In statistics, the Q-function is the tail distribution function of the standard normal distribution.[1][2] In other words, is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations. Equivalently, is the probability that a standard normal random variable takes a value larger than .
If is a Gaussian random variable with mean and variance , then is standard normal and
where .
Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.[3]
Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics.
The Q-function can be expressed in terms of the error function, or the complementary error function, as[2]
An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:[4]
This expression is valid only for positive values of x, but it can be used in conjunction with Q(x) = 1 − Q(−x) to obtain Q(x) for negative values. This form is advantageous in that the range of integration is fixed and finite.
Bounds and approximations
The Q-function is not an elementary function. However, the bounds, where is the density function of the standard normal distribution,
become increasingly tight for large x, and are often useful.
Using the substitutionv =u2/2, the upper bound is derived as follows:
The function finds application in digital communications. It is usually expressed in dB and generally called Q-factor:
where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for QPSK in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to y.
Values
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(0.0)
0.500000000
1/2.0000
Q(0.1)
0.460172163
1/2.1731
Q(0.2)
0.420740291
1/2.3768
Q(0.3)
0.382088578
1/2.6172
Q(0.4)
0.344578258
1/2.9021
Q(0.5)
0.308537539
1/3.2411
Q(0.6)
0.274253118
1/3.6463
Q(0.7)
0.241963652
1/4.1329
Q(0.8)
0.211855399
1/4.7202
Q(0.9)
0.184060125
1/5.4330
Q(1.0)
0.158655254
1/6.3030
Q(1.1)
0.135666061
1/7.3710
Q(1.2)
0.115069670
1/8.6904
Q(1.3)
0.096800485
1/10.3305
Q(1.4)
0.080756659
1/12.3829
Q(1.5)
0.066807201
1/14.9684
Q(1.6)
0.054799292
1/18.2484
Q(1.7)
0.044565463
1/22.4389
Q(1.8)
0.035930319
1/27.8316
Q(1.9)
0.028716560
1/34.8231
Q(2.0)
0.022750132
1/43.9558
Q(2.1)
0.017864421
1/55.9772
Q(2.2)
0.013903448
1/71.9246
Q(2.3)
0.010724110
1/93.2478
Q(2.4)
0.008197536
1/121.9879
Q(2.5)
0.006209665
1/161.0393
Q(2.6)
0.004661188
1/214.5376
Q(2.7)
0.003466974
1/288.4360
Q(2.8)
0.002555130
1/391.3695
Q(2.9)
0.001865813
1/535.9593
Q(3.0)
0.001349898
1/740.7967
Q(3.1)
0.000967603
1/1033.4815
Q(3.2)
0.000687138
1/1455.3119
Q(3.3)
0.000483424
1/2068.5769
Q(3.4)
0.000336929
1/2967.9820
Q(3.5)
0.000232629
1/4298.6887
Q(3.6)
0.000159109
1/6285.0158
Q(3.7)
0.000107800
1/9276.4608
Q(3.8)
0.000072348
1/13822.0738
Q(3.9)
0.000048096
1/20791.6011
Q(4.0)
0.000031671
1/31574.3855
Generalization to high dimensions
The Q-function can be generalized to higher dimensions:[7]
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 approximated arbitrarily well as becomes larger and larger.[8][9]
^Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal Res. Nat. Bur. Standards Sect. B. 66: 93–96.
^Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. arXiv:1603.04166. doi:10.1111/rssb.12162.