.. _discrete-bernoulli:

Bernoulli Distribution

A Bernoulli random variable of parameter :math:p takes one of only two values :math:X=0 or :math:X=1 . The probability of success ( :math:X=1 ) is :math:p , and the probability of failure ( :math:X=0 ) is :math:1-p. It can be thought of as a binomial random variable with :math:n=1 . The PMF is :math:p\left(k\right)=0 for :math:k\neq0,1 and

.. math:: :nowrap:

\begin{eqnarray*}
    p\left(k;p\right) & = & \begin{cases} 1-p & k=0\\ p & k=1\end{cases}\\
    F\left(x;p\right) & = & \begin{cases} 0 & x<0\\ 1-p & 0\le x<1\\ 1 & 1\leq x\end{cases}\\
    G\left(q;p\right) & = & \begin{cases} 0 & 0\leq q<1-p\\ 1 & 1-p\leq q\leq1\end{cases}\\
    \mu & = & p\\ \mu_{2} & = & p\left(1-p\right)\\
    \gamma_{3} & = & \frac{1-2p}{\sqrt{p\left(1-p\right)}}\\
    \gamma_{4} & = & \frac{1-6p\left(1-p\right)}{p\left(1-p\right)}
\end{eqnarray*}

.. math::

M\left(t\right) = 1-p\left(1-e^{t}\right)

.. math::

\mu_{m}^{\prime}=p

.. math::

h\left[X\right]=p\log p+\left(1-p\right)\log\left(1-p\right)

Implementation: scipy.stats.bernoulli