- #1
ergospherical
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- 1,305
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.