Formulae for Calculation of Normal Probability

Abstract

Various approximate formulae for calculation of the normal probability integral

$$ P(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{0}^{x}e^{-t^2/2}dt,\ \ x\ge 0, $$

are given. Our formulae provide a good approximation to P(x) over the entire range 0 < x < ∞, hence they can be used in practice instead of the usual numerical tables. Error bounds based on the Peano kernel technique are proved.

keywordsNormal probability, error function, quadrature formulae, Peano kernels