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## 3.12 Probability-Related Functions

### 3.12.1 Normal Cumulative Distribution Functions

The error function erf is related to the standard normal cumulative distribution function $$\Phi$$ by scaling. See section normal distribution for the general normal cumulative distribution function (and its complement).

R erf(T x)
error function, also known as the Gauss error function, of x

R erfc(T x)
complementary error function of x

R Phi(T x)
standard normal cumulative distribution function of x

R inv_Phi(T x)
standard normal inverse cumulative distribution function of p, otherwise known as the quantile function

R Phi_approx(T x)
fast approximation of the unit (may replace Phi for probit regression with maximum absolute error of 0.00014, see (Bowling et al. 2009) for details)