Introduction
==================

Metrics
----------------

* **Expected value/Espérance**: Noted :math:`\mathbb{E}[X]`, is a **theorical value**. For example, when playing coin
  flipping, the expected value for getting heads or tails is 0.5.

Variance
^^^^^^^^^^^^^^

Variance can be seen as the expected squared deviation from the expected value of a random variable :math:`X`.

.. math::
   \mathbb{V}[X]=\mathbb{E}[X-\mathbb{E}[X]]^2=\frac{\sum_{i=1}^n (x_i - \mathbb{E}[X])^2}{n}=\mathrm{Cov}(X,X)

Covariance
^^^^^^^^^^^^^^

Covariance is a way to quantify the relationship between two random variables :math:`X` and
:math:`Y` (`source <https://www.youtube.com/watch?v=qtaqvPAeEJY>`_). Covariance **DOES NOT**
quantify how strong this correlation is! If covariance is:

Positive
    For large values of :math:`X`, :math:`Y` is also taking large values
Negative
    For large values of :math:`X`, :math:`Y` is also taking low values
Null
    No correlation

.. math::
   \mathrm{Cov}(X,Y)=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]=\frac{\sum_{i=1}^n (x_i - \mathbb{E}[X])(y_i - \mathbb{E}[Y])}{n}