...I wish Google Maps had an "Avoid Ghetto" routing option. More often than not, when someone is telling me a story all I can think about is that I can’t wait for them to finish so that I can tell my own story that’s not only better, but also more directly involves me. Nothing sucks more than that moment during an argument when you realize you're wrong. I don't understand the purpose of the line, "I don't need to drink to have fun." Great, no one does. But why start a fire with flint and sticks when they've invented the lighter? Have you ever been walking down the street and realized that you're going in the complete opposite direction of where you are supposed to be going? But instead of just turning a 180 and walking back in the direction from which you came, you have to first do something like check your watch or phone or make a grand arm gesture and mutter to yourself to ensure that no one in the surrounding area thinks you're crazy by randomly switching directions on the sidewalk. I totally take back all those times I didn't want to nap when I was younger. The letters T and G are very close to each other on a keyboard. This recently became all too apparent to me and consequently I will never be ending a work email with the phrase "Regards" again. Do you remember when you were a kid, playing Nintendo and it wouldn't work? You take the cartridge out, blow in it and that would magically fix the problem. Every kid in America did that, but how did we all...
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...you have heard rumours about the board meeting this morning. The future of the four of you was discussed and a decision was taken to release one of you. I can’t tell you who - the Chairman is writing to you all immediately. However, I can say that we were unable to make the decision on merit, so we made it at random. The Chairman took one of four cards, each of which had one of your names on the back!” “So, I have a one in four chance of being out of work after Christmas”, said Alain. He felt relieved, as he had thought the odds might be worse. “But of course you know definitely who it is”, he added. Alain felt he could leach rather more out of the partner. “Since we both know that at least two out of Belén, Carlos or Dawood will definitely have a job here next year, if you tell me the name of two of them who will keep their jobs, it will still leave me in the dark as to my own fate. It would be pleasant to contemplate over Christmas the two who will continue in their jobs. Clearly, I’m not going to be able to see any of them before the Chairman’s letters arrive. If all three are to keep their jobs, just choose two names at random to tell me.” Xavier thought about this for a...
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...James W. Taylor February 2012 Imperial College EMBA 2012 Quantitative Methods Individual Assignment This assignment consists of two parts. Part A is worth 50% of the marks, and Part B is worth the remaining 50%. Your report for the two parts should consist of no more than 1,500 words. Part A – Blanket Systems Blanket Systems is developing and testing a new computer workstation, OB1, which it will introduce to the market in the next 6 months. OB1 will be sold under a three-year warranty covering parts and labour. The company has decided to subcontract the service support for the warranty and has entered negotiations about the support contract with Fixit Inc. Fixit has proposed two different pricing schemes for the subcontract. The first involves the payment of a fixed fee of $1,000,000 and the second a variable fee of $250 per workstation sold, subject to a minimum fee of $350,000. Under both schemes, the payment will be made one year after the introduction of the workstation to the market at which point the product will be replaced by newer models not covered by the warranty service subcontract. At the moment, there is uncertainty about the sales potential of the new workstation. Sales of OB1 are expected to come from two sources: (i) the successful closure by senior management of a major purchase of 2000 units by a long standing customer, (ii) the efforts of regional sales offices. Given the state of the negotiations with the long-standing customer, the current estimate of...
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...can begin to use probabilistic ideas in statistical inference and modelling, and the study of stochastic processes. Probability axioms. Conditional probability and independence. Discrete random variables and their distributions. Continuous distributions. Joint distributions. Independence. Expectations. Mean, variance, covariance, correlation. Limiting distributions. The syllabus is as follows: 1. Basic notions of probability. Sample spaces, events, relative frequency, probability axioms. 2. Finite sample spaces. Methods of enumeration. Combinatorial probability. 3. Conditional probability. Theorem of total probability. Bayes theorem. 4. Independence of two events. Mutual independence of n events. Sampling with and without replacement. 5. Random variables. Univariate distributions - discrete, continuous, mixed. Standard distributions - hypergeometric, binomial, geometric, Poisson, uniform, normal, exponential. Probability mass function, density function, distribution function. Probabilities of events in terms of random variables. 6. Transformations of a single random variable. Mean, variance, median, quantiles. 7. Joint distribution of two random variables. Marginal and conditional distributions. Independence. iii iv 8. Covariance, correlation. Means and variances of linear functions of random variables. 9. Limiting distributions in the Binomial case. These course notes explain the naterial in the syllabus. They have been “fieldtested” on the class of 2000. Many of the examples...
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...mathematical expression which gives the sum of four rolls of a die. To do this, we could let Xi , i = 1, 2, 3, 4, represent the values of the outcomes of the four rolls, and then we could write the expression X 1 + X 2 + X 3 + X4 for the sum of the four rolls. The Xi ’s are called random variables. A random variable is simply an expression whose value is the outcome of a particular experiment. Just as in the case of other types of variables in mathematics, random variables can take on different values. Let X be the random variable which represents the roll of one die. We shall assign probabilities to the possible outcomes of this experiment. We do this by assigning to each outcome ωj a nonnegative number m(ωj ) in such a way that m(ω1 ) + m(ω2 ) + · · · + m(ω6 ) = 1 . The function m(ωj ) is called the distribution function of the random variable X. For the case of the roll of the die we would assign equal probabilities or probabilities 1/6 to each of the outcomes. With this assignment of probabilities, one could write P (X ≤ 4) = 1 2 3 2 CHAPTER 1. DISCRETE PROBABILITY DISTRIBUTIONS to mean that the probability is 2/3 that a roll of a die will have a value which does not exceed 4. Let Y be the random variable which represents the toss of a coin. In this case, there are two possible outcomes, which we can label as H and T. Unless we have reason...
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...positive information from the seller regarding the copier reliability. The price of the primary copier is $12,000. Terri, one of the owners of JET Copier, received a loan for her family to purchase the copier. After talking to someone from the dean’s office of business, JET Copies discovered that the copier is not as reliable as they thought. To prevent loss of revenue, JET Copies uses several simulations in an attempt to estimate the loss of revenue—repair days, time between breakdowns, and number of copies per day. The price of the second copier is $8,000. If revenue lost for a year is greater than or equal to $8000, then JET should purchase. Below are my calculation after setting up and running simulations based on the information provided within the case study. 1. In Excel, use a suitable method for generating the number of days needed to repair the copier, when it is out of service, according to the discrete distribution shown. The owners of JET Copies decided to purchase a copier similar to the one used in the college of business at State. The company they purchased the copier from touted the copier reliability. The owners of JET Copies thought they made a good decision since the copier was so reliable. After purchasing the copier, Ernie, one of the owners of JET Copies discovered from someone in the dean’s office that the copier wasn’t as reliable the company had touted. Ernie discovered the copier broke down frequently and when it did, it took...
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...9. SAMPLING AND STATISTICAL INFERENCE We often need to know something about a large population. Eg: What is the average number of hours per week devoted to online social networking for all US residents? It’s often infeasible to examine the entire population. Instead, choose a small random sample and use the methods of statistical inference to draw conclusions about the population. But how can any small sample be completely representative? We can’t act as if statistics based on small samples are exactly representative of the entire population. Why not just use the sample mean x in place of μ? For example, suppose that the average hours for 100 randomlyselected US residents was x = 6.34. Can we conclude that the average hours for all US residents (μ) is 6.34? Can we conclude that μ > 6? Fortunately, we can use probability theory to understand how the process of taking a random sample will blur the information in a population. But first, we need to understand why and how the information is blurred. Sampling Variability Although the average social networking hours for all US residents is a fixed number, the average of a sample of 100 residents depends on precisely which sample is taken. In other words, the sample mean is subject to “sampling variability”. The problem is that by reporting x alone, we don’t take account of the variability caused by the sampling procedure. If we had polled different residents, we might have gotten a different average social networking hours. In general...
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...of this assignment I decided to use the method like the dice. Using the repair time and probability given to us in the book. I computed the cumulative Probability by adding cells A & B; lines 5- 8 which gave me the Cum. Prob. for each day it would take for repair of the copiers. Next, I took the Cumulative Probability and the Repair Time in Days, named it Repairs. This will assist me later in finding the random numbers for the number of days the possibility of the copier breaking down. I choose to do 30 days because all months except one have 30 days in them. I also used the process of VLOOKUP, the equation =VLOOKUP (F5, Repair,2). Taking the random function (=rand), I received this for the first day randomized number as well as the number of days it took to repair in the event of a breakdown. This step I used for the next 29 days. As you can see the maximum number of days that they will have in between repair cycles is 4 and they would happen closer to the end of the month. The number of breakdowns I based on the weeks, 52 weeks in a year. Using the same random function (=rand), I received the randomized number, within the 6 (=6*SQRT(I5); weeks intervals the square root gave me the time in between those breakdowns. My next step was to find the average of the square rooted number, lines J5- J56 of the weeks in between, which gave me that average. I also did an average of the days the repair time would take, lines G5-G34, which gave me that average. Of course these...
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...examples. A reader just learning probability should aim at mastering these basic problems in the sense of being able to recognize them in various settings and solve them by carry out the required analysis and computations. A good way to study each problem is to create and solve one or more examples of the problem analogous to those below. By creating new examples this way, one will actually own a piece of the subject as well as understanding it. 1.1 Probabilities of Events A description of events and their probabilities requires a framework for representing events of interest in terms of a random experiment. Our textbook tells us how to describe such an experiment in terms of a set of outcomes Ω called a sample space, and to represent events as subsets of Ω. A basic problem in this regard is as follows. Problem 1 Sample Spaces and Events as Sets Given a verbal description of a random experiment and certain events of interest, represent the sample space as a set Ω, and identify the events as subsets of Ω. There may be several natural choices for Ω; select the simplest one that contains enough information to describe...
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...CHAPTER 9 RISK AND RETURN FOCUS Our initial focus is on defining risk in financial terms and understanding how that concept fits into portfolio theory. As we gain a more sophisticated understanding of risk, we're able to focus on the concept of beta and how to apply it through the SML. PEDAGOGY The study of Risk and Return presents the biggest pedagogical challenge in basic finance. Therefore motivating the study and developing ideas patiently is especially important. Students are easily confused early in the discussion by the transition from the everyday notion of risk to its financial representation as variation in return. We therefore take pains to present these ideas carefully through an intuitive illustration. Risk and Return is also the area in which textbook treatments using mathematical statistics get students who aren't good at math into the most trouble. The approach used here presents statistical concepts graphically and in words to overcome this pedagogical roadblock. It's worth noting that while we minimize the statistics used in the theoretical development of the CAPM, we don't skimp on the algebraic math required to apply the SML. TEACHING OBJECTIVES Instruction should begin motivating the study of risk and return by explaining that higher long-term returns are available on equity than on debt but that there's an associated risk. Point out that the objective of investing is to take advantage of the high...
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...BUSINESS RESEARCH METHODS PROJECT ON GALLUP INTRODUCTION Gallup, Inc. is a research-based, global performance-management consulting company. Founded by George Gallup in 1935, the company became famous for its public opinion polls, which were conducted in the United States and other countries. Gallup works with major businesses and organizations around the world. In 1988, four years after George Gallup died, Selection Research (SRI) purchased the company from its heirs. SRI wanted the Gallup name to use on its polls, which gave them more credibility and higher response rates. Today the poll is used to gain visibility. Some of Gallup's key practice areas are employee engagement, customer engagement, talent management, and well-being. Gallup has nearly 40 offices in more than 20 countries. World headquarters are in Washington, D.C. Operational headquarters are in Omaha, Nebraska. Gallup's current Chairman and CEO is Jim Clifton. Gallup delivers forward-thinking research, analytics, and advice to help leaders solve their most pressing problems. Combining more than 75 years of experience with its global reach, Gallup knows more about the attitudes and behaviors of the world's constituents, employees, and customers than any other organization. Gallup consultants help private and public sector organizations boost organic growth through measurement tools, strategic...
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...The Inaugural Coase Lecture An Introduction to Regression Analysis Alan O. Sykes* Regression analysis is a statistical tool for the investigation of relationships between variables. Usually, the investigator seeks to ascertain the causal effect of one variable upon another—the effect of a price increase upon demand, for example, or the effect of changes in the money supply upon the inflation rate. To explore such issues, the investigator assembles data on the underlying variables of interest and employs regression to estimate the quantitative effect of the causal variables upon the variable that they influence. The investigator also typically assesses the “statistical significance” of the estimated relationships, that is, the degree of confidence that the true relationship is close to the estimated relationship. Regression techniques have long been central to the field of economic statistics (“econometrics”). Increasingly, they have become important to lawyers and legal policy makers as well. Regression has been offered as evidence of liability under Title VII of the Civil Rights Act of , as evidence of racial bias in death penalty litigation, as evidence of damages in contract actions, as evidence of violations under the Voting Rights Act, and as evidence of damages in antitrust litigation, among other things. In this lecture, I will provide an overview of the most basic techniques of regression analysis—how they work, what they assume, Professor of Law, University of Chicago...
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...How We Know What Isn't So The Fallibility of Human Reason in Everyday Life Thomas Gilovich THE FREE PRESS A Division of Macmillan, Inc. NEW YORK To Karen and liana Contents Acknowledgments 1. Introduction vn 1 PART ONE Cognitive Determinants of Questionable Beliefs 2. Something Out of Nothing: The Misperception and Misinterpretation of Random Data 3. Too Much from Too Little: The Misinterpretation of Incomplete and Unrepresentative Data 4. Seeing What We Expect to See: The Biased Evaluation of Ambiguous and Inconsistent Data 9 29 49 PART TWO Motivational and Social Determinants of Questionable Beliefs 5. Seeing What We Want to See: Motivational Determinants of Belief 6. Believing What We are Told: The Biasing Effects of Secondhand Information 7. The Imagined Agreement of Others: Exaggerated Impressions of Social Support 75 88 112 Contents PART THREE Examples of Questionable and Erroneous Beliefs 8. Belief in Ineffective "Alternative" Health Practices 9. Belief in the Effectiveness of Questionable Interpersonal Strategies 10. Belief in ESP 125 146 Acknowledgments 156 PART FOUR Where Do We Go from Here? 11. Challenging Dubious Beliefs: The Role of Social Science Notes Index 185 195 214 Four people made unusually significant contributions to this work and deserve special thanks. Lee Ross commented on drafts of many of the chapters and provided a number of his uniquely...
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...Probability and Statistics for Finance The Frank J. Fabozzi Series Fixed Income Securities, Second Edition by Frank J. Fabozzi Focus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. Abate Handbook of Global Fixed Income Calculations by Dragomir Krgin Managing a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. Fabozzi Real Options and Option-Embedded Securities by William T. Moore Capital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. Fabozzi The Exchange-Traded Funds Manual by Gary L. Gastineau Professional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. Fabozzi Investing in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia Pilarinu Handbook of Alternative Assets by Mark J. P. Anson The Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry The Handbook of Financial Instruments edited by Frank J. Fabozzi Collateralized Debt Obligations: Structures and Analysis by Laurie S. Goodman and Frank J. Fabozzi Interest Rate, Term Structure, and Valuation Modeling edited by Frank J. Fabozzi Investment Performance Measurement by Bruce J. Feibel The Handbook of Equity Style Management edited by T. Daniel Coggin and Frank J. Fabozzi The Theory and Practice of Investment Management edited by Frank J. Fabozzi and Harry M. Markowitz Foundations of Economic Value Added, Second Edition by James L. Grant Financial Management and Analysis, Second Edition...
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...PORTFOLIO SELECTION* HARRY ARKOWITZ M The Rand Corporation THEPROCESS OF SELECTING a portfolio may be divided into two stages. The first stage starts with observation and experience and ends with beliefs about the future performances of available securities. The second stage starts with the relevant beliefs about future performances and ends with the choice of portfolio. This paper is concerned with the second stage. We first consider the rule that the investor does (or should) maximize discounted expected, or anticipated, returns. This rule is rejected both as a hypothesis to explain, and as a maximum to guide investment behavior. We next consider the rule that the investor does (or should) consider expected return a desirable thing and variance of return an undesirable thing. This rule has many sound points, both as a maxim for, and hypothesis about, investment behavior. We illustrate geometrically relations between beliefs and choice of portfolio according to the "expected returns-variance of returns" rule. One type of rule concerning choice of portfolio is that the investor does (or should) maximize the discounted (or capitalized) value of future returns.l Since the future is not known with certainty, it must be "expected" or "anticipatded7'returns which we discount. Variations of this type of rule can be suggested. Following Hicks, we could let "anticipated" returns include an allowance for risk.2 Or, we could let the rate at which we capitalize the returns from particular...
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