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I can't seem to find a plain English name for the CDF of the normal distribution $\Phi(x)$. However, I am aware of several other related functions that have a name, so I feel like this one should as well.

  • The CDF of the logistic distribution $\sigma(x) = \frac{1}{1+\exp(-x)}$ is known as the logistic function. (It also happens to be sigmoidal like this one.)
  • The inverse normal CDF $\Phi^{-1}(x)$ is known as the probit function.

Doesn't this function have a name as well?

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If $X$~$N$(0, 1), then the cdf $\Phi (x)$ is related to the Error Function by:

$\Phi (x) = \frac{1}{2}+ \frac{1}{2} \operatorname{erf} \left(x/ \sqrt{2}\right)$

For more detail on the error function, see for instance: http://en.wikipedia.org/wiki/Error_function

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    $\begingroup$ Sorry, but no one will take me seriously if I call $\Phi(x)$ the error function. $\endgroup$ – Andrew Mao Oct 8 '13 at 4:42

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