From ee62e13342af25af50d569d3b8e20bfbd3959ac6 Mon Sep 17 00:00:00 2001
From: Loic Guegan <loic.guegan@mailbox.org>
Date: Mon, 16 Oct 2023 12:13:52 +0200
Subject: [PATCH] Minor changes

---
 source/statistics/bessel_correction.rst | 42 ++++++++++++++++++++++++-
 source/statistics/notations.rst         |  4 +++
 2 files changed, 45 insertions(+), 1 deletion(-)

diff --git a/source/statistics/bessel_correction.rst b/source/statistics/bessel_correction.rst
index e24d727..c9ef0a9 100644
--- a/source/statistics/bessel_correction.rst
+++ b/source/statistics/bessel_correction.rst
@@ -1,7 +1,47 @@
 .. _bessel_correction:
 
+This page is inpired by `Wikipedia <https://en.wikipedia.org/wiki/Bessel%27s_correction>`__.
+
 Bessel's Correction
 -----------------------
 
-Bessel's correction is the use of :math:`n-1` instead of :math:`n` in the formula for the sample
+Bessel's correction is the use of :math:`n-1` instead of :math:`n` in the formulas for sample
 variance and sample standard deviation.
+In fact, using :math:`n` as a denominator leads to a biased estimator.
+This variance estimator is noted :math:`s^2_n`.
+Lets compute the discrepency between population variance and the biased sample variance:
+
+.. math::
+   \mathbb{E}[\sigma^2-s_n^2] &= \mathbb{E} \left[ \frac{1}{n} \sum_{i=1}^n(x_i - \mu)^2 - \frac{1}{n}\sum_{i=1}^n (x_i - \overline{x})^2 \right]
+   
+   &=\mathbb{E}\left[ \frac{1}{n} \sum_{i=1}^n\left((x_i^2 - 2 x_i \mu + \mu^2) - (x_i^2 - 2 x_i \overline{x} + \overline{x}^2)\right) \right]
+
+   &=\mathbb{E}\left[ \frac{1}{n} \sum_{i=1}^n\left(\mu^2 - \overline{x}^2 + 2 x_i (\overline{x}-\mu) \right) \right]
+   
+   &=\mathbb{E}\left[ \frac{1}{n} \sum_{i=1}^n \left(\mu^2 - \overline{x}^2 \right) + \frac{1}{n} \sum_{i=1}^n 2 x_i (\overline{x} - \mu) \right]
+
+   &=\mathbb{E}\left[  \mu^2 - \overline{x}^2 + \frac{1}{n} \sum_{i=1}^n 2 x_i (\overline{x} - \mu) \right]
+
+   &=\mathbb{E}\left[  \mu^2 - \overline{x}^2 + 2\overline{x}(\overline{x} - \mu) \right]
+
+   &=\mathbb{E}\left[  \mu^2 - 2 \overline{x} \mu + \overline{x}^2 \right]
+
+   &= \mathbb{E}\left[  (\overline{x}   - \mu)^2 \right]
+
+   &= \mathrm{Var}[\overline{x}]
+
+   &= \frac{\sigma^2}{n}
+
+
+This result shows us that the discrepency between the population and sample variance is :math:`\frac{\sigma^2}{n}`.
+From this result we can deduce how :math:`S_n^2` must be adjusted:
+
+.. math::
+   \mathbb{E} \left[ s^2_n \right] = \sigma^2 - \frac{\sigma^2}{n} = \frac{n-1}{n} \sigma^2
+
+Thus, the adjustment factor is :math:`\frac{n-1}{n}`. As such, an unbiased estimator of :math:`\sigma^2` is:
+
+.. math::
+   s^2 = \frac{s^2_n}{\frac{n-1}{n}} = \frac{n}{n-1} s_n^2 &= \frac{n}{n-1} \frac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2
+
+   &= \frac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2
diff --git a/source/statistics/notations.rst b/source/statistics/notations.rst
index d111887..90ab557 100644
--- a/source/statistics/notations.rst
+++ b/source/statistics/notations.rst
@@ -26,15 +26,19 @@ Two different notation conventions are used. The one to use depends if you are w
     * - Metric
       - Population
       - Sample
+      - Notes
     * - Sample mean
       - :math:`\mu`
       - :math:`\overline{x}`
+      - 
     * - Variance
       - :math:`\sigma^2`
       - :math:`s^2`
+      - :math:`s^2_n` without `Bessel's Correction <bessel_correction>`__
     * - Standard deviation
       - :math:`\sigma`
       - :math:`s`
+      - :math:`s_n` without `Bessel's Correction <bessel_correction>`__
 
 To determine the metric of a population (say :math:`\mu`) using a sample, we use an estimator.
 In the case of :math:`\mu`, we use :math:`\overline{x}` as an estimator.