I want to use inverse transform sampling to generate some random numbers, which all fall into a given interval $(0,x_{max})$. The numbers are not necessarily distributed evenly but can be "skewed". I do not know the true distribution, all I know is that there can be some skew.
So I need a cumulative distribution function, which reaches 1 at $x_{max}$ and which has an inverse which can be written down as a formula. It should have a parameter, which allows to adjust the "skew", and which generates a uniform distribution (a straight line) as a special case.
These are quite humble requirements, but I cannot find such a function.
You might want to try the family $(F_\alpha)$ indexed by $\alpha\gt0$ and defined by $F_\alpha(x)=(x/x_{\mathrm{max}})^\alpha$ for every $x$ in $[0,x_{\mathrm{max}}]$.
Exercise: For every real number $\gamma$ there exists some value of $\alpha$ such that the skewness of $F_\alpha$ is $\gamma$.