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Chapter 1 Discrete Probability Distributions 1.1 Simulation of Discrete Probabilities Probability In this chapter, we shall first consider chance experiments with a finite number of possible outcomes ω 1 , ω 2 , , ω n . For example, we roll a die and the possible outcomes are 1, 2, 3, 4, 5, 6 corresponding to the side that turns up. We toss a coin with possible outcomes H (heads) and T (tails). It is frequently useful to be able to refer to an outcome of an experiment. For example, we might want to write the mathematical expression which gives the sum of four rolls of a die. To do this, we could let X i , 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 + X 4 for the sum of the four rolls. The X i ’s are called random variables. A random vari- able 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 ω j a nonnegative number m(ω j ) in such a way that m(ω 1 )+m(ω 2 )+···+ m(ω 6 )=1. The function m(ω j ) 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) = 2 3 1 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 to suspect that the coin comes up one way more often than the other way, it is natural to assign the probability of 1/2 to each of the two outcomes. In both of the above experiments, each outcome is assigned an equal probability. This would certainly not be the case in general. For example, if a drug is found to be effective 30 percent of the time it is used, we might assign a probability .3 that the drug is effective the next time it is used and .7 that it is not effective. This last example illustrates the intuitive frequency concept of probability. That is, if we have a probability p that an experiment will result in outcome A, then if we repeat this experiment a large number of times we should expect that the fraction of times that A will occur is about p. To check intuitive ideas like this, we shall find it helpful to look at some of these problems experimentally. We could, for example, toss a coin a large number of times and see if the fraction of times heads turns up is about 1/2. We could also simulate this experiment on a computer. Simulation We want to be able to perform an experiment that corresponds to a given set of probabilities; for example, m(ω 1 )=1/2, m(ω 2 )=1/3, and m(ω 3 )=1/6. In this case, one could mark three faces of a six-sided die with an ω 1 , two faces with an ω 2 , and one face with an ω 3 . In the general case we assume that m(ω 1 ), m(ω 2 ), , m(ω n ) are all rational numbers, with least common denominator n.Ifn>2, we can imagine a long cylindrical die with a cross-section that is a regular n-gon. If m(ω j )=n j /n, then we can label n j of the long faces of the cylinder with an ω j , and if one of the end faces comes up, we can just roll the die again. If n = 2, a coin could be used to perform the experiment. We will be particularly interested in repeating a chance experiment a large num- ber of times. Although the cylindrical die would be a convenient way to carry out a few repetitions, it would be difficult to carry out a large number of experiments. Since the modern computer can do a large number of operations in a very short time, it is natural to Deterministic Probability - The Marble Game Example Use the information in the following setting to answer questions through 7: While in Romania last year I noticed a group of children playing a game with a set of marbles The rules seemed a bit complicated (especially because I did not speak the local language) but the game pieces were interesting enough to motivate this example There were 20 marbles in the set Six of the colors - red, blue, purple, orange, green, and brown - came in three patterns The possible patterns were solid, striped, and speckled; so there was one solid red, one striped red and one speckled red etc In addition, there was a single speckled grey marble as well as a solid black one Consider that all of these marbles were placed into a bag and one was selected at random Of interest is the color and/or pattern of the selected marble Note that this sampling scheme is called a Simple Random Sampling design (SRS) …but more on that later 1) Briefly describe the sample space (list all outcomes): 3) Find the probabilities of the following events: A) P(red) = B) P(red and speckled) = C) P(brown ∩ solid) = D) P(purple or blue) = E) P(grey ∪ speckled) = F) P(grey ∩ solid) = G) P(black or striped) = H) P( orange or solid) = I) P(green | striped) = J) P(striped | green) = K) P(solid | black) = 4) From within the results of parts 3A through 3K identify any “impossible” events 5) From within the results of parts 3A through 3K identify any “certain” events 6) Support or oppose the following and defend your choice in one or two sentences and appropriate calculations: “Color and pattern are independent” (extra credit) 7) Support or oppose the following and defend your choice in one or two sentences and appropriate calculations: “The events ‘grey’ and ‘solid’ are mutually exclusive” (extra credit) L E C T U R E 1 : I N T R O D U C T I O N T O P R O B A B I L I T Y Probability and Computer Science Logistic details MS. Quoc Le Dr. Van Khanh Nguyen Time: 7-8 weeks Textbook: A first course in probability (Sheldon Ross) Probability and Computing – Randomized Algorithm and Probabilistic Analysis (Mitzenmacher and Upfal) E-books for Both can be found @ gigapedia.com Introduction to Probability Mathematical tools to deal with uncertain events. Applications include: Web search engine: Markov chain theory Data Mining, Machine Learning: Data mining, Machine learning: Stochastic gradient, Markov chain Monte Carlo, Stochastic gradient, Markov chain Monte Carlo, Image processing: Markov random fields, Design of wireless communication systems: random matrix theory, Optimization of engineering processes: simulated annealing, genetic algorithms, Finance (option pricing, volatility models): Monte Carlo, dynamic models, Design of atomic bomb (Los Alamos): Markov chain Monte Carlo. Plan of the course Combinatorial analysis; i.e counting Axioms of probability Conditional probability and inference Discrete & continuous random variables Multivariate random variables Multivariate random variables Properties of expectation, generating functions Additional topics: Poisson and Markov processes Simulation and Monte Carlo methods Applications Combinatorial (Counting) Many basic probability problems are counting problems. Example: Assume there are 1 man and 2 women in a room. You pick a person randomly. What is the probability P1 that this is a man? What is the probability P1 that this is a man? If you pick two persons randomly, what is the probability P2 that these are a man and woman Answer: … Both problems consists of counting the number of different ways that a certain event can occur. Basic Principle of Counting Basic Principle of Counting: Suppose that two experiments are to be performed. Experiment 1 can result in any one of n1 possible outcomes For each outcome of experiment 1, there are n2 possible outcomes of experiment 2, Then there are n1 . n2 possible outcomes of the two experiments. Example: A football tournament consists of 14 teams, each of which has 11 players. If one team and one of its players are to be selected as team and player of the year, how many different choices are possible? Answer: 14 . 11 = 154 Generalized Principle of Counting Generalized Principle of Counting: If r experiments that are to be performed are such that the 1st one may result in any of n1 possible outcomes; and if, for each of these n1 possible outcomes, there are n2 possible outcomes of the 2nd experiment; and if, for each of the n1 n2 possible outcomes of the first two experiments, there are n3 possible outcomes of the 3rd experiment; and experiments, there are n3 possible outcomes of the 3rd experiment; and if , then there is a total of n1 x n2 x nr possible outcomes of the r experiments. Example: A university committee consists of 4 undergrads, 5 grads, 7 profs and 2 non-university persons. A sub-committee of 4, consisting of 1 person from each category, is to be chosen. How many different subcommittees are possible? Answer: 4 . 5 . 7 . 2 = 280 More examples Example: How many different 6-place license plates are possible if the first 3 places are to be occupied by letters and the final 3 by numbers. Example: How many different 6-place license plates are possible if the first 3 places are to be occupied by are possible if the first 3 places are to be occupied by letters, the final 3 by numbers and if repetition among letters were prohibited? repetition among numbers were prohibited repetition among both letters and numbers were prohibited? Permutations Example: Consider the acronym UBC. How many different ordered arrangements of the letters U, B and C are possible? 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I believe not The most composed and well-armed opponents win wars The same is true for traders In... instrument: the stock In fact, thousands of stocks are More free books @ www.BingEbook.com THE OPTION TRADER’S GUIDE TO PROBABILITY, VOLATILITY, AND TIMING TEAMFLY Team-Fly ® More free books @ www.BingEbook.com Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States. With offices in North America, Europe, Australia, and Asia, Wiley is globally commit- ted to developing and marketing print and electronic products and services for our customers’ professional and personal knowl- edge and understanding. The Wiley Trading Series features books by traders who have been able to work with the market’s ever changing temperament and prosper—some by reinventing systems, others by getting back to basics. For the novice trader, professional, or somewhere in between, these books will provide the advice and strategies needed to prosper today and well into the future. For a list of available titles, please visit our Web site at www.WileyFinance.com. More free books @ www.BingEbook.com A Marketplace Book THE OPTION TRADER’S GUIDE TO PROBABILITY, VOLATILITY, AND TIMING Jay Kaeppel John Wiley & Sons, Inc. More free books @ www.BingEbook.com Copyright © 2002 by Jay Kaeppel. All rights reserved. Published by John Wiley & Sons, Inc., New York. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, e-mail: PERMREQ@WILEY.COM. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional person should be sought. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products visit our Web site at www.wiley.com. ISBN: 0-471-22619-X Printed in the United States of America. 10987654321 More free books @ www.BingEbook.com In loving memory of Arthur H. Kaeppel More free books @ www.BingEbook.com More free books @ www.BingEbook.com Contents Chapter 1 Introduction 1 Who Can Benefit from This Book 1 What Sets This Book Apart 2 Can Options Really Be Simplified? 2 What This Book Provides 2 Overview of Option Trading 3 Understanding Risk by Using Risk Curves 6 Asking the Right Question 9 Analyzing Risk: What Separates the Winners from the Losers 11 Summary 18 Chapter 2 The Basics of Options 21 Option Definitions 21 Options on a Specific Security 26 Intrinsic Value versus Extrinsic Value 30 In-the-Money versus Out-of-the- Money Options 32 Summary 35 Chapter 3 Reasons to Trade Options 37 The Three Primary Uses of Options 37 Leveraging an Opinion on Market Direction 39 vii More free books @ www.BingEbook.com Hedging an Existing Position (and Generating Income) 41 Taking Advantage of Neutral Situations 44 Summary 46 Chapter 4 Option Pricing 47 Theoretical Value 47 Examples of Theoretical Option Pricing 49 Overvalued Options versus Undervalued Options 50 Summary: Theory versus Reality 52 Chapter 5 Time Decay 55 The Effect of Time Decay on the Price of an Option 56 Implications of Time Introduction to Probability Charles M. Grinstead Swarthmore College J. Laurie Snell Dartmouth College To our wives and in memory of Reese T. Prosser Contents 1 Discrete Probability Distributions 1 1.1 Simulation of Discrete Probabilities . . . . . . . . . . . . . . . . . . . 1 1.2 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . . . 18 2 Continuous Probability Densities 41 2.1 Simulation of Continuous Probabilities . . . . . . . . . . . . . . . . . 41 2.2 Continuous Density Functions . . . . . . . . . . . . . . . . . . . . . . 55 3 Combinatorics 75 3.1 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3 Card Shuffling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4 Conditional Probability 133 4.1 Discrete Conditional Probability . . . . . . . . . . . . . . . . . . . . 133 4.2 Continuous Conditional Probability . . . . . . . . . . . . . . . . . . . 162 4.3 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5 Distributions and Densities 183 5.1 Important Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.2 Important Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6 Exp ect ed Value and Variance 225 6.1 Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 6.2 Variance of Discrete Random Variables . . . . . . . . . . . . . . . . . 257 6.3 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . 268 7 Sums of Random Variables 285 7.1 Sums of Discrete Random Variables . . . . . . . . . . . . . . . . . . 285 7.2 Sums of Continuous Random Variables . . . . . . . . . . . . . . . . . 291 8 Law of Large Numbers 305 8.1 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . 305 8.2 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . 316 v vi CONTENTS 9 Central Limit Theorem 325 9.1 Bernoulli Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 9.2 Discrete Independent Trials . . . . . . . . . . . . . . . . . . . . . . . 340 9.3 Continuous Independent Trials . . . . . . . . . . . . . . . . . . . . . 356 10 Generating Functions 365 10.1 Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 365 10.2 Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 10.3 Continuous Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 11 Markov Chains 405 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 11.2 Absorbing Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . 416 11.3 Ergodic Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . 433 11.4 Fundamental Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 447 11.5 Mean First Passage Time . . . . . . . . . . . . . . . . . . . . . . . . 452 12 Random Walks 471 12.1 Random Walks in Euclidean Space . . . . . . . . . . . . . . . . . . . 471 12.2 Gambler’s Ruin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 12.3 Arc Sine Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Preface Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two prob- lems from games of chance. Problems like those Pascal and Fermat solved continued to influence such early researchers as Huygens, Bernoulli, and DeMoivre in estab- lishing a mathematical theory of probability. Today, probability theory is a well- established branch of mathematics that finds applications in every area of scholarly activity from music to physics, and in daily experience from weather prediction to predicting the risks of new medical treatments. This text is designed for an introductory probability course taken by sophomores, juniors, and seniors in mathematics, the physical and so c ial sciences,