\[X \sim {\rm Beta}\left(\frac{1}{2},\frac{1}{2}\right)\] \[f_X(x) = \frac{1}{\pi\sqrt{x(1-x)}}\]
\[F(x) = \frac{2}{\pi}\arcsin\left(\sqrt{x}\right)\]
\[X \sim {\rm Beta}\left(\alpha,\beta\right)\]
\[f_X(x|\alpha,\beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}\] \[=\frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}\]
\[F_X(x|\alpha,\beta) = \frac{B(x|\alpha,\beta)}{B(\alpha,\beta)} \]
\[X \sim{\rm Uniform}(a,b)\] \[f_X(x)=\begin{cases} \frac{1}{b - a} & \mathrm{for}\ a \le x \le b, \\[8pt] 0 & \mathrm{for}\ x<a\ \mathrm{or}\ x>b \end{cases}\]
\[ F_X(x|a,b)= \begin{cases} 0 & \text{for }x < a \\[8pt] \frac{x-a}{b-a} & \text{for }a \le x \le b \\[8pt] 1 & \text{for }x > b \end{cases}\]
\[X \sim{\rm Kumaraswamy}(a,b)\] \[f_X(x|a,b) = a b x^{a-1}{ (1-x^a)}^{b-1}, \ \ \mbox{where} \ \ x \in (0,1).\] \[F_X(x|a,b)=1-(1-x^a)^b\]
\[X \sim{\rm RaisedCosine}(\mu,s)\] \[f_X(x|\mu, s)=\frac{1}{2s} \left[1+\cos\left(\frac{x-\mu}{s}\,\pi\right)\right]\,=\frac{1}{s}\operatorname{hvc}\left(\frac{x-\mu}{s}\,\pi\right)\,\]
\[F_X(x|\mu,s)= \frac{1}{2}\left[1+\frac{x-\mu}{s} +\frac{1}{\pi}\sin\left(\frac{x-\mu}{s}\,\pi\right)\right]\]
\[X\sim{\rm Log-Uniform}(a,b)\] \[ f_X( x| a,b ) = \frac{ 1 }{ x [ \ln( b ) - \ln( a ) ]} \quad \text{ for } a \le x \le b \text{ and } a > 0.\] \[F_X(x|a,b)=\frac{ \ln( x ) - \ln( a ) }{ \ln( b ) - \ln( a ) } \quad \text{ for } a \le x \le b.\]
\[X\sim{\rm Truncated-Normal}(\mu,\sigma,a,b)\] \[\xi=\frac{x-\mu}{\sigma},\ \alpha=\frac{a-\mu}{\sigma},\ \beta=\frac{b-\mu}{\sigma}\] \[Z=\Phi(\beta)-\Phi(\alpha)\]
\[f_X(x|\mu,\sigma,a,b) = \frac{1}{\sigma}\, \frac{\phi(\frac{x - \mu}{\sigma})}{\Phi(\frac{b - \mu}{\sigma}) - \Phi(\frac{a - \mu}{\sigma}) }\] \[F_X(x|\mu,\sigma,a,b) = \frac{\Phi(\xi) - \Phi(\alpha)}{Z}\]