I needed a probability density function which worked on the interval $[0,1]$, had kind of a bell shape, and had an adjustable mode / peak $p$.
I thought of a pdf $f(x|p)$, given by \begin{equation} f(x|p) = \left\{ \begin{array}{l l l } \frac{ (x^{- \ln 2/\ln p})^2 \cdot (1-x^{- \ln 2/\ln p})^2 }{ \log(p) \left( \frac{x^{1-\frac{4 \log 2}{\log p}}}{\log(p) - 4\log(2)} -\frac{2 x^{1-\frac{3 \log 2}{\log p}}}{\log(p) - 3\log(2)} +\frac{x^{1-\frac{2 \log 2}{\log p}}}{\log(p) - 2\log(2)} \right) } &\quad \text{ for } 0<x<1 \\ 0 &\quad \text{ otherwise} \end{array} \right. \end{equation} which
- has a peak at $x=p$ for $0<p<1$.
- $P(X\le 0) = P(X\ge 1) = 0$
- has a shape similar to the bell shape
- looks skewed to the left for $p>1$ and to the right for $0<p<1$
Or equivalently: \begin{equation} f(x|p) = \left\{ \begin{array}{l l l } \frac{ (x^a)^2 \cdot (1-x^a)^2 } { (4a+1)^{-1} - 2(3a+1)^{-1} + (2a+1)^{-1} } &\quad \text{ for } 0<x<1 \\ 0 &\quad \text{ otherwise} \end{array} \right. \end{equation} which
which has its peak at $p=-\frac{\log 2}{\log x}$
Is there a similar pdf (or exactly this one) used in literature? What is it called?
PS: note that the given pdf is not symmetric: $f(x|1-p) \neq f(1-x |p) $