Q-function

In statistics, the Q-function is the tail distribution function of the standard normal distribution.

is the probability that a normal (Gaussian) random variable will obtain a value larger than

is the probability that a standard normal random variable takes a value larger than

is a Gaussian random variable with 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.

Craig's formula was later extended by Behnad (2020)[5] for the Q-function of the sum of two non-negative variables, as follows: 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 (BER) of the digitally modulated signal under analysis.

For instance, for quadrature phase-shift keying (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.

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.

The Q-function can be generalized to higher dimensions:[14] where

follows the multivariate normal distribution with covariance

As in the one dimensional case, there is no simple analytical formula for the Q-function.