychology 790 Lecture #4-9/05/2006 Slide 1 of 26 Today s Lecture Today s Lecture Where this Fits central tendency/location

### Sect Properties of Real Numbers and Simplifying Expressions

Sect 1.6 - Properties of Real Numbers and Simplifying Expressions Concept #1 Commutative Properties of Real Numbers Ex. 1a.34 + 2.5 Ex. 1b 2.5 + (.34) Ex. 1c 6.3(4.2) Ex. 1d 4.2( 6.3) a).34 + 2.5 = 6.84

### 15.062 Data Mining: Algorithms and Applications Matrix Math Review

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

### Topic 4: Multivariate random variables. Multiple random variables

Topic 4: Multivariate random variables Joint, marginal, and conditional pmf Joint, marginal, and conditional pdf and cdf Independence Expectation, covariance, correlation Conditional expectation Two jointly

### Chapters 5. Multivariate Probability Distributions

Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are

### 2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

### Holding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10?

Return, Risk, and Risk Aversion Holding Period Return Ending Price - Beginning Price + Intermediate Income Return = Beginning Price R P t+ t+ = Pt + Dt P t An Example You bought IBM stock at \$40 last month.

### Joint Distribution and Correlation

Joint Distribution and Correlation Michael Ash Lecture 3 Reminder: Start working on the Problem Set Mean and Variance of Linear Functions of an R.V. Linear Function of an R.V. Y = a + bx What are the properties

### MULTIVARIATE PROBABILITY DISTRIBUTIONS

MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined

### Data reduction and descriptive statistics

Data reduction and descriptive statistics dr. Reinout Heijungs Department of Econometrics and Operations Research VU University Amsterdam August 2014 1 Introduction Economics, marketing, finance, and most

### Biostatistics: A QUICK GUIDE TO THE USE AND CHOICE OF GRAPHS AND CHARTS

Biostatistics: A QUICK GUIDE TO THE USE AND CHOICE OF GRAPHS AND CHARTS 1. Introduction, and choosing a graph or chart Graphs and charts provide a powerful way of summarising data and presenting them in

### ECE302 Spring 2006 HW7 Solutions March 11, 2006 1

ECE32 Spring 26 HW7 Solutions March, 26 Solutions to HW7 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

### Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### The Simple Linear Regression Model: Specification and Estimation

Chapter 3 The Simple Linear Regression Model: Specification and Estimation 3.1 An Economic Model Suppose that we are interested in studying the relationship between household income and expenditure on

### Variances and covariances

Chapter 4 Variances and covariances 4.1 Overview The expected value of a random variable gives a crude measure for the center of location of the distribution of that random variable. For instance, if the

### Module 3: Correlation and Covariance

Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis

### Grade 7/8 Math Circles October 7/8, Exponents and Roots - SOLUTIONS

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles October 7/8, 2014 Exponents and Roots - SOLUTIONS This file has all the missing

### Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

### 5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0

a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a

### Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

### Correlation in Random Variables

Correlation in Random Variables Lecture 11 Spring 2002 Correlation in Random Variables Suppose that an experiment produces two random variables, X and Y. What can we say about the relationship between

### 1 Portfolio mean and variance

Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a one-period investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring

### Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance)

Concepts in Investments Risks and Returns (Relevant to PBE Paper II Management Accounting and Finance) Mr. Eric Y.W. Leung, CUHK Business School, The Chinese University of Hong Kong In PBE Paper II, students

### Lecture 14: Correlation and Autocorrelation Steven Skiena. skiena

Lecture 14: Correlation and Autocorrelation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Overuse of Color, Dimensionality,

### The Scalar Algebra of Means, Covariances, and Correlations

3 The Scalar Algebra of Means, Covariances, and Correlations In this chapter, we review the definitions of some key statistical concepts: means, covariances, and correlations. We show how the means, variances,

### DISCRETE RANDOM VARIABLES

DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced

### Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

### 99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm

Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the

### Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

### Order of Operations. 2 1 r + 1 s. average speed = where r is the average speed from A to B and s is the average speed from B to A.

Order of Operations Section 1: Introduction You know from previous courses that if two quantities are added, it does not make a difference which quantity is added to which. For example, 5 + 6 = 6 + 5.

### Correlation key concepts:

CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)

### Math 016. Materials With Exercises

Math 06 Materials With Exercises June 00, nd version TABLE OF CONTENTS Lesson Natural numbers; Operations on natural numbers: Multiplication by powers of 0; Opposite operations; Commutative Property of

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### Stat 704 Data Analysis I Probability Review

1 / 30 Stat 704 Data Analysis I Probability Review Timothy Hanson Department of Statistics, University of South Carolina Course information 2 / 30 Logistics: Tuesday/Thursday 11:40am to 12:55pm in LeConte

### An-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)

An-Najah National University Faculty of Engineering Industrial Engineering Department Course : Quantitative Methods (65211) Instructor: Eng. Tamer Haddad 2 nd Semester 2009/2010 Chapter 5 Example: Joint

### not to be republished NCERT Measures of Central Tendency

You have learnt in previous chapter that organising and presenting data makes them comprehensible. It facilitates data processing. A number of statistical techniques are used to analyse the data. In this

### Worked examples Multiple Random Variables

Worked eamples Multiple Random Variables Eample Let X and Y be random variables that take on values from the set,, } (a) Find a joint probability mass assignment for which X and Y are independent, and

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

### The Bivariate Normal Distribution

The Bivariate Normal Distribution This is Section 4.7 of the st edition (2002) of the book Introduction to Probability, by D. P. Bertsekas and J. N. Tsitsiklis. The material in this section was not included

### CHAPTER 6 RISK AND RISK AVERSION

CHAPTER 6 RISK AND RISK AVERSION RISK AND RISK AVERSION Risk with Simple Prospects Risk, Speculation, and Gambling Risk Aversion and Utility Values Risk with Simple Prospects The presence of risk means

### Repton Manor Primary School. Maths Targets

Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### Using Excel for inferential statistics

FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

### Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

### A BRIEF GUIDE TO ABSOLUTE VALUE. For High-School Students

1 A BRIEF GUIDE TO ABSOLUTE VALUE For High-School Students This material is based on: THINKING MATHEMATICS! Volume 4: Functions and Their Graphs Chapter 1 and Chapter 3 CONTENTS Absolute Value as Distance

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### Math 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304. jones/courses/141

Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Variance: expected squared deviation from the mean: Standard

### Basic Utility Theory for Portfolio Selection

Basic Utility Theory for Portfolio Selection In economics and finance, the most popular approach to the problem of choice under uncertainty is the expected utility (EU) hypothesis. The reason for this

### Probability and Statistical Methods. Chapter 4 Mathematical Expectation

Math 3 Chapter 4 Mathematical Epectation Mean of a Random Variable Definition. Let be a random variable with probability distribution f( ). The mean or epected value of is, f( ) µ = µ = E =, if is a discrete

### Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

### Exclusive OR (XOR) and hardware random number generators

Exclusive OR (XOR) and hardware random number generators Robert B Davies February 28, 2002 1 Introduction The exclusive or (XOR) operation is commonly used to reduce the bias from the bits generated by

### Mathematical Expectation

Mathematical Expectation Properties of Mathematical Expectation I The concept of mathematical expectation arose in connection with games of chance. In its simplest form, mathematical expectation is the

### 17.0 Linear Regression

17.0 Linear Regression 1 Answer Questions Lines Correlation Regression 17.1 Lines The algebraic equation for a line is Y = β 0 + β 1 X 2 The use of coordinate axes to show functional relationships was

### STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

### Econometrics Simple Linear Regression

Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### (Refer Slide Time: 00:00:56 min)

Numerical Methods and Computation Prof. S.R.K. Iyengar Department of Mathematics Indian Institute of Technology, Delhi Lecture No # 3 Solution of Nonlinear Algebraic Equations (Continued) (Refer Slide

### Objectives. By the time the student is finished with this section of the workbook, he/she should be able

QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

### Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

### STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

### Factoring Polynomials

Factoring Polynomials 4-1-2014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall

### 3 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

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### (Mathematics Syllabus Form 3 Track 3 for Secondary Schools June 2013) Page 1 of 9

(Mathematics Syllabus Form 3 Track 3 for Secondary Schools June 2013) Page 1 of 9 Contents Pages Number and Applications 3-4 Algebra 5-6 Shape Space and Measurement 7-8 Data Handling 9 (Mathematics Syllabus

### Chapter 3. Algebra. 3.1 Rational expressions BAa1: Reduce to lowest terms

Contents 3 Algebra 3 3.1 Rational expressions................................ 3 3.1.1 BAa1: Reduce to lowest terms...................... 3 3.1. BAa: Add, subtract, multiply, and divide............... 5

### A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

### REVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012

REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit

### Domain Essential Question Common Core Standards Resources

Middle School Math 2016-2017 Domain Essential Question Common Core Standards First Ratios and Proportional Relationships How can you use mathematics to describe change and model real world solutions? How

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### REVISED GCSE Scheme of Work Mathematics Higher Unit T3. For First Teaching September 2010 For First Examination Summer 2011

REVISED GCSE Scheme of Work Mathematics Higher Unit T3 For First Teaching September 2010 For First Examination Summer 2011 Version 1: 28 April 10 Version 1: 28 April 10 Unit T3 Unit T3 This is a working

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Generating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010

Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 Quasi-Monte