Calculating the inverse of a function involving the error function

[ergospherical]

I have a probability distribution over the interval ##[0, \infty)## given by $$f(x) = \frac{x^2}{2\sqrt{\pi} a^3} \exp\left(- \frac{x^2}{4a^2} \right)$$From this I want to derive a formula for the inverse cumulative density function, ##F^{-1}##. The cumulative density function is a slightly nasty-looking but doable integral involving the error function,$$F(x) = \mathrm{erf}\left( \frac{x}{2a} \right) - \frac{x}{\sqrt{\pi} a} \exp \left( -\frac{x^2}{4a^2} \right)$$So it remains to invert this. Ideally I would like to find an analytic expression, but I haven't had much success.

 

[anuttarasammyak]

For a=1 the plot is

So the expected plot of the inverse function is

It seems difficult to get the anaytical form if not impossible.

 

[ergospherical]

I think it might be. Was just checking if there is a clever way.

 

[Gavran]

The inverse function $$ F^{-1}(x) $$ of the cumulative density function $$ F(x) = erf(\frac{x}{2a}) – \frac{x}{\sqrt\pi a} exp(-\frac{x^2}{4a^2}) $$ can not be expressed in an analytic form.