I am trying to design a function $f_{\alpha} : \left[0,1\right] \rightarrow [0,+\inf)$ that looks like the following:
It should have the following properties:
- Its integral should be 1 for any value of $\alpha$.
- When $\alpha<0$ it should be monotonically decreasing
- When $\alpha=0$ it should be be be flat.
- When $\alpha>0$ it should be monotonically increasing.
- The greater $|\alpha|$ is, "the more monotonic" the function can be (not sure if there is a better way to define this).
- It should be easy to compute, for any value of $\alpha$ and $x$ (i.e. in terms of algorithmic complexity).
- Ideally these effects should be visible for $\alpha$ in the range $(-1,1)$. If helpful we can make sure the function is only defined for $\alpha \in (-1, 1)$.
I know there is obviously an infinite number of functions that can fit this profile, but I am hoping to find something algebraically simple that fits the above description.
Some thoughts:
- The closest that comes to mind is perhaps some sub-family of the Beta pdf. Is there an easy to restrict $(\alpha, \beta)$ to produce these shapes with a new single parameter, e.g. $\gamma$?