WEBVTT
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Just in order to get some more
familiarity with joint PDFs,
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let us look at independent
normals.
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Actually, this is an important
example because noise is often
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modeled by normal random
variables, and noise terms
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that show up at different
parts of a system, or at
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different times, are often
assumed to be independent.
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Suppose that we have two
standard normal random
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variables, X and Y, with zero
means and unit variances.
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If their independent, their
joint PDF is the product of
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the marginal PDFs and
takes this form.
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This is just the PDF of a
standard normal X and the PDF
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of a standard normal Y
and we multiply them.
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If we are to plot this joint
PDF we obtain this figure.
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It looks like a bell which is
centered at the origin--
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at the point with coordinates
zero, zero.
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One way to think about what is
going on here is to rewrite
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this expression as 1 over 2pi,
and then the exponential of
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minus 1/2 x squared
plus y squared.
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If we look at the unit circle in
xy space, which is the set
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of points at which x squared
plus y squared is equal to 1,
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then, on that circle, the PDF
takes a constant value because
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this quantity is constant
on that circle.
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And the same is true for
any other circle.
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On any circle the PDF takes a
constant value, of course, a
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different constant.
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So the circles centered at the
origin are the so-called
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contours of the joint PDF.
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On each contour the joint
PDF is a constant.
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Let us now generalize.
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Consider two independent normal
random variables, but
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with general means mu x and mu
y, and variances sigma x
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squared and sigma y squared.
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The joint is, again, the product
of the marginal PDFs
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and, therefore, takes
this form.
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This looks intimidating but, in
fact, it is pretty simple.
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This part is just a normalizing
constant.
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It is the constant that's needed
so that the joint PDF
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integrates to 1.
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What we have here is the
negative exponential of a
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quadratic function of x and y.
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Let us plot the contours
of this quadratic.
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Remember that contour is the
set of points where the
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quadratic takes a
constant value.
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And by consequence,
the joint PDF also
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takes a constant value.
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If you have set this quadratic
to a constant, what you have
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is the equation that describes
an ellipse.
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And it is an ellipse whose
principal axes run along the x
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and y directions, and those
ellipses are all centered at
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this particular point,
mu x, mu y.
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The joint PDF is largest when
the exponent is equal to zero.
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And this happens when x is
equal to mu x, and y
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is equal to mu y.
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That is, right at the center
of the ellipse.
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That's where the joint
PDF is largest.
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As you move to ellipses that are
further out on this outer
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ellipse, this expression
is a constant.
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It's the exponential
of the negative of
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some positive numbers.
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So you get a smaller value
for the joint PDF.
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If you move to a further ellipse
further out, then
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again, the joint PDF will be a
constant, but it's going to be
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a smaller constant.
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Now, for the case of standard
normals, the joint PDF was
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circularly symmetric.
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The contours were actually
circles, instead of ellipses.
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But this is not the
case in general.
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For example, suppose that the
variance of Y is bigger than
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the variance of X. Then you get
a shape as the one shown
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in this figure.
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Since the variance of Y is
larger, we expect Y to take
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values over a bigger range,
and to be larger typically
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than the values of X. And so
the bell shape that we have
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for the joint PDF is stretched
in the y direction.
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It extends further out in the
y direction than it does in
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the x direction.
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To conclude, the joint PDF of
two independent normals has
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the shape of a bell.
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The center of the bell is
determined by the means.
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Furthermore, the bell is
stretched in the x and y
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directions by an amount that is
determined by the variances
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of x and y.
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However, the stretching
is always along
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the coordinate axes.
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If you wanted a bell that
stretches in some diagonal
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direction, or if you have
contours that are ellipses but
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with some different kinds of
axes, then you will have
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dependence between the
two random variables.
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In that case, we will be dealing
with a so-called
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bivariate normal distribution,
but we will not pursue this
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any further at this point.