The normal distribution does not have a closed form cdf (per Wikipedia). Is there a way to find its intersection with the line x = y?
Currently I use statistical software (R) to search through a fine grid of x's until I find an x value such that $\Phi(x) = x$. Is there a better way, either solving it analytically or computationally but takes less time?
It exists a nice approximation formula for the cdf:
$$\large{\Phi (x) \approx 0.5 \cdot \Bigg( 1 + \sqrt{1-e^{- \left(\sqrt{\frac{\pi}{8}} \cdot x^2\right)}} \Bigg)}=x$$
This equation can not solved algebraically. But you can apply an approximation method, for instance the Newton-Raphson method. Wolfram alpha calculated the value $x=0.78215$. And $\Phi (0.78215)=0.78294$
The exact value, with 5 decimal places, is $0.78326$
The article where the formula is mentioned can be found here. It is the journal Applied Mathematical Sciences, Vol. 2, 2008, no. 9, 425 - 429
