In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations. Equivalently, Q(x) is the probability that a standard normal random variable takes a value larger than x. If Y is a Gaussian random variable with mean \mu and variance \sigma^2, 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 \Phi(x) is the 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(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) for the Q-function of the sum of two non-negative variables, as follows:

Bounds and approximations

Inverse Q

The inverse Q-function can be related to the inverse error functions: The function Q^{-1}(y) 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.

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.

Generalization to high dimensions

The Q-function can be generalized to higher dimensions: where follows the multivariate normal distribution with covariance \Sigma and the threshold is of the form for some positive vector and positive constant \gamma>0. 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 \gamma becomes larger and larger.