Mathematical formulae

This page lists all the formulae used along with their derivations. The PDF function is given by \(f(y)\), with \(\ell(y)\) representing the logPDF. Similarly, the CDF functions is given by \(F(y)\), with \(\mathcal{L}(y)\) representing the logCDF.

ArcSineDistribution

The PDF of ArcSineDistribution is as follows,

\[f(y) = \dfrac{1}{\pi\sqrt{y(1-y)}}\]

The logPDF can be derived as follows,

\[\begin{split}\begin{gather} \ell(y) = \ln\left(\dfrac{1}{\pi\sqrt{y(1-y)}}\right)\\ \ell(y) = \ln(1) - \ln\left(\pi\sqrt{y - y^2}\right)\\ \ell(y) = -\ln(\pi) - \ln(\sqrt{y-y^2}) \end{gather}\end{split}\]

The CDF of ArcSineDistribution is as follows,

\[F(y) = \left(\dfrac{2}{\pi}\right)\arcsin(\sqrt{y})\]

The logCDF can be derived as follows,

\[\begin{split}\begin{gather} \mathcal{L}(y) = \ln\left[\left(\dfrac{2}{\pi}\right)\arcsin(\sqrt{y})\right]\\ \mathcal{L}(y) = \ln\left(\dfrac{2}{\pi}\right) + \ln\left[\arcsin(\sqrt{y})\right] \end{gather}\end{split}\]