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Cosine Distribution¶
Approximation to the normal distribution. The support is \(\left[-\pi,\pi\right]\).
\begin{eqnarray*} f\left(x\right) & = & \frac{1}{2\pi}\left(1+\cos x\right)\\
F\left(x\right) & = & \frac{1}{2\pi}\left(\pi+x+\sin x\right)\\
G\left(q\right) & = & F^{-1}\left(q\right)\\
M\left(t\right) & = & \frac{\sinh\left(\pi t\right)}{\pi t\left(1+t^{2}\right)}\\
\mu=m_{d}=m_{n} & = & 0\\
\mu_{2} & = & \frac{\pi^{2}}{3}-2\\
\gamma_{1} & = & 0\\
\gamma_{2} & = & \frac{-6\left(\pi^{4}-90\right)}{5\left(\pi^{2}-6\right)^{2}}\end{eqnarray*}
\begin{eqnarray*} h\left[X\right] & = & \log\left(4\pi\right)-1\\
& \approx & 1.5310242469692907930.\end{eqnarray*}
Implementation: scipy.stats.cosine