WEBVTT
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In this brief segment,
we will discuss
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an important application
of the convolution formula.
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Suppose that X is a normal
random variable with a given
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mean and variance.
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So that the PDF of
X takes this form.
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And similarly, Y is normal
with a given mean and variance.
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So its PDF takes this form.
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We assume that X and
Y are independent.
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And we're interested in the
sum of the two random variables
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X and Y. And we wish
to derive the PDF of Z.
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Of course, the PDF of Z is given
by the convolution formula.
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And now we plug in here, the
form for the density of X.
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And here, we plug the
form of the density of Y.
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Except that instead
of the argument Y,
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we need to put in the
argument z minus x.
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So we obtain this
form, where here we
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have a z minus x instead of y.
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Now this is an integral that
looks pretty complicated.
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But it is not too hard to do.
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One just needs to be patient,
rearrange terms, collect terms.
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And the details of
the calculations
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are not as interesting.
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So we will skip them for now.
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And I will just tell you
that the final answer
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takes this form.
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What is this form?
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Well, it's
exponential of minus z
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minus something squared
divided by a constant.
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And we recognize
that this is the form
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of a normal random variable.
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It's a normal random
variable whose mean
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is given by this term
here, it's mu x plus mu y.
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And the variance of that
normal random variable
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is that constant that appears
next to the factor of 2
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in the denominator.
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So the sum of these two
normal random variables,
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these two independent
normal random variables,
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is also normal.
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The fact that this
is the mean and this
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is the variance of the sum,
of course, is not a surprise.
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What is important in this
result that we have here
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is that the sum is
actually normal.
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Now, we carried
out this argument
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for the case of the sum of
two normal random variables.
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But suppose that we
had the sum of three
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independent normal
random variables,
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what can we say about it?
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By the result that we just
discussed, this sum is normal.
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This is assumed to be normal.
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We assume that X, Y,
and W are independent.
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Therefore, this sum
is independent from W.
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So we're dealing
with the sum of two
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independent normal
random variables again.
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So this sum here is going
to be normal as well.
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And we continue this
argument by induction,
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and conclude that
more generally,
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the sum of any finite number
of independent normal random
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variables is normal.
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This is a very important,
but also useful fact.
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It means that when
we start working
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with normal random
variables, very often
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we stay within the realm
of normal random variables.
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We can form linear
functions of them,
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take linear
combinations of them,
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and still remain in the world
of normal random variables.