From 79dafeccba814010ecd17466e43c37ce72debb24 Mon Sep 17 00:00:00 2001 From: Paul Romano Date: Thu, 2 Aug 2012 13:10:03 -0400 Subject: [PATCH] Added discussion on central limit theorem to documentation. --- docs/source/methods/statistics.rst | 25 +++++++++++++++++++++++++ 1 file changed, 25 insertions(+) diff --git a/docs/source/methods/statistics.rst b/docs/source/methods/statistics.rst index 993d1deed3..19cbf73203 100644 --- a/docs/source/methods/statistics.rst +++ b/docs/source/methods/statistics.rst @@ -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