WEBVTT
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We will now go through an
example, which is essentially
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a drill, to consolidate our
understanding of the
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conditional expectation and
the conditional variance.
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Consider a random variable X,
which is continuous and is
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described by a PDF
of this form.
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Whenever we have a PDF that
seems to consist of different
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pieces, it's always useful
to divide and conquer.
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And the way we will do that
will be to consider two
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different scenarios.
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That X falls in this range.
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And in that scenario, we say
that the certain random
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variable Y is equal to 1.
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And another scenario in which X
falls in this range, and in
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that case, we say that
Y is equal to 2.
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Let us now look at the
conditional expectation of X
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given Y. What is it?
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Well, it is a random variable
which can take a different
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values depending on what Y is.
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If Y happens to take
a value of 1, then
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we are in this range.
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And the conditional PDF of X,
given that Y falls in this
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range, keeps the same
shape, it's uniform.
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And so it's mean is going to
be equal to the midpoint of
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this interval, which is 1/2.
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And this is something that
happens when Y is equal to 1.
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What is the probability
of this happening?
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The probability that Y is equal
to 1 is the area under
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the PDF in this range.
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And since the height of
the PDF is 1/2, this
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probability is 1/2.
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The alternative scenario is
that Y happens to take the
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value of 2.
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In which case, X lives
in this interval.
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Given that X has fallen in this
interval, the conditional
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expectation of X is the midpoint
of this interval.
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And the midpoint of this
interval is at 2.
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And this is an event that,
again, happens with
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probability 1/2, because the
area under the PDF in this
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region is equal to 1/2.
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So the conditional expectation
is a random variable that
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takes these values with
these probabilities.
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Since we now have a complete
probabilistic description of
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this random variable, we're
able to calculate the
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expectation of this
random variable.
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What is it?
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With probability 1/2, the random
variable takes the
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value of 1/2.
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And with probability 1/2,
it takes a value of 2.
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And so the expected value
of the conditional
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expectation is 5/4.
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But the law of iterated
expectations tells us that
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this quantity is also the same
as the expected value of X. So
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we have managed to find the
expected value of X by the
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divide and conquer method, by
considering different cases.
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Let us now turn to the
conditional variance of X
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given Y. Once more, this
quantity is a random variable.
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The value of that quantity
depends on what Y
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turns out to be.
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And we have, again, the same
two possibilities.
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Y could be equal to 1, or
Y could be equal to 2.
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And these possibilities happen
with equal probabilities.
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If Y is equal to 1, conditional
on that event, X
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has a uniform PDF on this
range, on an interval of
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length one.
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And we know that the variance
of a uniform PDF on an
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interval of length
one is 1/12.
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If on the other hand, Y takes
a value of 2, then X is a
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uniform random variable on
an interval of length 2.
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And the variance in this case
is 2 squared, where this 2
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stands for the length of the
interval, divided by 12, which
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is the same as 4/12.
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So we now have a complete
probabilistic description of
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the conditional variance
as a random variable.
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It's a random variable that
with these probabilities,
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takes these two particular
values.
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Since we know the distribution
of this random variable, we
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can certainly calculate
its expected value.
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And the expected value
is found as follows.
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With probability 1/2, the random
variable of interest
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takes a value of 1/12.
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And with probability 1/2,
this random variable
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takes a value of 4/12.
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And this number happens
to be 5/24.
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Finally, let us calculate the
variance of the conditional
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expectation.
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Since we have complete
information about the
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distribution of the conditional
expectation,
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calculating its variance is
not going to be difficult.
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So what is it?
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With probability 1/2, the
conditional expectation takes
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a value of 1/2.
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We subtract from this is the
mean of the conditional
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expectation, which is 5/4.
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And we take the square
of that.
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So this term is the square or
the deviation of the value of
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the random variable of
1/2 from the mean
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of that random variable.
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And we get a similar term.
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If Y happens to be equal to 2.
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With probability 1/2 half, our
random variable takes a value
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of 2, which is so much
away from the mean
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of the random variable.
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And then we square
that quantity.
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If we carry out the algebra,
the answer turns out
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to be 9 over 16.
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And now we can go back to the
law of the total variance and
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calculate that the total
variance is equal to the
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expected value of the variance,
which is 5/24.
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And then we have the variance
of the expected
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value, which is 9/16.
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And this number evaluates
to 37/48.
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So we have managed to find the
variance of this random
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variable using the divide and
conquer methods and the law of
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the total variance.