In this lecture, we develop the weak law of large numbers.
Loosely speaking, the weak law of large numbers says that if
we have a sequence of independent random variables
with the same distribution, then the average of these
random variables, which is called the sample mean,
approaches the expected value of the distribution.
In this sense, it reinforces our interpretation of the
expected value as some kind of overall average.
The weak law of large numbers is the
reason why polling works.
By asking many people about the value of some attribute,
and by taking the average of the responses, we can get a
good estimate of the average over the entire population.
On the mathematical side, in order to derive the weak law
of large numbers, we will first need to develop some
inequalities, namely the Markov and Chebyshev
inequalities.
Both of them tell us something about tail probabilities.
Suppose that a is a number.
Then it is reasonable to expect that the probability
that the random variable exceeds a will be small when a
is very large.
But how small?
The Markov and Chebyshev inequalities give us some
answers to this question, based only on knowledge of the
mean and the variance of the distribution.
Finally, we will have to deal with a technical issue.
The weak law of large numbers talks about the convergence of
a random variable to a number.
For this to make sense, we need to define an appropriate
notion of convergence.
We will introduce one such notion that goes under the
name of convergence in probability.
And we will see that in many respects, it is similar to the
common notion of convergence of numbers.