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In this lecture, we
introduce the
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notion of a random variable.
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A random variable is, loosely
speaking, a numerical quantity
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whose value is determined by the
outcome of a probabilistic
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experiment.
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The weight of a randomly
selected
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student is one example.
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After giving a general
definition, we will focus
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exclusively on discrete
random variables.
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These are random variables that
take values in finite or
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countably infinite sets.
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For example, random variables
that take
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integer values are discrete.
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To any discrete random variable
we will associate a
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probability mass function, which
tells us the likelihood
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of each possible value of
the random variable.
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Then we will go over a few
examples and introduce some
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common types of random
variables.
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And finally we will introduce
a new concept- the expected
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value of the random variable,
also called the
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expectation or mean.
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It is a weighted average of
the values of the random
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variable, weighted according
to their respective
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probabilities, and has an
intuitive interpretation as
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the average value we expect to
see if we repeat the same
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probabilistic experiment
independently a
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large number of times.
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Expected values play a central
role in probability theory.
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We will look into some
of their properties.
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And we will also calculate the
expected values of the example
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random variables that we
will have introduced.