*A tightAnother approximation of <math>Q(x)</math> for <math>x \in [0,\infty)</math> is given by Karagiannidis & Lioumpas (2007)<ref>Karagiannidis, G. K., & Lioumpas, A. S. [http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf ''An improved approximation for the Gaussian Q-function'']. 2007. Communications Letters, IEEE, 11(8), pp. 644-646.</ref> who showed for the appropriate choice of parameters <math>\{A, B\}</math> that
:: <math>f(x; A, B) = \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x} \approx \operatorname{erfc} \left(x\right).</math>
Revision as of 16:41, 26 October 2018
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,[5]
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:[8]
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.[9][10]
^P.O. Borjesson and C-E.W. Sundberg, Simple approximations of the error function Q(x) for communications applications ]. 1979. Transactions on Communications, IEEE, 27(3), pp. 639-643.
^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.