Is there a name for the normal CDF function $\Phi(\cdot)$?


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:


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Andrew Mao
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Andrew Mao

Updated on August 01, 2022


  • Andrew Mao
    Andrew Mao over 1 year

    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?

  • Andrew Mao
    Andrew Mao about 10 years
    Sorry, but no one will take me seriously if I call $\Phi(x)$ the error function.
  • electronpusher
    electronpusher almost 2 years
    @AndrewMao I don't think that's what he was suggesting. But your function is a linear transformation of the error function.