Electronic Transactions on Numerical Analysis
its book of mathematics on numerical analysis.
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.
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,
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.
15 Counting - Free eBook 15 Counting - Download ebook 15 Counting free