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Added discussion on central limit theorem to documentation.

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Paul Romano 2012-08-02 13:10:03 -04:00
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docs/source/methods/statistics.rst
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@ -37,6 +37,27 @@ X_n}{n}` `converges in probability`_ to the true mean, i.e. for all
Central Limit Theorem
---------------------
The `central limit theorem`_ (CLT) is perhaps the most well-known and ubiquitous
statistical theorem that has far-reaching implications across many
disciplines. The CLT is similar to the law of large numbers in that it tells us
the limiting behavior of the sample mean. Whereas the law of large numbers tells
us only that the value of the sample mean will converge to the expected value of
the distribution, the CLT says that the distribution of the sample mean will
converge to a `normal distribution`_. As we defined before, let :math:`X_1, X_2,
\dots, X_n` be an infinite sequence of independent, identically-distributed
random variables with expected values :math:`E(X_i) = \mu` and variances
:math:`\text{Var} (X_i) = \sigma^2 < \infty`. Note that we don't require that
these random variables take on any particular distribution -- they can be
normal, log-normal, Weibull, etc. The central limit theorem states that as
:math:`n \rightarrow \infty`, the random variable :math:`\sqrt{n} (\bar{X}_n -
\mu)` `converges in distribution`_ to the standard normal distribution:
.. math::
:label: central-limit-theorem
\sqrt{n} \left ( \frac{1}{n} \sum_{i=1}^n X_i - \mu \right ) \xrightarrow{d}
\mathcal{N} (0, \sigma^2)
------------------------------------------
Estimating Statistics of a Random Variable
------------------------------------------
@ -220,3 +241,7 @@ so the idea is to determine the new multiplicative and additive constants in
.. _expected value: http://en.wikipedia.org/wiki/Expected_value
.. _converges in probability: http://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_probability
.. _normal distribution: http://en.wikipedia.org/wiki/Normal_distribution
.. _converges in distribution: http://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_distribution