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Welcome to the first lecture
of this class.
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You may be used to having a
first lecture devoted to
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general comments and motivating
examples.
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This one will be different.
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We will dive into the heart
of the subject right away.
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In fact, today we will
accomplish a lot.
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By the end of this lecture, you
will know about all of the
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elements of a probabilistic
model.
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A probabilistic model is a
quantitative description of a
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situation, a phenomenon,
or an experiment
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whose outcome is uncertain.
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Putting together such a model
involves two key steps.
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First, we need to describe the
possible outcomes of the
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experiment.
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This is done by specifying
a so-called sample space.
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And then, we specify a
probability law, which assigns
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probabilities to outcomes or
to collections of outcomes.
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The probability law tells us,
for example, whether one
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outcome is much more likely
than some other outcome.
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Probabilities have to satisfy
certain basic properties in
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order to be meaningful.
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These are the axioms of
probability theory.
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For example probabilities
cannot be negative.
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Interestingly, there will be
very few axioms, but they are
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powerful, and we will see
that they have lots of
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consequences.
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We will see that they imply
many other properties that
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were not part of the axioms.
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We will then go through a couple
of very simple examples
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involving models with
either discrete
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or continuous outcomes.
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As you will be seeing many times
in this class, discrete
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models are conceptually
much easier.
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Continuous models involve some
more sophisticated concepts,
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and we will point out some of
the subtle issues that arise.
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And finally, we will talk a
little bit about the big
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picture, about the role of
probability theory, and its
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relation with the real world.