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In this lecture, we develop the
weak law of large numbers.

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Loosely speaking, the weak law
of large numbers says that if

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we have a sequence of
independent random variables

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with the same distribution,
then the average of these

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random variables, which is
called the sample mean,

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approaches the expected value
of the distribution.

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In this sense, it reinforces
our interpretation of the

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expected value as some kind
of overall average.

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The weak law of large
numbers is the

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reason why polling works.

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By asking many people about the
value of some attribute,

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and by taking the average of
the responses, we can get a

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good estimate of the average
over the entire population.

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On the mathematical side, in
order to derive the weak law

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of large numbers, we will first
need to develop some

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inequalities, namely the
Markov and Chebyshev

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inequalities.

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Both of them tell us something
about tail probabilities.

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Suppose that a is a number.

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Then it is reasonable to expect
that the probability

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that the random variable exceeds
a will be small when a

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is very large.

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But how small?

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The Markov and Chebyshev
inequalities give us some

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answers to this question, based
only on knowledge of the

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mean and the variance
of the distribution.

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Finally, we will have to deal
with a technical issue.

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The weak law of large numbers
talks about the convergence of

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a random variable to a number.

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For this to make sense, we need
to define an appropriate

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notion of convergence.

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We will introduce one such
notion that goes under the

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name of convergence
in probability.

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And we will see that in many
respects, it is similar to the

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common notion of convergence
of numbers.