WEBVTT
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Besides PMFs and PDFs, we can
also describe the distribution
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of a random variable, as
we know, using a CDF.
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A CDF is always well-defined.
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And for the case of a continuous
random variable,
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the CDF can be found by
integrating the PDF.
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And conversely, we can recover
the PDF from the CDF by
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differentiating.
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There is something similar that
happens for the case of
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multiple random variables,
as well.
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We can define the joint CDF as
the probability that X and Y,
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the pair X-Y, takes values
that are below certain
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numbers, little x
and little y.
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So we are talking about the
probability of the blue set in
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this diagram.
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This probability can be found
by integrating the joint PDF
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over the blue set.
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And, since we're using x and y
to be some specific numbers,
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let us use some different dummy
variables to carry out
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the integration.
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What is the range of
the integration?
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The first variable, which is s
in this integral, ranges from
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minus infinity up to x.
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And the other variable, which
is the one that we're
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integrating with respect to,
in the outer integral--
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the t variable--
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ranges from minus
infinity to y.
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Now, let us see what happens if
we start taking derivatives
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of this expression.
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If we take the derivative of
this expression with respect
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to y, what is left is
the inner integral.
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And if we take, now, a
derivative with respect to x
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of this inner integral,
we will be left with
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just the joint PDF.
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And it will be the joint PDF
evaluated at the particular
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limits of the integration.
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So, it's going to be f sub
xy at little x, little y.
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So, we have this particular
formula.
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By taking derivative with
respect to x, and then with
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respect to y, or maybe in
the opposite order.
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It doesn't matter.
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This particular derivative
gives us back the PDF.
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Let us look at an example.
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Suppose that we have a uniform
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distribution on the unit square.
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So the PDF is equal to 1
on this green square.
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And is equal to 0 otherwise.
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So, in this example, if we take
some x and y, so that the
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xy pair falls inside the
rectangle, the probability of
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the blue set is going to be just
the probability of that
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little rectangle here.
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Because everything outside
has zero probability.
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With a uniform joint PDF,
which is equal to 1, the
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probability is just the area
of the set that we are
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considering.
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And since this set that we are
considering is a rectangle
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with [sides]
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x and y, the joint CDF is
equal to x times y.
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Now, if we take the derivative
of this expression with
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respect to x, and then with
respect to y, then we're left
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just with a constant
equal to 1--
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which is as it should be, so
that it integrates to one.
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So, we have seen that CDFs
also apply to the case of
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multiple random variables, and
that we can recover the joint
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PDF from the joint CDF.