Approximation Formula for the Gaussian Error Integral, Q(x) 2.3 2003/03/13 2003/03/18 Nick Kingsbury ngk10@cam.ac.uk Liqun Wang liqun@rice.edu Nick Kingsbury ngk10@cam.ac.uk Gaussian Error Integral Gaussian pdf Q function This module introduces approximation formulae for the Gaussian error Integral A Gaussian pdf with unit variance is given by: f x 1 2 x 2 2 The probability that a signal with a pdf given by f x lies above a given threshold x is given by the Gaussian Error Integral or Q function: Q x u x f u There is no analytical solution to this integral, but it has a simple relationship to the error function, erf x , or its complement, erfc x , which are tabulated in many books of mathematical tables. erf x 2 u 0 x u 2 and erfc x 1 erf x 2 u x u 2 Therefore, Q x 1 2 erfc x 2 1 2 1 erf x 2 Note that erf 0 0 and erf 1 , and therefore Q 0 0.5 and Q x 0 very rapidly as x becomes large. It is useful to derive simple approximations to Q x which can be used on a calculator and avoid the need for tables. Let v u x , then: Q x v 0 f v x 1 2 v 0 v 2 2 v x x 2 2 x 2 2 2 v 0 v x v 2 2 Now if ≫ x 1 , we may obtain an approximate solution by replacing the v 2 2 term in the integral by unity, since it will initially decay much slower than the v x term. Therefore Q x x 2 2 2 v 0 v x x 2 2 2 x This approximation is an upper bound, and its ratio to the true value of Q x becomes less than 1.1 only when x 3 , as shown in . We may obtain a much better approximation to Q x by altering the denominator above from ( 2 x ) to ( 1.64 x 0.76 x 2 4 ) to give: Q x x 2 2 1.64 x 0.76 x 2 4 This improved approximation gives a curve indistinguishable from Q x in and its ratio to the true Q x is now within ± 0.3 % of unity for all x 0 as shown in . This accuracy is sufficient for nearly all practical problems.

Q x and the simple approximation of .

The ration of the improved approximation of Q x in to the true value, obtained by numerical integration.