Closed form solutions to a Gaussian equation











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Let $phi(t) := frac{1}{sqrt{2pi}}exp{-t^2/2}$ 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 $x$?






probability gaussian-integral transcendental-equations







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  • If you still interested in an answer give a reply.
    – callculus
    Nov 16 at 12:40










  • Looks like $x=39.9$ (wolframalpha.com/input/…).
    – StubbornAtom
    Nov 16 at 15:14















up vote
0
down vote

favorite












Let $phi(t) := frac{1}{sqrt{2pi}}exp{-t^2/2}$ 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 $x$?






probability gaussian-integral transcendental-equations







share|cite|improve this question
























  • If you still interested in an answer give a reply.
    – callculus
    Nov 16 at 12:40










  • Looks like $x=39.9$ (wolframalpha.com/input/…).
    – StubbornAtom
    Nov 16 at 15:14













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $phi(t) := frac{1}{sqrt{2pi}}exp{-t^2/2}$ 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 $x$?






probability gaussian-integral transcendental-equations







share|cite|improve this question















Let $phi(t) := frac{1}{sqrt{2pi}}exp{-t^2/2}$ 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 $x$?






probability gaussian-integral transcendental-equations




probability gaussian-integral transcendental-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 15 at 19:25









Alejandro Nasif Salum

3,629117




3,629117










asked Nov 15 at 19:08









Yining Wang

774315




774315












  • If you still interested in an answer give a reply.
    – callculus
    Nov 16 at 12:40










  • Looks like $x=39.9$ (wolframalpha.com/input/…).
    – StubbornAtom
    Nov 16 at 15:14


















  • If you still interested in an answer give a reply.
    – callculus
    Nov 16 at 12:40










  • Looks like $x=39.9$ (wolframalpha.com/input/…).
    – StubbornAtom
    Nov 16 at 15:14
















If you still interested in an answer give a reply.
– callculus
Nov 16 at 12:40




If you still interested in an answer give a reply.
– callculus
Nov 16 at 12:40












Looks like $x=39.9$ (wolframalpha.com/input/…).
– StubbornAtom
Nov 16 at 15:14




Looks like $x=39.9$ (wolframalpha.com/input/…).
– StubbornAtom
Nov 16 at 15:14















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