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Revision as of 16:43, 31 October 2016 by Paolostar(talk | contribs)(The Q-function is also called Q-factor in digital communications.)
In statistics, the Q-function (also called Q-factor) is the tail probability of the standard normal distribution.[1][2] In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations above the mean.
If the underlying random variable is y, then the proper argument to the tail probability is derived as:
which expresses the number of standard deviations away from the mean.
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.
Improved exponential bounds and a pure exponential approximation are [5]
A tight approximation of for is given by Karagiannidis & Lioumpas (2007)[6] who showed for the appropriate choice of parameters that
The absolute error between and over the range is minimized by evaluating
Using and numerically integrating, they found the minimum error occurred when which gave a good approximation for
Substituting these values and using the relationship between and from above gives
Inverse Q
The inverse Q-function can be trivially related to the inverse error function:
Values
The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as R, 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]
^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 (Statistical Methodology). doi:10.1111/rssb.12162.