Formally, the Q-function is defined as Thus, where is the cumulative distribution function of the standard normal Gaussian distribution. The Q-function can be expressed in terms of the error function, or the complementary error function, as An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as: This expression is valid only for positive values of x, but it can be used in conjunction with Q = 1 − Q to obtain Q for negative values. This form is advantageous in that the range of integration is fixed and finite.
Improved exponential bounds and a pure exponential approximation are
Another approximation of for is given by Karagiannidis & Lioumpas who showed for the appropriate choice of parameters that
A tighter and more tractable approximation of for positive arguments is given by López-Benítez & Casadevall based on a second-order exponential function:
The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, MATLAB and Mathematica. Some values of the Q-function are given below for reference.
Q
0.500000000
1/2.0000
Q
0.460172163
1/2.1731
Q
0.420740291
1/2.3768
Q
0.382088578
1/2.6172
Q
0.344578258
1/2.9021
Q
0.308537539
1/3.2411
Q
0.274253118
1/3.6463
Q
0.241963652
1/4.1329
Q
0.211855399
1/4.7202
Q
0.184060125
1/5.4330
Q
0.158655254
1/6.3030
Q
0.135666061
1/7.3710
Q
0.115069670
1/8.6904
Q
0.096800485
1/10.3305
Q
0.080756659
1/12.3829
Q
0.066807201
1/14.9684
Q
0.054799292
1/18.2484
Q
0.044565463
1/22.4389
Q
0.035930319
1/27.8316
Q
0.028716560
1/34.8231
Q
0.022750132
1/43.9558
Q
0.017864421
1/55.9772
Q
0.013903448
1/71.9246
Q
0.010724110
1/93.2478
Q
0.008197536
1/121.9879
Q
0.006209665
1/161.0393
Q
0.004661188
1/214.5376
Q
0.003466974
1/288.4360
Q
0.002555130
1/391.3695
Q
0.001865813
1/535.9593
Q
0.001349898
1/740.7967
Q
0.000967603
1/1033.4815
Q
0.000687138
1/1455.3119
Q
0.000483424
1/2068.5769
Q
0.000336929
1/2967.9820
Q
0.000232629
1/4298.6887
Q
0.000159109
1/6285.0158
Q
0.000107800
1/9276.4608
Q
0.000072348
1/13822.0738
Q
0.000048096
1/20791.6011
Q
0.000031671
1/31574.3855
Generalization to high dimensions
The Q-function can be generalized to higher dimensions: where follows the multivariate normal distribution with covariance and the threshold is of the form for some positive vector and positive constant. As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, the Q-function can be as becomes larger and larger.
