Q-function

[Image:Q-function.png|thumb|right|400px|A plot of the Q-function.]
In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, is the probability that a normal 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.
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.

Definition and basic properties

Formally, the Q-function is defined as
Thus,
where is the [Standard normal distribution#Cumulative distribution function|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.
Craig's formula was later extended by Behnad for the Q-function of the sum of two non-negative variables, as follows:

Bounds and approximations

The Q-function is not an elementary function. However, it can be upper and lower bounded as,
Tighter bounds and approximations of can also be obtained by optimizing the following expression
The Chernoff bound of the Q-function is
Improved exponential bounds and a pure exponential approximation are
The above were generalized by Tanash & Riihonen, who showed that can be accurately approximated or bounded by
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:
A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments was introduced by Abreu based on a simple algebraic expression with only two exponential terms:

These bounds are derived from a unified form, where the parameters and are chosen to satisfy specific conditions ensuring the lower and upper bounding properties. The resulting expressions are notable for their simplicity and tightness, offering a favorable trade-off between accuracy and mathematical tractability. These bounds are particularly useful in theoretical analysis, such as in communication theory over fading channels. Additionally, they can be extended to bound for positive integers using the binomial theorem, maintaining their simplicity and effectiveness.

Inverse ''Q''

The inverse Q-function can be related to the inverse error functions:
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 of the digitally modulated signal under analysis. For instance, for quadrature phase-shift keying 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	0.500000000	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.