WEBVTT
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In the previous lecture we
introduced random variables,
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probability mass functions
and expectations.
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In this lecture we continue
with the development of
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various concepts associated
with random variables.
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There will be three
main parts.
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In the first part we define
the variance of a random
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variable, and calculate
it for some of our
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familiar random variables.
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Basically the variance is a
quantity that measures the
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amount of spread, or the
dispersion of a probability
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mass functions.
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In some sense, it quantifies the
amount of randomness that
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is present.
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Together with the expected
value, the variance summarizes
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crisply some of the qualitative
properties of the
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probability mass function.
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In the second part we discuss
conditioning.
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Every probabilistic concept or
result has a conditional
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counterpart.
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And this is true for probability
mass functions,
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expectations and variances.
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We define these conditional
counterparts and then develop
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the total expectation theorem.
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This is a powerful tool that
extends our familiar total
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probability theorem and allows
us to divide and conquer when
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we calculate expectations.
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We then take the opportunity
to dive deeper into the
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properties of geometric random
variables, and use a trick
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based on the total expectation
theorem to
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calculate their mean.
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In the last part we show how to
describe probabilistically
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the relation between multiple
random variables.
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This is done through a so-called
joint probability
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mass function.
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We take the occasion to
generalize the expected value
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rule, and establish a further
linearity property of
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expectations.
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We finally illustrate the power
of these tools through
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the calculation of the expected
value of a binomial
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random variable.