Closed form solutions to a Gaussian equation

Asked Nov 15 '18 at 19:08

Active Nov 15 '18 at 20:03

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Let $\phi(t) := \frac{1}{\sqrt{2\pi}}\exp\{-t^2/2\}$ be the standard Gaussian pdf function and $\Phi(t) := \int_{-\infty}^t \phi(u)du$ be the Gaussian CDF function. Consider equation $$ \Phi(x) + \phi(x) = 1. $$

I'm wondering whether such an equation has simple closed-form solutions $x$?

edited Nov 15 '18 at 19:25

Alejandro Nasif Salum

5,703

asked Nov 15 '18 at 19:08

Yining Wang

1,279
7
23

If you still interested in an answer give a reply. – callculus42 Nov 16 '18 at 12:40
Looks like $x=39.9$ (https://www.wolframalpha.com/input/?i=N%5BPDF%5BNormalDistribution%5B0,+1%5D,+39.9%5D%5D%2BN%5BCDF%5BNormalDistribution%5B0,+Sqrt%5B1%5D%5D,+39.9%5D%5D). – StubbornAtom Nov 16 '18 at 15:14

Closed form solutions to a Gaussian equation

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