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We will now go through
a derivation of the

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law of total variance.

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This particular derivation
is not insightful.

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It will not really give you any
intuition as to why the

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law of total variance
is correct.

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On the other hand, it involves
some interesting manipulations

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that will be useful to be able
to follow, and understand the

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kinds of objects that they're
being moved around, and why

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each step is valid.

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Our derivation relies on the
standard formula that we have

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on how to calculate variances.

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And our first step is to apply
this formula to the

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conditional variance.

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Now, the conditional variance is
like an ordinary variance,

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except that it is calculated
in a conditional universe.

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So we apply this formula, except
that the expectation of

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X squared is the expectation
calculated in

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the conditional universe.

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And similarly, for the next term
it is the square of the

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expected value of X. But it's
the expected value of X as

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calculated in the conditional
universe.

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So this is an equality
between numbers.

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What does it translate to?

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This has been defined as a
random variable that takes

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this value when capital Y
is equal to little y.

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What is the random variable
that takes this value when

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capital Y is little y?

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Well, this random variable here
is a random variable that

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takes this value when capital
Y is equal to little y.

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And this random variable here
is a random variable that

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takes this numerical value
when capital Y is

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equal to little y.

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So to summarize, this is the
random variable that takes

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this numerical value
when capital Y is

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equal to little y.

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And this is a random variable
that takes this value when

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capital Y is equal
to little y.

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This expression, the left hand
side is equal to the right

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hand side for all y's.

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And therefore, this random
variable and that random

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variable always take the same
numerical values no matter

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what y happens to be.

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So these are identical
random variables.

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And so we have this equality
between random variables.

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The next step as we're working
towards calculating this first

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term here in the law of total
variance is to take the

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expectation of this
expression.

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What is it?

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We take the expectation
of the first term.

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It's the expectation of a
conditional expectation.

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And according to the law of
iterated expectations, it is

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the same as the unconditional
expectation.

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And then we have the expected
value of the next term.

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Next, we want to make some
progress towards calculating

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this second quantity in the
law of total variance.

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And the way to calculate it is
to just apply this general

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property of variances to the
special case where X gets

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replaced by the expected
value of X given Y.

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So the first term will be the
expected value of our random

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variable squared.

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Our random variable is the
expected value of X given Y.

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And the second term involves
the expected value of the

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random variable whose variance
we're considering.

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So it's the expected value
of this random variable.

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So it's the expected value of
the conditional expectation.

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And everything gets squared.

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What is this term?

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By the law of iterated
expectations, the expected

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value of a conditional
expectation is the same as the

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unconditional expectation.

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So this last term here
is of this form.

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What we will do next is to take
this expression here and

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that expression here, and
add them together.

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When we add them, we notice that
this term and that term

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are the same.

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So they cancel out.

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And we're left with the expected
value of X squared

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minus the square of the
expected value.

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But we know that this is the
same as the variance of X. So

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we have proved that the sum of
these two terms, which are the

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two terms up here, give
us the variance of X.