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I am looking for a family of functions to model some data.
I found this question, and the functions I need are quite similar: I need to define a family (one parameter) of monotonic curves

Just as in the linked question, I am looking for a one parameter family where the parameter gives the curvature of the functions. I also want to have a straight line for some middle parameter, a straight line for $a = 1/2$ and $a$ ranging from $0$ to $1$ would be totally fine.

Now for the difference with the linked question: In the linked question, we have two points given, $(0,Y_0)$ and $(X_0,0)$. These two span a rectangle and everything should be inside this rectangle.
Unfortunately, I can't do that, as I don't have these points given. If $f_a$ is the function, then what I have is a certain point $X_1$ with $0 < X_1 < X_0$ and I know the fraction of areas (not the areas themselves)

$$\frac{\int_0^{X_1} f_a(x) dx}{\int_0^{X_0} f_a(x) dx}.$$

For a fixed $a$, I want to be able to find $X_0$ from this information. I already managed to solve this problem for the case of a straight line, but I currently don't know which family of functions to choose so that it is possible to find $X_0$ from the given fraction and $a$ properly describes the curvature.

Any suggestions are highly appreciated.

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