WEBVTT
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Let us now look at an example.
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Consider a piecewise constant
PDF of the form
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shown in this diagram.
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Suppose that we condition on the
event that x lies between
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a plus b over 2, which
is here, and b.
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So we're conditioning
on x lying in this
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particular red interval.
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What is the conditional PDF?
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The conditional PDF is going
to be 0 outside of the
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interval on which we
are conditioning.
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So the conditional PDF is 0 in
this range, and also, it is 0
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in this range.
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Within the range of values of
x that are allowed given the
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conditioning information, the
conditional PDF must retain
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the same shape as the
unconditional one.
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And the unconditional one is
constant in that range.
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So the conditional PDF will
also be a constant.
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Because in this case the length
of this interval is
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half of the distance
between b minus a--
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so the length of this interval
is b minus a over 2--
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in order for the area under this
curve to be equal to 1,
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it means that the height of this
curve has to be equal to
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2 over b minus a.
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The conditional expectation in
this example is just the
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ordinary expectation
but applied to
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the conditional model.
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Since the conditional PDF is
uniform, the conditional
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expectation will be the midpoint
of the range of this
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conditional PDF.
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And in this case, the midpoint
is 1/2 the left end of the
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interval, which is a plus b over
2 plus 1/2 the right end
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point of the interval,
which is b.
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And so this evaluates to 1/4
times a plus 3/4 times b.
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We can also calculate the
expected value of X squared in
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the conditional model using
the expected value rule.
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According to the expected value
rule, it's going to be
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an integral of the conditional
PDF, which is 2 over b minus a
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multiplied by x squared.
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And this integral runs over
the range where the
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conditional PDF is actually
non-zero.
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So it's an integral that
ranges from a plus b
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over 2 up to b.
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And this an integral which is
not too hard to evaluate, and
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there's no point in carrying out
the evaluation to the end.