| |
Why do you need a course in Measure and Integration Theory?
|
It is difficult to learn the curriculum of this course. And it is difficult to
understand precisely the need for all the messy mathematical details -- especially if you
are taking the course to study an applied mathematical subject. This
document was written to try at least to explain why it is worth focusing on
learning the stuff.
In all situations where we need to learn new things we always have to
make a compromise between focusing on the motivation to learn and the actual
learning. To master chess, say, we need first to learn the movements of the pieces,
then how to combine the movements to play a sensible game, and then the really hard
part is to understand how to analyze the huge number of different situations you can
encounter in the game to make the right (winning) choices of moves.
The motivation for leaning is to play better chess. It is, however, impossible
to learn everything right away. We need to focus first on the movements of the
pieces and then gradually build up an understanding
for the entire game.
The motivation for learning mathematics, and in this case probability and measure and
integration theory, is much like learning chess. The overall goal is to be able
to master (or at least work reasonably well with) the concepts we use in
probability theory to solve a range of probabilistic questions. The type of
relevant questions depends on the applications in mind. Many of these
applications are really complicated, and just as it would be rather futile to
explain the details of a chess game played by masters to a rookie chess player,
it is futile to try to explain all the mathematical details in some of the
probabilistic applications to persons who are just about to learn the very
fundamental concepts involved.
This does not mean that we should not point out the directions we are going
and what we are aiming for. When we learn chess it is because we want to
be able to play it. So what are we aiming for?
Statistics: The fundamental question is statistics is how we use
observations (empirical data) to infer knowledge that go beyond the data.
This is based on building probabilistic models and applying mathematical
principles to decide how to transform the data into a model. To start doing so
we need the probability theory to formulate the models as well as the methods.
Econometrics is essentially the use of statistics to deal with questions raised in
economy by analysis of empirical data.
Finance: A fundamental question in finance is how to price a derivative
like an option. Another is how to put together a good portfolio of
investments. Satisfactory answers to these questions are based on probabilistic
models of the price fluctuations of a stock, say. To really understand the
models used, one needs to be able to understand probability measures on "quite
wild" sets, namely the sets of sample paths for the price over time. These measures
are out of reach in the current course, but will be handled in later courses.
One part of actuarial mathematics also deals a lot with "financial math"
questions, but also more classical things as the modeling of occurrences of
events, like car accidents or burglary, over time are very important, so that
one can calculate a reasonable price for the insurance -- and so that we can understand the risk
that the company is taking.
We are clearly aiming for something -- like the ability to win when playing
chess! The course(s) Probability Theory 1 or Measure and Integration Theory
correspond to learn fundamentally how to play good moves in a game of chess. We learn from the
masters how to build up the theory, just like we would learn from the
masters how to make good moves. It takes time and hard work to appreciate the
the fundamental concepts, but it is definitely better to be guided by the
existing expert knowledge then to try to deduce everything all over again.
And what is perhaps more important, the understanding of the arguments for a good
and a bad move in chess is not just whether we win or loose the game. Of course
that is the aim, but in the choice of a move you need to analyze the game, what
is the current position, how can I improve my position etc., but we cannot
predict the outcome of the game. Arguments for or against different moves are
very difficult to understand, and depend on an analysis of the game.
If you were supposed to play an average game of chess, then maybe it is not
worth focusing so much on all the messy details, but instead on the bigger
picture. This may result in becoming a reasonable player in a shorter
time. But eventually, if we want to improve our abilities, we need to focus
on the messy details more carefully and become better at analyzing the game.
This is like probability theory. You have some prior knowledge and intuition,
and it is the decision of the study board that now is the time to study some of the messy
details. It is also a decision that the educations in statistics,
mathematical economy and actuarial sciences should educate people to a high
level on the mathematical details to produce candidates in the end that are
able to carry out their own analysis of probabilistic models, say, and that are
able to develop new models themselves.
So based on these consideration I find that you should focus on the how rather
than the why. The why is above. You need to be able to work with probability
theory. Period. In this course, I teach you how to do that. I teach you the
fundamental concepts. What is a probability, how can I compute with
probabilities, what are the standard probability measures in use. It happens
so that these questions are mathematically intertwined with questions such
as, what is the area of a set, how do I compute with areas, volumes etc. and
how do I integrate functions. Therefore we have one course covering both the
general measure and integration theory and the fundamental probability
theory. Personally I find the theory to very nice and complete, and once you
learn more about it you will also learn to appreciate it I am sure.
|
| |
