WEBVTT
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By now, we have introduced
all sorts of PMFs for
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the discrete case.
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The joint PMF, the
conditional PMF--
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given an event--
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and the conditional PMF of one
random variable given another.
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And we're moving along with
the program of defining
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analogous concepts for
the continuous case.
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We have already discussed the
joint PDF and the conditional
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PDF, given an event.
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The next item in our menu is to
define a conditional PDF of
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one random variable, given
another random variable.
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We proceed by first looking
at the definition for the
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discrete case.
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A typical entry of the
conditional PMF is just a
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conditional probability, but
in different notation.
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And using the definition of
conditional probabilities,
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this is equal to the ratio of
the joint divided by the
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probability of the conditioning
event.
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Unfortunately, in the continuous
case, a definition
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of this form would be
problematic, because the event
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that Y takes on a specific value
is an event that has 0
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probability.
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And we know that we cannot
condition on a
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0 probability event.
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However, we can take this
expression as a guide on how
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to define a conditional PDF
in the continuous case.
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And this is the definition,
which just mimics the formula
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that we have up here.
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Notice that this conditional
PDF-- defined this way-- is
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well defined, as long
as the denominator
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is a positive quantity.
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Let us now try to make sense
of this definition.
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Let us first recall the
interpretation of the
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conditional PDF, given an event,
A, that has positive
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probability.
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We know that the PDF is used to
determine the probability
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of a small interval.
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And similarly, the conditional
PDF is used to calculate the
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conditional probability of a
small interval given the
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conditioning event.
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We would like to do something
similar for the conditional
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PDF, where we would like to
take the event A to be
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something like the event that Y
is equal to some particular
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value-- little y.
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But as we said, this is
problematic, because this
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event does not have positive
probability.
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So instead, we can take A to
be the event that Y is
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approximately equal to
a certain value.
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So we're dealing with a little
interval around this value,
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little y, which in general would
be an event of positive
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probability.
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And we can try to have a
similar interpretation.
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Let us see how this works out.
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So what does it mean that Y is
approximately equal to some
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particular value, little y?
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We interpret that as follows.
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We're told that the random
variable, Y, takes a value
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that is within epsilon--
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where epsilon is a
small number--
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of a given value, little y.
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And given this conditioning
information, we want to
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calculate the probability
of a small interval.
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How do we do that?
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Well, here--
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because this, in general,
will be a
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positive probability event--
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we can use the definition of
conditional probabilities.
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And it would be equal to the
probability of both events
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happening, divided by the
probability of the
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conditioning event.
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What is the probability of
both events happening?
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This is a probability of a small
rectangle in xy space.
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At that rectangle, the joint
PDF, has a certain value.
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And because we're integrating
over that rectangle--
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and that rectangle has
dimensions delta and epsilon--
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of that probability, that
small rectangle, is
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approximately equal to this.
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Then we need the denominator,
which is the probability of
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the conditioning event.
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And this is approximately equal
to the density of Y
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evaluated at that point,
times the length
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of the small interval.
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We cancel the epsilons.
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And then we notice that the
ratio we have here is what we
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defined as the conditional
PDF.
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So we get this relation
times delta.
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So what do we see?
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We see that the probability of a
small interval is equal to a
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PDF times the length of
the small interval.
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However, because we are
conditioning on Y being
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approximately equal to a certain
value, we end up using
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a corresponding conditional PDF,
where the conditional PDF
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is defined this way.
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So we now have an interpretation
of the
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conditional PDF in terms of
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probabilities of small intervals.
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Now that we have an intuitive
interpretation of the
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conditional PDF, we can also
use it to calculate
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conditional probabilities
of more general
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events, not just intervals.
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And we do this as follows.
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In general, for continuous
random variables, we can find
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the probability that X belongs
to a certain set by
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integrating a PDF
over that set.
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Because here we're dealing with
a conditional situation
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where we're given the value of
Y, we use the conditional PDF
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instead of the true PDF.
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And this way, we calculate the
conditional probability.
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Now, the difficulty is that this
conditional probability
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is not a well-defined quantity
according to what we did early
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on in this class.
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We cannot condition on zero
probability events.
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But we can get the around this
difficulty as follows.
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This quantity is well-defined.
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And we can use this quantity
as the definition of this
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conditional probability.
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And so we have managed to
provide definition of
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conditional probabilities, given
a 0 probability event of
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a certain type.
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It turns out that this
definition is sound and
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consistent with everything
else that we are doing.
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But when we're dealing with
particular problems and
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applications, we can generally
forget about all of these
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subtleties that we have
been discussing here.
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The bottom line is
that we will be
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treating conditional PDFs--
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given the value of a random
variable, Y--
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just as ordinary PDFs, but given
the information that
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this random variable took
on a specific value.
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And in that conditional
universe, we will calculate
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probabilities the usual way,
by using conditional PDFs
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instead of ordinary PDFs.