I Calculating the inverse of a function involving the error function

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The discussion centers on deriving the inverse cumulative density function, F^{-1}, from the given probability distribution function, f(x). The cumulative density function, F(x), involves an integral that includes the error function, making it complex. Participants note that finding an analytic expression for the inverse function appears to be very challenging, if not impossible. The consensus is that there may not be a clever method to achieve an analytic form for F^{-1}(x). Overall, the difficulty lies in the nature of the error function and the resulting complexity of the cumulative density function.
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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.
 
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For a=1 the plot is
1712272242615.png


So the expected plot of the inverse function is

1712272420660.png

It seems difficult to get the anaytical form if not impossible.
 
I think it might be. Was just checking if there is a clever way.
 
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.
 
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