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30 Classification of States: In a Markov chain, each state can be placed in one of the three classifications.1 Since each state falls into one and only one category, these categories partition the states. The secret of categorizing the states is to find the communicating classes. The states of a Markov chain can be partitioned into these communicating classes. Two states communicate if and only if it is possible to go from each to the other. That is, states A and B communicate if and only if it is possible to go from A to B and from B to A.

Date Added: 8/6/2008 | Visits: 32220

17 Stirling Numbers of the Second Type: Again: a partition of n objects is a division of these objects into separate classes. Each object must be in one and only one class and partitions with empty classes are not allowed.

Date Added: 8/6/2008 | Visits: 32674

28 Bayes' Rule: Bayes' rule is a probability benchmark. Once you understand it, it indicates that you have moved on to a new level. Suppose that we have partitioned events into mutually exclusive and exhaustive cases E1, E2, ..., En. That is, exactly one of E1, E2, through, En will occur. Ei might be the weather (such as temperature will be between 10 and 30 degrees). Suppose also, that for any Ei we know the probability of X (which might be the event that the Raiders win). That is,

Date Added: 8/6/2008 | Visits: 30532

27 Binary Arithmetic: In the previous section we looked at the binomial distribution. The binomial distribution is essentially the mathematics of repeatedly flipping a coin (and there is no requirement that the coin be unbiased). These coin flips are known as Bernoulli trials. Anytime you repeat an experiment with two possible outcomes and with your experiments independent of each other, you are performing Bernoulli trials. Frequently we desire to simulate such experiments on the computer.1

Date Added: 8/6/2008 | Visits: 31025

24 The Bonferroni Inequality: 1By far its most useful application is in joint confidence intervals. The inequality gives you a confidence interval without assuming independence of the various parameters. It usually turns out at around 95% confidence that the confidence region isnít much smaller than with the assumption of independence.

Date Added: 8/6/2008 | Visits: 32425

21 PIE: The law of addition of Section 20 is a probabilistic interpretation of the principal of inclusion and exclusion. In general, the problem of counting things is to count each object once and only once. Suppose we have N objects, some of which have property a and some which have property b and some of which have both properties.

Date Added: 8/6/2008 | Visits: 32342

20 Probability and the Law of Addition: We are interested in the probabilities of events. Whatever events are, they are not numbers. Therefore, given events A and B, if I speak of the event A + B, it does not mean A plus B. Rather it is shorthand for A or B. Elsewhere,

Date Added: 8/6/2008 | Visits: 32465

19 Boolean Algebra: It has been known for sometime in mathematics that set algebras and Boolean1 algebras are different perspectives on the same thing. The treatment of sets here is informal and is known as naive set theory. Most of the time naive set theory is sufficient for the purposes of even professional mathematicians.

Date Added: 8/6/2008 | Visits: 32169

18 Matrices: Matrix algebra is also known as linear algebra and it is one of the most important disciplines of mathematics. It pervades virtually all of mathematics, but it could be argued to be closest to geometry. In this chapter, we will look at the relation between matrices and graphs.

Date Added: 8/6/2008 | Visits: 32547

31 Geometric Series: Geometric series are a basic artifact of algebra that everyone should know.1 I am teaching them here because they come up remarkably often with Markov chains. The finite geometric series formula is at the heart of many of the fundamental formulas of financial mathematics. All students of the mathematical sciences should be intimately familiar with this topic and have all the formulas memorized.

Date Added: 8/6/2008 | Visits: 31692

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