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4 Algorithms: An algorithm is the abstract version of a computer program. In essence a computer
program is the codification of an algorithm, and every algorithm can be coded as a computer
program. Date Added: 8/6/2008  Visits: 30970 
29 Markov Chains: Markov chains are one of the most fun tools of probability; they give a lot of power for
very little effort. We will restrict ourselves to finite Markov chains. This is not much of a
restriction. We get much of the most interesting cases and avoid much of the theory.1 This
chapter defines Markov chains and gives you one of the most basic tools for dealing with them:
matrices. Date Added: 8/6/2008  Visits: 30761 
Introduction to Methods of Applied Mathematics: For the past few years I have been working on an open source textbook. It contains material on calculus, functions of a complex variable, ordinary differential equations, partial differential equations and the calculus of variations. The text is still under development, but I believe that the current version will be useful for students and instructors. Date Added: 8/6/2008  Visits: 30044 
37 A Little Game Theory: Game theory is one of the most interesting topics of discrete mathematics. The principal
theorem of game theory is sublime and wonderful. We will merely assume this theorem and use
it to achieve some of our early insights. To appreciate the theorem it is not necessary to know
the proof. Do not let any math pedant tell you otherwise.1 By a game mathematics refers to a
conflict between individuals (or entities) with conflicting goals. Date Added: 8/6/2008  Visits: 31079 
36 Poisson Processes: We are going to look at a random process that typifies what most of us think of as
random. This process has the added virtues that it is easy to work with and it is used a great deal
in mathematical modeling. In fact, it may be used more than any other process of its type. In
particular what I am talking about is the Poisson1 process which is described by both the Poisson
distribution and the exponential distribution. Date Added: 8/6/2008  Visits: 32545 
35 Mixed Chains: In this chapter we learn how to analyze Markov chains that consists of transient and
absorbing states. Later we will see that this analysis extends easily to chains with (nonabsorbing)
ergodic states. Date Added: 8/6/2008  Visits: 30188 
34 Ergodic Chains: Until this point in the book, results have been derived or proven, or at least the nature of
the proof has been indicated. However, proofs of the techniques of Sections 34, 35 and 36 as
well as the core concept of Chapter 37 are beyond this book and have been omitted. Do not let
this deter you. Date Added: 8/6/2008  Visits: 30643 
33 The Gambler's Ruin: There are many variations of the gambler's ruin problem, but a principal one goes like
this. We have i dollars out of a total of n. We flip a coin which has probability p of landing
heads, and probability q = 1 ! p of landing tails. If the coin lands heads we gain another dollar,
otherwise we lose a dollar. Date Added: 8/6/2008  Visits: 32605 
32 Averages: You should remember the arithmetic average. Given n data points, their arithmetic
average is their sum divided by n. Now suppose that we have the average of n numbers, An. We
are given a new data point x and we would like to compute the new average of all n + 1
numbers, An + 1. Many people simply add up all n + 1 numbers and then divide by n + 1.
However, Date Added: 8/6/2008  Visits: 32187 
23 The Law of Multiplication: Probability trees are one of the most useful
tools in probability and applied mathematics. We
will use them throughout the rest of this text. The
graph in Figure 1 is a probability tree. Date Added: 8/6/2008  Visits: 30477 
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