science-notes/source/statistics/notations.rst

Notations
---------------

This page was inspired from the following `Wikipedia page <https://en.wikipedia.org/wiki/Notation_in_probability_and_statistics>`_.


Sample and Population
=====================

A *population* is the entire population that you want to analyze. It is a exhaustive set. A *sample* is a subset of the population. It is a non-exhaustive set.

.. figure:: figures/population_sample.svg
    :align: center
    :width: 300px

*Inferential statistics* allows you draw conclusions (with a certain degree of confidence) on a population using samples.

A Tale of Schizophrenia
================================
Two different notation conventions are used. The one to use depends if you are working on a *population* or a *sample*:

.. list-table:: Notation
    :align: center
    :header-rows: 1

    * - 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.

.. note::
   Estimators can also be denoted with the hat symbol.
   For example :math:`\hat{\mu}\equiv\overline{x}`.
        
Operators
==========

Expected value
^^^^^^^^^^^^^^^

The expected value (*espérance*) of a random variable :math:`X` is noted :math:`\mathbb{E}[X]`.
It as the following linearity  properties:

.. math::
   \mathbb{E}[X+Y]=\mathbb{E}[X] + \mathbb{E}[Y]
.. math::
   \mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]

Variance
^^^^^^^^

The variance operator of a random variable :math:`X` is noted :math:`\mathbb{V}[X]` or :math:`\mathrm{Var}[X]`.

Coraviance
^^^^^^^^^^
The covariance operator of a random variable :math:`X` and :math:`Y`
is noted :math:`\mathrm{Cov}[X,Y]`.