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Here is a simple application
of the law of iterated

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expectations.

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We revisit the stick-breaking
example, which we have seen

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sometime in the past.

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So in this example, we start
with a stick that has a

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certain length and which we
break at a point that's chosen

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uniformly at random throughout
the length of the stick.

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And we call the point at which
we cut the stick capital Y.

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So the random variable Y has a
uniform distribution on the

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interval from 0 to l,
and is described by

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this particular PDF.

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Then we take the piece of the
stick that's left and we break

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it at a point that's chosen
uniformly over the length of

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the stick that's left.

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So the stick that was left has
a length Y, and the place at

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which we cut it, X, is chosen
uniformly over that interval.

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So in particular, X-- or
rather the conditional

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distribution of X given Y--

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is uniform on that interval.

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So in this example, what is the
expected value of X if I

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tell you the value of Y?

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Well, given the value of Y,
the random variable X is

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uniform on that range.

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So the expected value is going
to be at the midpoint that is

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equal to y over 2.

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This is an equality
between numbers.

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For any particular
number, little y,

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we have this equality.

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Now let us convert this concrete
equality between

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numbers to a more abstract
equality

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between random variables.

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This object is a random variable
that takes this value

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whenever capital
Y is little y.

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So this is an object that takes
the value little y over

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2 whenever the random variable
capital Y happens

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to be little y.

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But that's the same as
the random variable

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capital Y over 2.

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This is a random variable that
takes this value whenever

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capital Y happens to be
the same as little y.

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So the conditional
expectation--

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the abstract conditional
expectation is a random

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variable because its value is
determined by the random

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variable capital Y, and it is
this particular function of

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the random variable capital Y.

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And now we can proceed and
calculate the expected value

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of X using the law of iterated
expectations.

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The law of iterated expectations
takes this form.

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We have already calculated what
this random variable is.

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It is the random variable that's
equal to Y over 2.

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So this is the same as 1/2 the
expected value of Y. And since

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Y is uniform in the range from
0 to l, the expected value of

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Y is equal to l over 2,
which gives us an

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answer of l over 4.

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This is the same as the answer
that we got in the past where

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we actually found it using the
total expectation theorem.

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The calculations were exactly
the same as what went on here

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except that here we carry out
the calculation in a more

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abstract form.

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And what is important to
appreciate from this example

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is the distinction between
these two lines.

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This is an equality between
numbers, which is true for any

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specific little y.

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Whereas this is an equality
between random variables.

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This quantity is random and this
quantity is also random,

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meaning that their values are
not known until the experiment

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is carried out and the
specific value of

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capital Y is realized.