Gaussian Distribution CDF in terms of Error Function

Theorem

Let $\map \Phi x$ denote the cumulative distribution function of the standard Gaussian distribution.

Then:

$\ds \map \Phi x = \dfrac 1 2 \paren {1 - \map \erf {\dfrac x {\sqrt 2} } }$

where $\erf$ denotes the error function.

Proof

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Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error function