WEBVTT
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In this lecture, we will deal
with a single topic.
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How to find the distribution,
that is, the PMF or PDF of a
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random variable that is defined
as a function of one
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or more other random variables
with known distributions.
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Why is this useful?
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Quite often, we construct a
model by first defining some
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basic random variables.
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These random variables usually
have simple distributions and
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often they are independent.
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But we may be interested in the
distribution of some more
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complicated random variables
that are defined in terms of
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our basic random variables.
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In this lecture, we will develop
systematic methods for
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the task at hand.
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After going through a warm-up,
the case of discrete random
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variables, we will see that
there is a general, very
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systematic 2-step procedure
that relies on cumulative
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distribution functions.
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We will pay special attention
to the easier case where we
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have a linear function of a
single random variable.
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We will also see that when
the function involved is
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monotonic, we can bypass CDFs
and jump directly to a formula
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that is easy to apply.
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We will also see an example
involving a function of two
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random variables.
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In such examples, the
calculations may be more
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complicated but the basic
approach based on CDFs is
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really the same.
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Let me close with
a final comment.
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Finding the distribution of the
function g of X is indeed
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possible, but we should only do
it when we really need it.
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If all we care about is the
expected value of g of X we
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can just use the expected
value rule.