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Week 47
Measures and Integrals
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Monday Lecture
In the first lecture we introduce how probablity theory and the
modeling of experiments and observations can be phrased using
abstract measure theory (Chapter 13). A probability measure is
simply a measure whose total mass is 1. Probability theory can thus
be viewed as a special case of general measure theory, but
it gets its distinct flavor from the interpretations. We
present the frequentistic and the Bayesian interpretations of
probability measures. A link between the probabilistic concept of
expectations and the concept of integrals is
established.
We also introduce the transformations or image
measures, Section 10.1, which will play a central role. As we
will see, this construction is the key to understanding how
probability models transform either by our computations with data or
by our limited ability to make observations.
The ContaminatedSoil case (not in the book) on
quantitative risk assessment and decision making based on
probability theory is introduced, and this case will follow us
throughout the course as we develop the necessary theory
to gradually solve questions and expand on various aspects of the case.
To prepare yourself you should read Chapter 2 as a brush up on the
fundamental concepts and results that were taught on An2. We
will only briefly cover aspects of this chapter in the lecture, but
you will also work with Chapter 2 material in the exercise
class.
In addition, it will be a really good idea to consult the appendices A, B
and C for background material on sets, countability and properties
of the real numbers that we will use in the course.
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Monday Exercises
2.2, 2.3, 2.7, 2.22
2.22 is relatively technical, but it is a quite good exercise for
training fundamental concepts like nullsets and sigma-algebras.
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Wednesday Lectures
Expectations are integrals, and we show the abstract result on
integration w.r.t. image measures in Section 10.3. The image of a
probability measure is the probability distribution obtained by a
transformation, e.g. a squaring, of the observable. Theorem
10.8 gives the abstract formula for how integrals and measures
transform.
Probability measures, and other measures for that matter, are often
given in terms of a density (Chapter 11). Typically in applications the
density is w.r.t. the Lebesgue measure. In this case many practical
computations with probability measures can be carried out by more or
less standard Riemann integration techniques.
In the lectures we develop the theory for integration w.r.t. a
measure given by a density (Chapter 11) and how we transform measures given by a
density w.r.t. the Lebesgue measure (Section 12.2).
We skip Lemma 11.5 but Lemma 12.4 (in Section 12.1) is also covered
this week.
The first assignment is made available on Absalon.
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Wednesday Exercises
10.1, 10.2, 10.3, 13.1, 13.2, 13.3
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