## 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)

### References

Bowling, Shannon R., Mohammad T. Khasawneh, Sittichai Kaewkuekool, and Byung Rae Cho. 2009. “A Logistic Approximation to the Cumulative Normal Distribution.” Journal of Industrial Engineering and Management 2 (1): 114–27.