Probability density function within [0,1] with specifiable mode - Cross Validated most recent 30 from stats.stackexchange.com 2019-10-22T01:17:24Z https://stats.stackexchange.com/feeds/question/89186 https://creativecommons.org/licenses/by-sa/4.0/rdf https://stats.stackexchange.com/q/89186 1 Probability density function within [0,1] with specifiable mode Angelorf https://stats.stackexchange.com/users/31221 2014-03-07T16:58:39Z 2017-02-10T09:36:48Z <p>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 $p$.</p> <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) } &amp;\quad \text{ for } 0&lt;x&lt;1 \\ 0 &amp;\quad \text{ otherwise} \end{array} \right. \end{equation} which </p> <ul> <li>has a peak at $x=p$ for $0&lt;p&lt;1$.</li> <li>$P(X\le 0) = P(X\ge 1) = 0$</li> <li>has a shape similar to the bell shape</li> <li>looks skewed to the left for $p&gt;1$ and to the right for $0&lt;p&lt;1$</li> </ul> <p>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} } &amp;\quad \text{ for } 0&lt;x&lt;1 \\ 0 &amp;\quad \text{ otherwise} \end{array} \right. \end{equation} which </p> <p>which has its peak at $p=-\frac{\log 2}{\log x}$</p> <p>Is there a similar pdf (or exactly this one) used in literature? What is it called?</p> <p><strong>PS</strong>: note that the given pdf is not symmetric: $f(x|1-p) \neq f(1-x |p)$</p> https://stats.stackexchange.com/questions/89186/probability-density-function-within-0-1-with-specifiable-mode/89192#89192 2 Answer by Tom Minka for Probability density function within [0,1] with specifiable mode Tom Minka https://stats.stackexchange.com/users/2074 2014-03-07T17:34:17Z 2014-03-07T17:34:17Z <p>That pdf is similar to the <a href="http://en.wikipedia.org/wiki/Generalized_Beta_distribution" rel="nofollow">generalized Beta distribution</a> of the first kind, which is obtained by raising a Beta variable to a power.</p>