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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?

Maesumi
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1 Answers1

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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.

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

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

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

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

enter image description here

  • Two-component mix of Triangular and Uniform

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

enter image description here

wolfies
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