Alpha distribution

The probability density function for alpha distribution is:

.. math::

f(x,a) = \frac{1}{x^2\Theta(a)\sqrt{2\pi}}\cdot\exp\left(-\frac{1}{2}\left(a-\frac{1}{x}\right)^2\right)

where :math:\Theta is the normal CDF, :math:x>0 and :math:a>0. Here :math:a is the shape parameter.

The cumulative distribution function is:

.. math::

F(x,a) = \frac{\Theta(a - 1/x)}{\Theta(a)}

The following plots show the pdf and cdf:

.. image:: ../../../../tests/imgs/stats/continuous/alpha/alphapdf.png :align: center :width: 480

.. image:: ../../../../tests/imgs/stats/continuous/alpha/alphacdf.png :align: center :width: 480