A Gaussian pdf with unit variance is given by: 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: 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. and Therefore, Note that erf0=0 erf 0 0 and erf∞=1 erf 1 , and therefore Q0=0.5 Q 0 0.5 and Qx→0 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=u-x v u x , then: Now if x≫1 ≫ x 1 , we may obtain an approximate solution by replacing the ⅇ-v22 v 2 2 term in the integral by unity, since it will initially decay much slower than the ⅇ-vx v x term. Therefore 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: 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 x≥0 x 0 as shown in Figure 2. This accuracy is sufficient for nearly all practical problems.