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20 Probability and the Law of Addition



EBook Name20 Probability and the Law of Addition
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EBook Description: 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,


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Electronic Transactions on Numerical Analysis

its book of mathematics on numerical analysis.

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.

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.

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.

25 Sigma Notation

The sign 3, called sigma, is a glorified symbol for addition that can be quite useful. It is utilized throughout mathematics, statistics, computer science and all other mathematical disciplines. With the 3 there is usually an index that typically is an i or j.

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

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,

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.

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.

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.

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