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15 Counting

EBook Name 15 Counting
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Ebook Added On Added On: 6-Aug-2008
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EBook Category Category: Mathematics eBooks

EBook Description: Let us examine a real-life problem. As a university professor I feel that teaching twenty students for one hour a week is too much of a demand on my time and that it cuts into my research (on sour mash). Therefore, in order to cut back on my teaching load in future classes, I have decided to flunk most of my students this semester. Specifically, I am going to give one A, one B, one C, one D, and I'll flunk everyone else.

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Electronic Transactions on Numerical Analysis

its book of mathematics on numerical analysis.

16 Multinomials

Let us slightly generalize the problem of selecting r out of n objects without respect to order. Suppose, for example, that I have 20 students.

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.

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.

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.

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,

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.

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