# The second nonlinear model that I ll describe are called Markov Processes. The name

Save this PDF as:

Size: px
Start display at page:

Download "The second nonlinear model that I ll describe are called Markov Processes. The name"

## Transcription

1 Chapter 5 Markov Processes History is a cyclic poem written by time upon the memories of man. Percy Bysshe Shelley The second nonlinear model that I ll describe are called Markov Processes. The name sounds kind of scary, but they re easy to understand. To frame my presentation of Markov Processes, I pick up on a point raised at the end of the previous chapter: the shortcomings of linear extrapolation. At present, we see a general trend toward more democratic countries. And one might well think that eventually, everyone will live in democracy. That might be pretty to think, but unfortunately given some rather mild assumptions the trend to democracy should slow, and in the long run, the entire world probably won t be democratic. Just like, in the long run, we won t all be living in California. The democratization example provides an entree into thinking about linkages between states of the world, the key assumption in Markov theory. When a modeler mentions a state of the world she does not mean Kazakhstan or even Idaho. She means the relevant features of the world. So, the state of a country might be democratic or dictatorship. The state of a boat might be at dock, a sea and afloat, sinking, or sunk. Markov processes consist two 69

2 70 CHAPTER 5. MARKOV PROCESSES parts: (1) a finite number of states and (2) transition probabilities for moving between those states. For example, on a given day, a person might be happy, sad, or feel a sense of ennui. These would be the states of the Markov process. The transition probabilities represent the likelihood of moving between those states from a state of happiness on one day to a state of ennui the next. Markov processes provide a powerful lens for viewing the world. Provided a small number of assumptions are met: a fixed set of states, fixed transition probabilities, and the possibility of getting from any state to another through a series of transitions, a Markov process converges to a unique distribution over states. This means that what happens in the long run won t depend on where the process started or on what happened along the way. What happens in the long run will be completely determined by the transition probabilities the likelihoods of moving between the various states. Endowed with knowledge of Markov Processes, we have a new way to think about the benefits of taking action. If a system follows a Markov Process, then initial conditions, interventions, and history itself have no bearing on the long run distribution over states. To put this in the context of an example, if your mental state can be characterized as a Markov Process, then any e ort to move you from sadness to happiness will only have temporary e ects. In the long run, you ll settle into a steady distribution over happy, sad, and ennui that s una ected by bursts of joy. Similarly, if social processes are Markov Processes, then redistribution of resources would also have no long term e ect. When I teach Markov Processes, students often undergo a period of initial confusion as they think I am saying that history doesn t matter. I m not saying that at all. What I am saying is that if events follows a Markov process, then history doesn t matter. So, If we want to believe (as I do by the way), that history does matter in the social world, then one of the assumptions of the Markov model must not hold. Thus, the model provides a framework to which we can compare the world. And it o ers up a challenge: If the world fits the

3 71 assumptions of the model, then what happens in the long run is set in stone. If we believe history matters, then it s incumbent upon us to show that at least one of the assumptions of the model doesn t hold. Otherwise, we re ignoring some pretty basic logic. In what follows. I first show the data on democratization, which looks pretty compelling. I then describe a Markov Process that includes only two states and present some general theoretical results. I then go back to the data on democratization and view the trends through the lens of Markov theory. From the perspective of the Markov model, we should expect democratization to level out. I then lay describe the general Markov model. I conclude with some comments on the many uses of the Markov approach and also return to, and eventually leave, California. Democratic Vistas First, the data on democratization: Freedom House categories countries as either free, partly free or not free. The figure below shows the percentage of countries in each category over the past thirty-five years. The figure shows a clear trend toward increased democratization. It s easy to make a linear approximation of any one of these lines project into the future. In the past thirty-five years, the percentage of free countries has risen approximately twenty percent. If that trend were to continue, by 2045, two thirds of all countries will be free and by 2080, nearly eight out of every nine countries will be free. And by the turn of the next century, almost every country will be free. This is a happy thought. Yet, a closer look at the data reveals that the transition go in both directions: some countries move from partly free to free and from not free to partly free, but some move from the free category to the partly free and some even move from partly free to not free. These movements back and forth between categories suggest that

4 72 CHAPTER 5. MARKOV PROCESSES Free Partly Free Not Free Figure 5.1: Freedom House Democratization Data democratization shouldn t be seen as a simple linear trend. But how do we model it? Happy or Sad: a Two State Markov Process Good modeling requires starting simple. I begin with simplest possible Markov Process. In amarkovprocess,thethingbeingconsidered,beitaneconomy,aperson,amachine,oran ecosystem, takes on one of some finite number of states. In the types of Markov Processes that I consider here time moves in discrete steps. These steps could be counted out in seconds, years, or days. 1 In each time step, the state of the system updates according to a transition mapping. Isaythatthestateupdatesratherthansaythatitchangesbecauseit s possible that the state doesn t change. This transition mapping is typically probabilistic not deterministic, though nothing in what follows precludes deterministic transition mappings. What s important is that the transition probabilities stay fixed. They never change. 1 In more general models this assumption can be relaxed. Throughout the book, I use the least technical versions of the models that still carry some weight.

5 73 The process I describe here has only two states. Each state describes a person s mental mood. I ll call these states happy and sad. On any given day, any person will be in exactly one of these states. Further, the person s state tomorrow will depend only on her state today. Suppose that I ve gathered a lot of data and found the following transitions between states. A person who is happy today has a ninety percent chance of being happy tomorrow and a ten percent chance of being sad. In contrast, an unhappy person today has only a thirty percent chance of being happy tomorrow and a seventy percent chance of being unhappy. Given these assumptions, if I start on a Monday with fifty happy people (represented by the yellow circle in the figure below) and fifty sad people (represented by the blue circle). On Tuesday, five of the happy people (ten percent of them) will become sad, and fifteen of the sad people (thirty percent) will become happy. This creates sixty happy people and forty sad people. Similar calculations yield that on Wednesday, sixty-six people are happy. On Thursday, approximately seventy people are happy. And on Friday about seventy-two people are happy Figure 5.2: The first two steps of a Markov Process

6 74 CHAPTER 5. MARKOV PROCESSES Suppose instead, that I begin with one hundred happy people on Monday. On Tuesday, ninety people will be happy. On Wednesday, eighty-four will be happy. On Thursday, approximately eighty will be happy, and on Friday, seventy-eight will be happy. This number is not too far from the seventy-two that were happy when I began with equal numbers of happy and sad people. In fact, if I were to continue to iterate through these two examples, I would find that the process converges on exactly seventy-five happy people and twenty-five unhappy people. This holds more generally. If I were to start with any number of happy people, any number from zero to one hundred, eventually, I ll have seventy-five happy people. This distribution of seventy-five happy people and twenty-five unhappy people is called an equilibrium. Once the processes reaches those numbers, it stays there. To see why this is an equilibrium, we need only apply our transition probabilities as shown in the figure below. If seventy-five people are happy, then 7.5 will become unhappy next period. And, if twenty-five people are unhappy, then 7.5 willbecomehappy. Thesetwoe ectsbalance, so the result is an equilibrium. As equilibria go, this one seems rather odd. Each of the hundred people moves back and forth between happy and sad. So, the system churns constantly. However, the expected number of people in each state doesn t change, so by convention, we say the system reaches a statistical equilibrium. Thismeansthatsomestatistic, inthiscasethenumberofhappy people, reaches an equilibrium, even though the process keeps churning. In equilibrium, each day, seven and a half people move from happy to sad, and seven and a half people move from sad to happy. The process creates a cyclic poem. 2 2 If the half person concept bothers you, think of these as percentages of people.

7 The Ergodic Theorem Figure 5.3: Equilibrium of a Markov Process Our toy model of happy and sad people has four important features. First, each person had to be in one of a finite number of states. Second, the probability of moving from one state to another depended only on a person s current state and remained fixed throughout. And third, it was possible to move from any state to any other. Fourth, the process didn t produce a cycle in which every person alternated back and forth from happy to sad. The toy model exhibited an interesting property. It converged to a unique statistical equilibrium regardless of where the process was begun. Mathematicians have shown a logical connection between these four assumptions and the convergence to a unique equilibrium. This connection can be made formal in what I will call the Ergodic Theorem.

8 76 CHAPTER 5. MARKOV PROCESSES The Ergodic Theorem AMarkovProcesshasthefollowingattributes 1. The state of the process at any point in time belongs to a finite set of possible states. 2. The state of the process at the next point in time depends only on its current state and does so according to fixed transition probabilities. 3. It is possible through a series of transitions to get from any one state to any other. 4. The system does not produce a deterministic cycle through a sequence of states. The Ergodic Theorem Any Markov Process converges to a unique statistical equilibrium that does not depend on the initial state of the process or any one time changes to the state during the history of the process. The Ergodic Theorem might also be called the history doesn t matter theorem. If its assumption are satisfied, then what has happened in the past doesn t influence the long run equilibrium. Yet, most of us believe that history does matter. So what gives? Let s look at the assumptions. Finite number of states? Seems pretty innocuous. No simple cycles? That makes sense too. Can get from any state to any other? Well, that could be problematic in some cases. Israeli and Palestinian people may not be able to get to the we re best friends state in light of events over the past few thousand years. So, there s one opening. Fixed transition probabilities? Aha! Probably not. Here s where Markov processes don t fit history. Transition probabilities do change. When someone gets involved in a

9 77 mutually supportive, constructive relationship their transitions from sad to happy become more probable. This observation doesn t mean that Markov processes are irrelevant to the study of historical processes. To the contrary, they tell us that events matter not if they change the state, but if they change the transition probabilities once we re in that state. Democratic Redux Now that I ve covered the basics of Markov theory, I want to use the model applied to the Freedom House data to show that every country probably won t become democratic. To do so, I have to parse that data a little more finely. In what follows, I approximate the probabilities that countries move between categories in any five year period. For the purposes of making the larger point, these crude approximations simplify the numerical calculations. If I were writing a research article, I would nail down the transition probabilities as closely as possible. I d also test to see if they appeared to be fixed over the time period. Those caveats aside, let s assume the transition probabilities shown in the table below: State in Next Period Free Partly Not Free Free 95% 5% 0% Current State Partly 10% 80% 10% Not Free 5% 15% 80% If I seed the model with the data from 1975: twenty-seven percent free states, thirty-two percent partly free states, and forty-one percent not free states, then (assuming five year time steps) in 2010, I have forty-eight percent free states, thirty-one percent partly free states, and twenty-one percent not free states. The actual percentages are forty-six, thirty, and twenty-four respectively.

10 78 CHAPTER 5. MARKOV PROCESSES Figure 5.4: Markov Model vs Actual Here s where the power of the Markov model enters. If we continue to apply these fixed transition probabilities, what happens? Well, in about fifty years, we should see only fiftyeight percent free countries, twenty-seven percent partly free, and fifteen percent not free. And in the long long run, we should expect sixty-two and a half percent free, twenty-five percent partly free, and twelve and a half percent not free. Iamnotstakingaclaimthatthesepredictionswillbecorrect. Iammerelystatingthatif countries continue to move between categories as they have been, we might more accurately expect about two-thirds of countries to be democratic then for all countries to be democratic. That doesn t mean that transition probabilities won t change. In fact, if we want a larger percentage of democracies, then we must figure out a way to change those probabilities.

11 79 Fertility of the Markov Model Lave and March emphasize that good models must be fertile they must apply to multiple contexts. The Markov model proves especially fertile. It can be been applied in almost any setting in which a system moves probabilistically between states. Consider a person s health. We can categorize a person s health as being in some finite number of states. And, we can probably also assume that a person moves probabilistically between those states. At least in the short run, we can assume those probabilities remain nearly fixed, and thus, we can use the model to predict equilibrium health distributions. Now, consider interventions such as drug protocols, behavioral changes, and surgeries. Typically, these change those transition probabilities. Evaluating the economic benefit of an intervention can be simplified by constructing a Markov model of a patient s health status both with and without the intervention. 3 If the intervention produces a better equilibrium, it s worth pursuing. If not, then the intervention should be abandoned. Of course, the challenge in constructing a useful model in such settings lies in how one defines the states. AMarkovmodelcanalsocapturestockprices,wherethepriceofastockequalsthe state. Notice that here the implicit assumption is that the price in the next period does not depend on history. It only depends on the current price. In point of fact, stock prices may have some historisis past prices may matter. If so, the Markov model can still be useful. By comparing actual price streams to those produced by the model, we can see how much history really matters. Markov models can be used to recognize patterns in international crises (Schrodt 1998). The transitions that lead to war might look very di erent from the transitions that produce peace and reconciliation. To perform this sort of analysis, two di erent Markov models must be fit. The first uses data from cases where crises led to war. The second uses data in 3 See Briggs and Sculper (1998) for an overview.

12 80 CHAPTER 5. MARKOV PROCESSES which crises were settled. If the two Markov models di er significantly, then we can look at acurrentpattern: bombing,hostagetaking,noexchangeofprisoners,escalatedpostering..., and ask which process was more likely to produce it: the process that leads to war or the process that doesn t. Outside the Box Once we see that Markov models can be used to discriminate patterns a whole world of outside the box applications opens up. I ll describe just one: how Markov models can be used to help settle competing claims of authorship. That may seem an outrageous claim, but it s true. To get to that point, I first show how Markov Models can construct words, then get to sentences and books. Claude Shannon, a developer of information theory, constructed amarkovmodelthatconstructswords. Insimplestform,thelettersplusa space comprise the twenty-seven states of the model. The transition probabilities tell which letters are likely to follow another letter. For example, the letter i would transition with high probability to the letter e and to the letter o. It would also have a high transition to space so that it could create the word I.. True, this simple Markov model wouldn t be a particularly good writer. It would be more likely to spit out a nonsense sequence than a word I athe pilly sared than something deep like whoso list to hunt, the opening words of Thomas Wyatt s famous sonnet to Anne Boelyn. Even so, this basic model reveals structure. It would encode the i before e rule by having the transition from i to e exceed the transition from e to i. And, if we make the model even richer, by letting the state denote the previous two letters, then the rule that i before eexceptafterc couldbecapturedbyhavingce transition to i with high probability, but ci not likely to transition to e. Thistwoletterstatemodelwouldhaveoversevenhundred states and would be even better at constructing words.

13 81 Let s take this idea of having a Markov model write texts up one level of abstraction. Instead of having letters as states, let s have words as states. So, after the word once we might expect the words in, upon, or twice, but we would not expect the word frequently. To construct this model, we d need a lot of computer memory (there are after all, a lot of words). And, we d need some way of determining the probabilities that one word follows another. Doing this by hand would take a long time. Fortunately, we could automate it. We could write a program that reads books and keeps track of what words likely follow others. In fact, this has been done. But which book? Di erent authors produce di erent patterns of word sequences. Thus, we could construct separate Markov model for Melville, Morrison, and Mao. This has been done as well. 4 These models wouldn t be su ciently sophisticated to produce new texts by Melville or Dickens, but they could predict whether or not a disputed text was likely to have been written by a given author. We ve now stumbled upon an unexpected use for Markov models: determining authorship. Take the case of the Federalist Papers, eighty-five essays written in 1787 and 1788 to convince New Yorkers to support the constitution. The authorship of twelve of these essays has long been a subject of dispute. Some say Hamilton wrote them and some attribute the works to Madison. The Markov model assigns them all to Madison. We cannot say for sure that the model predicts correctly, but it makes giving Hamilton credit a more di cult proposition. I want to be clear. Models aren t always correct. They re very often wrong. But, what they do accomplish is focusing our thinking on some aspects of a situation and thinking through the logic clearly. Here that logic points to Madison. 4 Khmelev and Tweedie (2001).

14 82 CHAPTER 5. MARKOV PROCESSES Takeaways We can now put the Markov model in our pocket. We know its assumptions: finite states, fixed transition probabilities, get to any state from any other, and no a simple cycle. We also know that given those assumptions that the process converges to a unique statistical equilibrium. That means that the process may keep churning but the statistical average remains unchanged. What do we do with this? Here are just a few takeaways. Takeaway #1 If we want to say that history matters, then we first have to say why history cannot be captured by a Markov model. Takeaway #2 In a system with many entities moving in an out of states, beware of inferring too much from initial linear trends. Pay attention to the long run equilibrium. We won t all be democracies. Takeaway #3 If a system is Markov, changing the current state has no long term impact. Making someone laugh won t change their long term demeanor, though it will give a temporary burst of joy. Takeaway #4 Fundamental change requires changing transition probabilities. It requires changing processes. Takeaway #5 Di erent processes produce distinct Markov model approximations. Therefore, we can fit Markov models to identify processes and to see the e ects of interventions on long run equilibria. To put a bow on this, California doesn t have one hundred million residents because while many many people did in fact move in, many others moved (including me!)

### Math. Rewriting Number Sentences. Answers. 1) 'J' times Twelve equals Four hundred Forty-Four. 2) 'K' times Fifteen equals One hundred Five

1) 'J' times Twelve equals Four hundred Forty-Four 1. 2) 'K' times Fifteen equals One hundred Five 2. 3) One thousand, Two hundred Eighty divided by 'X' equals Two 3. 4) Two plus 'Z' equals Eight 4. 5)

### 27) fifty thousand and two millionths 28) ninety thousand and six millionths

-2-23) ten thousand, nine and nine hundred thousand, nine 24) eighty thousand and four 25) ten thousand, one hundred and three hundred thousand, seven hundred eight 26) nine hundred five thousand and three

### Math. Writing Numbers through 1 Million. Answers. 1) five thousand, four hundred sixty-six. 2) seven thousand, six hundred eighty-two

1) five thousand, four hundred sixty-six 2) seven thousand, six hundred eighty-two 3) three thousand, seven hundred forty-five 4) one hundred twenty-three thousand, six hundred seventy-six 5) eighty-three

### Math. Writing Numbers as Words Through 1 Million. 1) 5,451 five thousand, four hundred fifty-one. 2) 32,617 thirty-two thousand, six hundred seventeen

1) 5,451 five thousand, four hundred fifty-one 2) 32,617 thirty-two thousand, six hundred seventeen 3) 995,333 nine hundred ninety-five thousand, three hundred thirty-three 4) 811,967 eight hundred eleven

### and so on (81,129,638,414,606,681,695,789,005,144,064) brad thiele 2011

and so on (81,129,638,414,606,681,695,789,005,144,064) brad thiele 2011 in zen they say: if something is boring after two minutes, try it for four. if still boring try it for eight, sixteen, thirty-two

### Write words with numbers

-1- Write words with numbers Write each as a numeral. 1) six twenty-four, five four 3) four, five thirty-seven 2) six ninety-nine, ninety-eight 4) three thirty, four eighteen 5) three, one ninety-six 6)

### Writing Numbers (A) Write each number in words. Math-Drills.com

2 020 Writing Numbers (A) 3 486 2 262 5 639 1 352 8 870 6 684 7 249 2 020 two thousand twenty Writing Numbers (A) Answers 3 486 three thousand four hundred eighty-six 2 262 two thousand two hundred sixty-two

### Place Value. Key Skills. Complete the daily exercises to focus on improving this skill.

Day 1 1 Write Four Million, Four Hundred and Twelve Thousand, Eight Hundred and Seventy One in digits 2 Write 6330148 in words 3 Write 12630 in words 4 Write 71643 in words 5 Write 28009 in words 6 Write

### Whole Number and Decimal Place Values

Whole Number and Decimal Place Values We will begin our review of place values with a look at whole numbers. When writing large numbers it is common practice to separate them into groups of three using

### ORDINANCE NO. 13625. AN ORDINANCE, repealing and superseding Ordinance No. 13403 adopted June

ORDINANCE NO. 13625 AN ORDINANCE, repealing and superseding Ordinance No. 13403 adopted June 14, 2012, and making appropriations for the current expenses of the District in the General Fund, the Water

### Exercise 4. Converting Numbers To Words And Words To Numbers. (This will help you to write cheques, stories and legal papers)

Exercise 4 Converting Numbers To Words And Words To Numbers. (This will help you to write cheques, stories and legal papers) At the end of this exercise you will: Be able to convert numbers to the written

### 118 One hundred Eighteen

1 2 3 4 5 6 7 8 9 10 ten 11 Eleven 12 Twelve 13 Thirteen 14 Fourteen 15 Fifteen 16 Sixteen 17 Seventeen 18 Eighteen 19 Nineteen 20 Twenty 21 Twenty 22 Twenty 23 Twenty 24 Twenty 25 Twenty 26 Twenty 27

### Can you spell these words?

a an as at am man if in is it sit tip pips on off of men peg pen can dad had back and big him his not got dot up mum but put us run I the to do into no go so 1 the this that then them will with went win

### Teacher Tam. Microsoft Word

Teacher Tam Microsoft Word Name Date Math Practice: Reading and Writing Numbers Directions: Write each number in written form. Example: 231 Two hundred thirty-one a. 19 b. 291 c. 9,215 d. 11,000 e. 114,378

### Four hundred and sixty eight 468. Six hundred and eighty one. Two hundred and sixty three 263. Four hundred and seventy five 475

Four hundred and sixty eight 468 Four hundred Six hundred and eighty one 681 One Two hundred and sixty three 263 Sixty Four hundred and seventy five 475 Seventy Three hundred and seventy two 372 Three

### One million, eight hundred forty-five thousand, twenty-seven dollars. 1, 8 4 5, 0 2 7

Section 1.1 Place Value Whole numbers appear in everyday situations. We encounter whole numbers in ATM machines when we withdraw money, conduct an inventory, carry out a census count, and when counting

### Numbers Galore using. Print. font

Numbers Galore! Numbers Galore using Print font 0 zero Draw Tally 0 0 zero 1 one Draw Tally 1 1 1 1 1 one one 2 two Draw Tally 2 2 2 2 two two 3 three Draw Tally 3 3 3 3 three 4 four Draw Tally 4 4 4 4

COPIA LOS SIGUIENTES NÚMEROS EN TU CUADERNO: 1.- one 2.-two 3.-three 4.-four 5.- five 6.- six 7.- seven 8.- eight 9.- nine 10.- ten 11.- eleven 12.- twelve 13.- thirteen 14.- fourteen 15.- fifteen 16.-

### PQRST 11 PUZZLE COMPETITION

PQRST PUZZLE COMPETITION PUZZLE 0 (0+0 points penalty for wrong answers) + points Wild Operations You re writing mathematical operations, using letters instead of digits to represent numbers. You take

### 1 One 2 Two 3 Three 4 Four 5 Five 6 Six 7 Seven 8 Eight 9 Nine The number you dialed cannot be reached at the moment; please try again later.

GXE5028 Voice Prompt Definition Version 2.0 System Prompt Number (For GS internal tracking) Speech Contents Other Language 0 Zero 1 One 2 Two 3 Three 4 Four 5 Five 6 Six 7 Seven 8 Eight 9 Nine The number

### Convert each problem to word form. Ex) seventy-three and two hundred seventy-three thousandths seven and five thousandths

Ex) 73.273 seventy-three and two hundred seventy-three thousandths 1) 7.005 seven and five thousandths 2) 983.53 nine hundred eighty-three and fifty-three hundredths 3) 26.17 twenty-six and seventeen hundredths

### Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how

### EduSahara Learning Center Assignment

06/05/2014 11:23 PM http://www.edusahara.com 1 of 5 EduSahara Learning Center Assignment Grade : Class VI, CBSE Chapter : Knowing Our Numbers Name : Words and Figures Licensed To : Teachers and Students

### Properties of Numbers

Properties of Numbers 1. 2. 3. 4. What is the next number? fifteen, twenty, twenty-five Is the next number in this sequence odd or even? Thirty, twenty-seven, twenty-four,... Write the number which comes

### Math in Focus Vocabulary. First Grade

Math in Focus Vocabulary First Grade Chapter Word Definition 1 zero 0 1 one 1 1 two 2 1 three 3 1 four 4 1 five 5 1 six 6 1 seven 7 1 eight 8 1 nine 9 1 ten 10 1 same equal to 1 more 4 eggs is more than

### A Short Course in Logic Example 8

A Short ourse in Logic xample 8 I) Recognizing Arguments III) valuating Arguments II) Analyzing Arguments valuating Arguments with More than one Line of Reasoning valuating If then Premises Independent

### Unit 2 Number and Operations in Base Ten: Place Value, Addition, and Subtraction

Unit 2 Number and Operations in Base Ten: Place Value, Addition, and Subtraction Introduction In this unit, students will review the place value system for reading and writing numbers in base ten. Students

### LEGISLATIVE BILL 310

LB 0 LB 0 LEGISLATURE OF NEBRASKA ONE HUNDRED THIRD LEGISLATURE FIRST SESSION LEGISLATIVE BILL 0 Introduced by Bolz,. Read first time January, 0 Committee: Business and Labor A BILL FOR AN ACT relating

### Permission is given for the making of copies for use in the home or classroom of the purchaser only.

Copyright 2005 Second Edition 2008 Teresa Evans. All rights reserved. Permission is given for the making of copies for use in the home or classroom of the purchaser only. Part 1 Math Card Games to Play

### The Use of Monte Carlo in Modern Portfolio Reality

WEALTHCARE CAPITAL MANAGEMENT: white paper The Use of Monte Carlo in Modern Portfolio Reality David B. Loeper, CIMA Chairman/CEO 600 East Main Street, Richmond, VA 23219 p 804.644.4711 f 804.644.4759 www.wealthcarecapital.com

### MUNICIPAL CODE OF THE TOWN OF NEENAH, WINNEBAGO COUNTY, WI CHAPTER 16 CONSTRUCTION AND EFFECT OF ORDINANCES

16.01 CONFLICT AND SEPARABILITY CONSTRUCTION AND EFFECT OF ORDINANCES (1) Conflict of Provisions. If the provisions of the different Chapters of this Code conflict with or contravene each other, the provisions

### Critical analysis. Be more critical! More analysis needed! That s what my tutors say about my essays. I m not really sure what they mean.

Critical analysis Be more critical! More analysis needed! That s what my tutors say about my essays. I m not really sure what they mean. I thought I had written a really good assignment this time. I did

### The Lobster Problem A Lesson for Fifth and Sixth Graders

The Lobster Problem A Lesson for Fifth and Sixth Graders by Carrie De Francisco and Marilyn Burns From Online Newsletter Issue Number 10, Summer 2003 Money is a useful model for helping students make sense

### Lecture 1: Asset Allocation

Lecture 1: Asset Allocation Investments FIN460-Papanikolaou Asset Allocation I 1/ 62 Overview 1. Introduction 2. Investor s Risk Tolerance 3. Allocating Capital Between a Risky and riskless asset 4. Allocating

### 4 9 7, 5 4 8, 6 0 1, 3 7 2.

1.1 Digits and Place Value 1. Understand Digits and Place Value Digits are mathematical symbols that are arranged in a specific order to represent numeric values. There are ten different digits in our

### The authors analyze the responses from 126 project manager interviews for general area-of-work and gender trends and report the results.

Lessons Learned from Interviewing Project Managers R. Anthony (Tony) MAI The Dow Chemical Company P.O. Box 8361 South Charleston, WV 25303 Eldon R. LARSEN Engineering and Computer Science Division College

### DEPARTMENT OF HEALTH CARE FINANCE & DEPARTMENT ON DISABILITY SERVICES PUBLIC NOTICE OF PROPOSED AMENDMENTS

DEPARTMENT OF HEALTH CARE FINANCE & DEPARTMENT ON DISABILITY SERVICES PUBLIC NOTICE OF PROPOSED AMENDMENTS Home and Community-Based Services Waiver for Persons with Intellectual and Developmental Disabilities

### Grade. Place Value II. By Monica Yuskaitis

rd 3 Grade Place Value II By Monica Yuskaitis California Standard Addressed Number Sense Third Grade 1.1 Count, read, and write whole numbers to 10,000. California Standard Addressed Number Sense Third

### Test One. 20p ½ litres p 16 9: How many twenty pence pieces are there in two pounds?

Test One twenty pence pieces are there in two pounds? 0p Imagine a triangular prism. edges does it have? Subtract thirty from one hundred and twenty. 0 The sequence of numbers follows the rule double and

### 16 Questions Sales Managers Must Ask

16 Questions Sales Managers Must Ask Here are 16 critical questions sales managers should learn to ask their salespeople about any pending sale. If managers make a habit of asking these questions during

### Place Value of Whole Numbers

Chapter 1 Review of Basic Arithmetic 3 1.1 Place Value of Numbers and Rounding Numbers Place value is the value of a digit as determined by its position in a number, such as ones, tens, hundreds, etc.

### Summary: Presenter: Start Time: Duration: Winifred Robinson: Les Matheson, Chief Executive of personal and business banking at RBS and NatWest:

Summary: Interview with Les Matheson, Chief Executive of Personal and Business Banking at RBS and NatWest. Presenter: Winifred Robinson Start Time: 12:18:15 Duration: 11 minutes Ministers have written

### NEW STUDENT SURVEY REPORT FOR FALL 2008 COHORT

NEW STUDENT SURVEY REPORT FOR FALL 2008 COHORT of NEW and TRANSFER STUDENTS Introduction The New Student Survey has been administered to new and new transfer students in all twelve Connecticut Community

### NS3-1: Place Value Ones, Tens, and Hundreds page 33

NS3-1: Place Value Ones, Tens, and Hundreds page 33 1. Write the place value of the underlined digit. a) 17 ones b) 98 REMEMBER: 3 7 5 c) 24 d) 63 e) 381 f) 972 g) 457 h) 79 i) 261 j) 8 2. Give the place

### A Short Course in Logic Zeno s Paradox

1 Grappling with Good Arguments A Short Course in Logic Zeno s Paradox We ve seen that if we decide that an argument is good then we should be inclined to believe that the ultimate conclusion is true.

### Club Accounts. 2011 Question 6.

Club Accounts. 2011 Question 6. Anyone familiar with Farm Accounts or Service Firms (notes for both topics are back on the webpage you found this on), will have no trouble with Club Accounts. Essentially

### Level I Math Black Line Masters Toolkit

Level I Math Black Line Masters Toolkit NSSAL (Draft) C. David Pilmer 2012 (Last Updated: February, 2013) This resource is the intellectual property of the Adult Education Division of the Nova Scotia Department

### NECESSARY AND SUFFICIENT CONDITIONS

Michael Lacewing Personal identity: Physical and psychological continuity theories A FIRST DISTINCTION In order to understand what is at issue in personal identity, it is important to distinguish between

### What is the wealth of the United States? Who s got it? And how

Introduction: What Is Data Analysis? What is the wealth of the United States? Who s got it? And how is it changing? What are the consequences of an experimental drug? Does it work, or does it not, or does

### Mental Mathematics Test Questions Here are previously asked SATS Mental Maths questions for you to practice (enjoy ) 5-Second Questions

Mental Mathematics Test Questions Here are previously asked SATS Mental Maths questions for you to practice (enjoy ) 5-Second Questions A. Money How many fifty-pence pieces in four pounds? How many twenty-pence

### Beating Roulette? An analysis with probability and statistics.

The Mathematician s Wastebasket Volume 1, Issue 4 Stephen Devereaux April 28, 2013 Beating Roulette? An analysis with probability and statistics. Every time I watch the film 21, I feel like I ve made the

### Formula for Theoretical Probability

Notes Name: Date: Period: Probability I. Probability A. Vocabulary is the chance/ likelihood of some event occurring. Ex) The probability of rolling a for a six-faced die is 6. It is read as in 6 or out

### The Three-Store Problem A Lesson with Fifth Graders

The Three-Store Problem A Lesson with Fifth Graders by Marilyn Burns From Online Newsletter Issue Number 18, Summer 2005 The December 2003 issue of Teaching Children Mathematics (volume 10, number 4) contained

### 1Targeting 2. 4Analysis. Introducing Marketing Automation. Best Practices for Financial Services and Insurance Organizations.

Introducing Marketing Automation Best Practices for Financial Services and Insurance Organizations 5 Marketing Technology 1Targeting 2 Engagement 4Analysis 3 Conversion 1 Marketing Automation = Marketing

### Retaining Teachers: The Principal as Motivating Factor

Retaining Teachers: The Principal as Motivating Factor Lawrence Allen Jr. People have asked me why, in my inner-city school of twelve hundred children 80 percent of them impoverished deemed in many eyes

### Mammon and the Archer

O. H e n r y p Mammon and the Archer OLD ANTHONY ROCKWALL, WHO HAD MADE millions of dollars by making and selling Rockwall s soap, stood at a window of his large Fifth Avenue house. He was looking out

### THE OPTIMIZER HANDBOOK:

THE OPTIMIZER HANDBOOK: LEAD SCORING Section 1: What is Lead Scoring? Picture this: you re selling vehicles, and you have a list that features over two hundred different leads, but you re only allowed

### Encoding Text with a Small Alphabet

Chapter 2 Encoding Text with a Small Alphabet Given the nature of the Internet, we can break the process of understanding how information is transmitted into two components. First, we have to figure out

### Module 1: The Career Planning Process: An Overview Transcript

Module 1: The Career Planning Process: An Overview Transcript Introduction (video clip 1) Hello, everyone. I m Jennifer from the Boise State University Career Center, and you re completing the first in

### Exploring Ones, Tens, and Hundreds with Base Ten Blocks A Lesson for Third Graders

Exploring Ones, Tens, and Hundreds with Base Ten Blocks A Lesson for Third Graders by Maryann Wickett and Marilyn Burns From Online Newsletter Issue Number 19, Fall 2005 In this lesson, excerpted from

### Proving the Value of Library Collections Part II: An Interdisciplinary Study Using Citation Analysis

Purdue University Purdue e-pubs Charleston Library Conference Proving the Value of Library Collections Part II: An Interdisciplinary Study Using Citation Analysis Amalia Monroe-Gulick University of Kansas

### How to Pass Physics 212

How to Pass Physics 212 Physics is hard. It requires the development of good problem solving skills. It requires the use of math, which is also often difficult. Its major tenets are sometimes at odds with

### NBT4-1 Place Value Ones, Tens, Hundreds, Page 24

NBT4-1 Place Value Ones, Tens, Hundreds, Page 24 and Thousands STANDARDS 4.NBT.A.2 Goals Students will identify the place value of digits in 2-, 3-, and 4-digit numbers. Vocabulary hundreds ones place

### Place Value of Whole Numbers (Through 999,999,999)

Place Value of Whole Numbers (Through 999,999,999) 1. Mercury is thirty-five million, nine hundred eighty-three thousand, ninety-five miles from the sun. Which is another way to write this number? A. 35,983,000,095.

### Which Planned Gift Gives Money To Charity Now? Friday, April 25, 2008, www.onphilanthropy.com By Jonathan Gudema, Esq.

Which Planned Gift Gives Money To Charity Now? Friday, April 25, 2008, www.onphilanthropy.com By Jonathan Gudema, Esq. Understanding Charitable Lead Trusts Which planned gift gives money to charity now?

### Guide to GUIDE TO TRAVEL HACKING WITH ALASKA MILES - INTENTIONAL TRAVELERS

Guide to 1 Who is this Guide for? I created this guide for the average person who simply wants to do more traveling without breaking the bank. It will be especially helpful for people who have an international

### An Innocent Investigation

An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

### Representing and Renaming Whole Numbers

Module 3 Representing and Renaming Whole Numbers Diagnostic...3 Representing Numbers to 10 000...7 Renaming Numbers to 10 000...15 Representing Numbers to 100 000...20 Renaming Numbers to 100 000...26

### LESSON 7. Leads and Signals. General Concepts. General Introduction. Group Activities. Sample Deals

LESSON 7 Leads and Signals General Concepts General Introduction Group Activities Sample Deals 330 More Commonly Used Conventions in the 21st Century General Concepts This lesson covers the defenders conventional

### UNITED STATES DISTRICT COURT -- SOUTHERN DISTRICT OF FLORIDA. Notice of Settlement of Class Action

UNITED STATES DISTRICT COURT -- SOUTHERN DISTRICT OF FLORIDA Notice of Settlement of Class Action If you performed at Scarlett's of Hallandale, Scarlett's of Ybor Strip or Scarlett's of Toledo as an Exotic

### HART RESEARCH ASSOCIATES Study #11127--page 1

HART RESEARCH ASSOCIATES Study #11127--page 1 1724 Connecticut Avenue, NW Interviews: 1,355 adults Washington, DC 20009 Dates: April 8-14, 2014 (202) 234-5570 FINAL Study #11127 48 Male 52 Female [109]

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### SESSION OF 1963. Act Nos. 180-181-182 329

SESSION OF 1963. Act Nos. 180-181-182 329 authorities fixing the tax rate for any year at a mill rate need include a statement expressing the rate of taxation in dollars and cents on each one hundred dollars

### Math Matters: Why Do I Need To Know This?

Math Matters: Why Do I Need To Know This? Bruce Kessler, Department of Mathematics Western Kentucky University Episode Fourteen 1 Annuities Investment strategies Objective: To illustrate how knowing the

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### CHAPTER 1. Compound Interest

CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.

### FOR. 14 Recommendations from a Top Futures Broker. Stuart A. Vosk. Center for Futures Education, Inc.

BASIC TRAINING FOR FUTURES TRADERS: 14 Recommendations from a Top Futures Broker Stuart A. Vosk Center for Futures Education, Inc. P.O. Box 309 Grove City, PA 16127 Tel.: (724) 458-5860 FAX: (724) 458-5962

### Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

### Massachusetts Institute of Technology Sloan School of Management System Dynamics II: Applications of System Dynamics. Professor Jim Hines

Massachusetts Institute of Technology Sloan School of Management System Dynamics II: Applications of System Dynamics Professor Jim Hines Guidelines for kickoff meeting and first breakout Group presentation:

### The Pain Free Guide to Cold Calling

The Pain Free Guide to Cold Calling The Pain Free Guide to Cold Calling - Introductory Notes This document is based on my experience doing telemarketing myself and teaching it to others. Today I advise

### Decree of Arab Republic Of Egypt By law no. (99) for year 2012 For Final account of the State s General Budget For The Fiscal Year 2011/2012

Decree of Arab Republic Of Egypt By law no. (99) for year 2012 For Final account of the State s General Budget For The Fiscal Year 2011/2012 President Of The Arab Republic Of Egypt And Constitution Declaration

### 1 Uncertainty and Preferences

In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package

### MATH 105: PRACTICE PROBLEMS FOR SERIES: SPRING Problems

MATH 05: PRACTICE PROBLEMS FOR SERIES: SPRING 20 INSTRUCTOR: STEVEN MILLER (SJM@WILLIAMS.EDU).. Sequences and Series.. Problems Question : Let a n =. Does the series +n+n 2 n= a n converge or diverge?

### ARE STOCK PRICES PREDICTABLE? by Peter Tryfos York University

ARE STOCK PRICES PREDICTABLE? by Peter Tryfos York University For some years now, the question of whether the history of a stock's price is relevant, useful or pro table in forecasting the future price

### Repair Your Credit. Step1: Start where you are. Rebuilding Good Credit

REPAIR YOUR CREDIT A free publication provided by Consolidated Credit Counseling Services of Canada, Inc., a registered charitable credit counselling and debt management organization. Consolidated Credit

### MINNESOTA STATE LOTTERY GAME PROCEDURES BINGO PROGRESSIVE PRINT-N-PLAY LOTTO GAME ADOPTED: AUGUST 23, 2010 AMENDED: SEPTEMBER 8, 2010

MINNESOTA STATE LOTTERY GAME PROCEDURES BINGO PROGRESSIVE PRINT-N-PLAY LOTTO GAME ADOPTED: AUGUST 23, 2010 AMENDED: SEPTEMBER 8, 2010 MINNESOTA STATE LOTTERY GAME PROCEDURES BINGO PROGRESSIVE PRINT-N-PLAY

### Race to 20. Game 9. Overview. Related Lessons. Key Questions. Time. Materials. Extension

Game 9 Race to 20 140 Time 20 minutes Materials dice, 1 per pair of students double ten-frame (Reproducible D), 1 per student counters, 25 each of two colors per student Extension dice with a small sticker

### The Truth About Medicare Supplemental Policies

The Truth About Medicare Supplemental Policies By David Belk MD If you have Medicare, or are about to get Medicare, then you re probably getting a barrage of phone calls and mail from insurance companies

### IN A SMALL PART OF THE CITY WEST OF

p T h e L a s t L e a f IN A SMALL PART OF THE CITY WEST OF Washington Square, the streets have gone wild. They turn in different directions. They are broken into small pieces called places. One street

### Mike: Alright welcome to episode three of Server Talk, I m here with Alexey. I m Mike. Alexey, how are things been going, man?

Mike: Alright welcome to episode three of Server Talk, I m here with Alexey. I m Mike. Alexey, how are things been going, man? Alexey: They re doing pretty good. Yeah, I don t know, we ve launched two

### Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

### Effectiveness and Satisfaction Survey Florida Nursing Resource Center

Effectiveness and Satisfaction Survey Florida Nursing Resource Center Centralized Clinical Placement System (CCPS) December 2008 Prepared for: Nursing Consortium of South Florida, Inc. South Miami, Florida

### The equivalence of logistic regression and maximum entropy models

The equivalence of logistic regression and maximum entropy models John Mount September 23, 20 Abstract As our colleague so aptly demonstrated ( http://www.win-vector.com/blog/20/09/the-simplerderivation-of-logistic-regression/

### MINIMIZING THE SOCIAL SECURITY WORKERS COMPENSATION OFFSET. James D. Leach 1

MINIMIZING THE SOCIAL SECURITY WORKERS COMPENSATION OFFSET James D. Leach 1 This article was published in The Practical Litigator in 1994. At the request of WILG, I updated it to take account of any changes

### R E PA I R YOUR CREDIT

R E PA I R YOUR CREDIT A free publication provided by Consolidated Credit Counseling Services of Canada, Inc., a registered charitable credit counselling and debt management organization. Consolidated

### BBN MEDIA Television and Radio Transcription Service Email: info@bbn.ie Tel: 01 4911935

BBN MEDIA Television and Radio Transcription Service Email: info@bbn.ie Tel: 01 4911935 TRANSCRIPT Station: RTE Radio One Programme: Drivetime Date: 09.02.12 Time: 15 minutes Ref: The Law Society Presenter

Sell Your House in DAYS Instead of Months No Agents No Fees No Commissions No Hassle Learn the secret of selling your house in days instead of months If you re trying to sell your house, you may not have