Electronic Transactions on Numerical Analysis
its book of mathematics on numerical analysis.
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 isn’t 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.
18 Matrices  Free eBook 18 Matrices  Download ebook 18 Matrices free
