Let $\Phi(x)$ be a the Standard Normal Univariate Gaussian CDF function, $\beta \in(0,1)$, and the standard deviations $\sigma_i$ be positive. Does anybody have intuition into solving the following for $x$? $$\dfrac{1}{N}\sum_{i=1}^N\Phi\left(\dfrac{x-m_i}{\sigma_i}\right) = \beta$$

I think there should be a unique solution here, so i was hoping to "invert" this sum. Any help is much appreciated. Thanks in advance!

  • $\begingroup$ I am afraid that the only solution would involve numerical methods. $\endgroup$ – Claude Leibovici Apr 8 at 4:01

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