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The weak law of large
numbers tells us

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that the sample mean--

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that is, the average of
independent identically

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distributed random
variables, Xi--

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converges, in a certain sense,
to a number, namely the

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expected value of the random
variables, Xi.

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But it does not tell us much
about the details of the

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distribution of the
sample mean.

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The central limit theorem
provides us exactly with this

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kind of detail.

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It tells us that the sum of many
independent identically

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distributed random variables
has approximately a normal

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distribution.

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The mean and variance of this
normal is easy to find if we

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know the mean and variance of
the original random variables.

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This enables us to carry out
approximate calculations

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rather quickly by using
the normal tables.

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We will start with a precise
statement of the central limit

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theorem, and we will emphasize
that it is a universal result.

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It holds no matter what the
distribution of the original

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random variables, and for this
reason, it is very useful.

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We will work through several
examples of the typical ways

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that the central limit
theorem is used.

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We will develop a refinement
that can be used when we are

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dealing with discrete
distributions, which provides

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us with even more accurate
approximations.

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And finally we will revisit
the polling problem, and

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inquire again about the number
of samples that are needed to

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obtain a certain accuracy with
a certain confidence.

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We will see that the central
limit theorem is much more

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informative, much less
conservative, compared to the

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conclusions that we had gotten
before based on the Chebyshev

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inequality.