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# Approximation Formulae for the Gaussian Error Integral, Q(x)

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces approximation formulae for the Gaussian error Integral

A Gaussian pdf with unit variance is given by:

fx=12πex22 f x 1 2 x 2 2
(1)
The probability that a signal with a pdf given by fx f x lies above a given threshold xx is given by the Gaussian Error Integral or QQ function:
Qx=xfud u Q x u x f u
(2)
There is no analytical solution to this integral, but it has a simple relationship to the error function, erfx erf x , or its complement, erfcx erfc x , which are tabulated in many books of mathematical tables.
erfx=2π0xeu2d u erf x 2 u 0 x u 2
(3)
and
erfcx=1erfx=2πxeu2d u erfc x 1 erf x 2 u x u 2
(4)
Therefore,
Qx=12erfcx2=12(1erfx2) Q x 1 2 erfc x 2 1 2 1 erf x 2
(5)
Note that erf0=0 erf 0 0 and erf=1 erf 1 , and therefore Q0=0.5 Q 0 0.5 and Qx0 Q x 0 very rapidly as xx becomes large.

It is useful to derive simple approximations to Qx Q x which can be used on a calculator and avoid the need for tables.

Let v=ux v u x , then:

Qx=0fv+xd v =12π0ev2+2vx+x22d v =ex222π0e(vx)ev22d v 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
(6)
Now if x1 x 1 , we may obtain an approximate solution by replacing the ev22 v 2 2 term in the integral by unity, since it will initially decay much slower than the e(vx) v x term. Therefore
Qx<ex222π0e(vx)d v =ex222πx Q x x 2 2 2 v 0 v x x 2 2 2 x
(7)
This approximation is an upper bound, and its ratio to the true value of Qx Q x becomes less than 1.11.1 only when x>3 x 3 , as shown in Figure 1. We may obtain a much better approximation to Qx Q x by altering the denominator above from ( 2πx 2 x ) to ( 1.64x+0.76x2+4 1.64 x 0.76 x 2 4 ) to give:
Qxex221.64x+0.76x2+4 Q x x 2 2 1.64 x 0.76 x 2 4
(8)
This improved approximation gives a curve indistinguishable from Qx Q x in Figure 1 and its ratio to the true Qx Q x is now within ±0.3% ± 0.3 % of unity for all x0 x 0 as shown in Figure 2. This accuracy is sufficient for nearly all practical problems.

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