WEBVTT
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The definition of the
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conditional PDF is very simple.
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It is just this formula, which
is analogous to the one for
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the discrete case.
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In all respects--
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mathematical and intuitive--
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it is very similar to
the conditional PMF.
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Even so, developing a solid
grasp of this concept does
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take some further thinking,
so we will now make some
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observations that should be
helpful in this respect.
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The first and obvious
observation is that the
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conditional PDF is
non-negative.
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It's defined when the
denominator is positive, the
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numerator is a non-negative
quantity, so it's always a
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non-negative quantity.
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A more interesting observation
is that for any given value of
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little y, the conditional
PDF looks like a slice
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of the joint PDF.
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Indeed, if you fix the value
of little y, then the
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denominator in this definition
is a constant, and we have a
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function that varies with x the
same way that the joint
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PDF varies with x.
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Pictorially, let us consider
this particular joint PDF, and
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let this be the x-axis and
let that be the y-axis.
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If we fix a certain value of y,
if we condition on Y having
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taken this particular value so
that our universe is now this
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particular line, on that
universe the value of the
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denominator in this definition
is a constant, and the
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conditional PDF is going to vary
according to the height
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of the joint on that
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particular conditional universe.
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So the height of the joint, if
we trace it, is one of those
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curves up here, and [then]
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goes down.
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So it is really a slice taken
out of the joint PDF.
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If we condition on a different
y, we get a different slice of
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the joint PDF, and so on.
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Actually, the conditional
is not exactly
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the same as the slice.
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We also have this term on the
denominator that serves as a
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scaling factor.
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It turns out that this scaling
factor is exactly what we need
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for the conditional PDF, given
a specific value of little y,
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to integrate to 1.
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Indeed, if we fix little y and
take the integral over all
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x's, using the definition, and
because this term is a
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constant and does not involve
x, we only need to integrate
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the numerator.
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And we recognize that the
numerator corresponds to our
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earlier formula for the marginal
distribution--
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the marginal PDF of Y. From
the joint, this is how we
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recover the marginal PDF of Y.
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So the numerator turns out
to be the same as the
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denominator, and so
we get a ratio 1.
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Therefore, the conditional PDF
for a given value of the
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random variable Y behaves
in all respects
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like an ordinary PDF.
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It is non-negative and
it integrates to 1.
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A last observation is that we
can take this definition and
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move the denominator to the
other side to obtain this
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formula, which has the
familiar form of the
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multiplication rule.
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The probability of two events
happening is the probability
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of the first times the
probability of the second
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given the first, except that
here we're not really dealing
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with probabilities, we're
dealing with densities.
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By symmetry, a similar formula
must also be true when we
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interchange the roles of X
and Y. So, algebraically,
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everything is similar to what
we have seen for the case of
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discrete random variables.
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It's the same form of the
multiplication rule, although
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the interpretation is a bit
different because densities
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are not probabilities.