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In this segment, we will first
discuss and compare different

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views of the sum of independent
identically

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distributed random variables.

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And then, we will conclude
with a statement of the

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central limit theorem.

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So let X1 up to Xn be
independent identically

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distributed random variables
that have a certain finite

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mean and finite variance
that we'll denote by

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mu and sigma squared.

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In order to have a concrete
example in our hands, let us

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assume that this random variable
has a distribution,

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let's say, a PDF, that ranges
from minus 1 to plus 1 and has

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a mean of 0.

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Let us look at the sum of
these random variables.

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The sum has a variance of n
times sigma squared, which

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goes to infinity.

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And correspondingly, the
standard deviation of this

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random variable also
grows to infinity.

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This random variable takes
values between

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minus n and plus n.

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And because the variance and
the standard deviation

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increase, this means that for
larger and larger n, the width

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of this distribution is going
to be larger and larger.

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We can obtain a different view
of this sum if we divided by n

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in, which case, we obtain
the sample mean.

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In this case, the variance
goes to 0

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as n goes to infinity.

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And as a consequence, the
distribution is highly

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concentrated around 0.

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This is also what the weak law
of large numbers tells us.

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The bulk of the distribution
is concentrated in an

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arbitrarily small interval
around 0.

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So this width becomes
smaller and smaller

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as n goes to infinity.

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So in this case, we obtain
a limiting distribution.

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But this limiting distribution
is trivial.

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It's degenerate.

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It's all concentrated
on a single point.

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How can we make it so that we
obtain a limiting distribution

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that is more interesting?

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The key is to divide not by
n, but to divide by the

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square root of n.

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This has the following effect.

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The variance of this ratio
is calculated as follows.

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We take the variance of the
numerator, which is n times

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sigma squared.

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And then we divide by
the square of this

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number, which is n.

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And therefore, the variance
is equal to sigma squared.

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What's important here is that
the variance stays constant.

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No matter what n is, the width
of this distribution is going

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to be more or less the same.

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The distribution itself might
change as n changes.

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But the distribution--

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at least in this case, where
we assume 0 mean, the

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distribution stays in place.

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It doesn't move to the
right or to the left.

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And its width stays the same.

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So one can wonder, in the limit,
as n goes to infinity,

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does this shape start
to approach a

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certain limiting shape?

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And if it does, what is the
limiting shape that it

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approaches?

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The central limit theorem
will give us the

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answers to these questions.

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The setting for the central
limit theorem will be pretty

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much the setting that we
were just discussing.

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So we will be looking into the
case where we divide the sum

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of the random variables by
square root of n, except for a

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few additional twists.

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Since this ratio has a variance
of sigma squared, it

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would help to divide by a
further factor of sigma here

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so that the variance is
going to become 1.

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And there's another issue.

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If the mean of the X's
is non-zero, then the

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distribution is centered at
a quantity that keeps

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changing with n.

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So the distribution will be
drifting away from 0.

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It's not staying in place.

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And so it wouldn't
have any hope of

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converging to something.

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For this reason, instead of
looking at this ratio in

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particular, what we do is we
first subtract the mean of the

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sum, which is n times
the mean.

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And then we divide by a further
factor of sigma.

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This random variable that
we obtain here has nice

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properties.

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The mean of this random variable
is equal to 0,

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because we did subtract
the mean of the X's.

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And the variance of this random
variable is going to be

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equal to 1.

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The reason is that the variance
is the variance of

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the numerator, which is n times
sigma squared, divided

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by the square of the
denominator, which is also n

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times sigma squared.

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So as n changes, the
distribution of the random

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variable Zn stays in place.

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It has a mean of 0.

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And its width, more or less,
stays the same, because we

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have a constant variance.

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We will compare this random
variable with a standard

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normal random variable that has
0 mean and unit variance.

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The central limit theorem states
the amazing fact that

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as n goes to infinity, the
distribution of this random

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variable converges to the
standard normal distribution

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in the following sense--

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that this probability here
converges to that probability

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for any choice of little z.

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Now, what we have here is just
the CDF of this random

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variable, Zn.

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So it tells us that the CDF
of the random variable Zn

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converges to the CDF of
a standard normal.

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And fortunately, for the
standard normal, the CDF is

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available in tables.

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So if we needed to calculate the
numerical value here, we

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can just look up the
normal tables.

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And this suggests an
approximation to this

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probability for the case where
n is finite but large.

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When n is large, we can
approximate this probability

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by this probability on the
right, which we can find from

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the normal tables.

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The central limit theorem is
a very important result.

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For this reason, we will spend
some time discussing how to

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interpret it, what it means, how
we use it, and we will go

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through a few examples to see
how we actually apply it.