In statistics, the Q-function is the tail distribution function of the standard normal distribution. is the probability that a normal (Gaussian) random variable.

is standard normal and

.

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,

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:

The above proper integral form of Q-function, has been incorrectly credited to Craig. This form of Q-function was implied in earlier works by Wiesten , and explicitly stated by Pawula, Rice and Roberts.

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

• The Q-function is not an elementary function. However, it can be upper and lower bounded as,

• is the density function of the standard normal distribution, and the bounds become increasingly tight for large x.

  Using the substitution v =u2/2, the upper bound is derived as follows:

  and the quotient rule,

  Solving for Q(x) provides the lower bound.

  The geometric mean:

• can also be obtained by optimizing the following expression

• with maximum absolute relative error of 1.17%.

• The Chernoff bound of the Q-function is

  Improved exponential bounds and a pure exponential approximation are

  The above were generalized by Tanash & Riihonen (2020), can be accurately approximated or bounded by

• that yield a minimax have been released to open access as a comprehensive dataset.

• is given by Karagiannidis & Lioumpas (2007) that

• is minimized by evaluating

  from above gives

  Alternative coefficients are also available for the above 'Karagiannidis–Lioumpas approximation' for tailoring accuracy for a specific application or transforming it into a tight bound.

• is given by López-Benítez & Casadevall (2011) based on a second-order exponential function:

• is trivial and does not alter the algebraic form of the approximation).

• was introduced by Abreu (2012) based on a simple algebraic expression with only two exponential terms:

using the binomial theorem, maintaining their simplicity and effectiveness.

Inverse Q

The inverse Q-function can be related to the inverse error functions:

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:

follows the multivariate normal distribution. 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 becomes larger and larger.

References