WEBVTT
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We now study a model that
involves the sum of
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independent random variables,
but with a twist.
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It's going to be the sum of a
random number of independent
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random variables, as opposed
to a fixed number.
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This is a model that shows up in
a variety of applications,
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but it will also help us fine
tune our command of the law of
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iterated expectations, and the
law of total variance.
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The story goes as follows--
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you go shopping and you visit
a number of stores, except
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that the number of stores that
you will visit, is itself a
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random variable.
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At each one of the stores,
you spend a
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certain amount of money.
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We denote it by Xi.
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And we make the assumption that
the Xi's are drawn from a
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certain distribution.
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They're identically
distributed.
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And they're independent
of each other.
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We also make the assumption that
the Xi's are independent
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of capital N. This means that no
matter how many stores you
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visit, the Xi, the amount of
money you spend in each one of
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the stores that you visit, is a
random variable that's drawn
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from a common distribution,
which does not change, no
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matter what capital N is.
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With these assumptions in place,
let us now focus on the
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total amount of money that
you're spending.
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This is the sum of random
variables, but with the extra
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twist that the index goes up to
capital N, which is itself
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a random variable.
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How do we deal with
this situation?
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One approach that's always worth
trying when faced with a
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complicated problem is to
try to condition on some
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information that will make
the problem easier.
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In this case, we can condition
on the value of capital N
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taking a fixed specific value
because in that case, we will
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be dealing with the sum of
a finite number of random
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variables where that number is
a fixed, specific number.
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And this is a situation we have
encountered before and
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know how to deal with it.
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So let us get started.
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Let us calculate the expected
value of Y, if we condition on
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the number of stores.
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Let's say, for example, someone
tells us that we
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visited five stores.
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Then, the expected value of Y
is going to be the expected
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value of the sum of the amount
of money you spent in each one
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of those five stores.
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In our instance, it's that
random variable, capital N.
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But since I told you that
capital N takes a specific
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numerical value, this means that
this instance of capital
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N, in the index of the
summation, can be
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replaced by little n.
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If I tell you that capital N
is equal to little n, then
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this number here, capital N,
becomes the same as little n.
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Here we use now the assumption
that capital N is independent
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from the Xi's.
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Here we have the sum of a fixed
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number of random variables.
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All of them are independent
of capital N.
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If I give you some information
on capital N, this does not
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change the distribution of the
Xi's, so the conditioning does
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not affect the answer.
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The conditional expectation is
going to be the same as the
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unconditional expectation.
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And now we have the expected
value of a
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sum of random variables.
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Each one of them has a common
expectation that's denoted
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with this notation.
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This is the common expected
value of all the Xi's, and
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we're adding n of them, so
we obtain n times this
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expectation.
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Now let us apply the total
expectation theorem.
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We take the familiar form of the
total expectation theorem,
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and in here, ' we can plug in
the expression that we have
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just found, which is n times
expected value of X. Now the
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expected value of X
is just a number.
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And then we have this summation
here, which we
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recognize to be just the
definition of the expected
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value of N.
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And so we come to the conclusion
that the expected
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amount of money that you will
be spending is equal to the
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following product--
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the expected number of stores
that you visit times the
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expected amount of money
that you will be
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spending in each store.
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This is a quite plausible
answer.
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It makes sense.
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On the average, the amount of
money you spend is equal to
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the average number of stores
times the average amount of
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money in each store.
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So it is intuitively what
you might expect.
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On the other hand, we know
that reasoning "on the
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average" does not always give
us the right answers.
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So it's important to corroborate
this particular
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formula by working out a
mathematical derivation.
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Now let us carry out a second
mathematical derivation using
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the law of iterated
expectations.
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To use the law of iterated
expectations, we need to put
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our hands on this random
variable--
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the abstract conditional
expectation.
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What is this object?
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It's a random variable that
takes this value whenever
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capital N is equal
to little n.
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So it's an object that takes
this value whenever capital N
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is equal to little n.
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But that object is the same as
this random variable because
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this is the random variable that
takes the value here when
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capital N is equal
to little n.
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Therefore, the abstract
conditional expectation takes
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this particular form here, which
we can substitute inside
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this expectation here.
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And now notice that the expected
value of X is a
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constant, so it can be pulled
outside this expectation.
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And we're left with a product
of the expected value of N
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times the expected value of
X. So this completes the
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derivation of the expected value
of the sum of a random
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number of random variables.