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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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00:00:23,130 --> 00:00:26,440
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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00:00:42,200 --> 00:00:48,010
So the conditional PDF is 0 in
this range, and also, it is 0

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00:00:48,010 --> 00:00:49,260
in this range.

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00:00:51,770 --> 00:00:57,420
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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00:01:05,590 --> 00:01:08,580
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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00:01:41,550 --> 00:01:44,789
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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00:01:56,970 --> 00:02:03,810
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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00:02:13,260 --> 00:02:24,390
And so this evaluates to 1/4
times a plus 3/4 times b.

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00:02:24,390 --> 00:02:28,079
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.