Name for probability density function supported on a closed interval and increasing I am looking for names/examples/references for probability density functions which are supported on a closed interval, say $[0,1]$, and increasing there.
If $f(x)$ is positive and increasing on $[a,b]$ and $I=\int_a^b f(x)dx$ then $g=f/I$ would do as such a PDF. But what are some examples where such functions show up in practice or applications as PDF?

Fun question. Here are a few distributions that come to mind ... This is illustrative, but should be relatively straightforward to derive first derivatives etc, if needed.



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*$\text{Beta}(a,b)$ distribution, with parameter $a > 1$  and $b\leq1$



In the above plot, $b=0.97$. The $b = 1$ case is plotted below separately as the Power Function.



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*$\text{Bradford}(\beta)$ distribution with parameter $-1<\beta<0$






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*$\text{PowerFunction}(a)$ with parameter $a>1$ (special case of Beta)






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*Two-component mix of Triangular and Uniform






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*Variation on a $\text{Leipnik}(\theta)$ distribution with parameter $0<\theta<1$



