Any one can help finding a distribution function?

Question

I am looking for a distribution function with the following characteristics:

F_x(x) is Continuous, differentiable and truncated on some [a,b]. I also need that that the density around the lower bound will be of measure zero, meaning: f_x(a)=0.

Answer 1

From extreme value theory we have the Frechét distribution that almost fulfills your requirements: It is continuous, differentiable and the density is zero at the left end-point. The distribution function is given by

$$ \Phi_\alpha(x) = e^{-x^{-\alpha}}, \quad x\geq 0 $$ for $\alpha>0$ with density $$ \varphi_\alpha(x) = \frac {{x}^{-\alpha}\alpha\,{{\rm e}^{-{x}^{-\alpha}}}}{x}, \quad x>0 $$ for which $\varphi_\alpha(x) \to 0$ for $ x\to 0$. If you want it to be upper bounded, you could truncate it but I suppose that would beat the purpose of the distribution.