Bernoulli distribution
======================

The Bernoulli distribution is a discrete distribution with boolean-valued outcome; 1 indicating *success* with probability :math:`p` and 0 indicating *failure*
with probability :math:`q = 1 -p`, where :math:`p \in [0, 1]`. The probability
mass function for :math:`k \in \{0, 1\}` is

.. math::

   f(k; p) = p^k (1-p)^{k-1} = \begin{cases} 1-p & \text{if } k = 0\\
   p & \text{if }k = 1, \end{cases}

and the cumulative distribution function is

.. math::

   F(k; p) = \begin{cases} 1-p & \text{if } k = 0\\
   1 & \text{if }k = 1. \end{cases}

The expected value and variance are as follows

.. math::

   \mathrm{E}[X] = p, \quad \mathrm{Var}[X]= p(1-p).  

The Bernoulli distribution is suitable for binary-outcome tests, for example, CRO (conversion rate) or CTR (click-through rate) testing.

.. autoclass:: cprior.models.BernoulliModel
   :members:
   :inherited-members:
   :show-inheritance:

.. autoclass:: cprior.models.BernoulliABTest
   :members:
   :inherited-members:  
   :show-inheritance:

.. autoclass:: cprior.models.BernoulliMVTest
   :members:
   :inherited-members:
   :show-inheritance: