In statistics, the Q-function is the tail probability 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 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.
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
where Φ(x) is the cumulative distribution function of the normal Gaussian distribution.
An alternative and more useful 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 finite.
- The Q-function is not an elementary function. However, the bounds
- become increasingly tight for large x, and are often useful.
- Using the substitution v =u2/2, the upper bound is derived as follows:
- Similarly, using φ′(u) = −uφ(u) and the quotient rule,
- Solving for Q(x) provides the lower bound.
- The Chernoff bound of the Q-function is
- A pure exponential approximation is given by Chiani, Dardari & Simon (2003)
- A tight approximation for the whole range of arguments is given by Karagiannidis & Lioumpas (2007) 
The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as Matlab and Mathematica. Some values of the Q-function are given below for reference.
Q(0.0) = 0.500000000
Q(1.0) = 0.158655254
Q(2.0) = 0.022750132
Q(3.0) = 0.001349898
|This article needs additional citations for verification. (September 2011)|
- The Q-function, from cnx.org
- Basic properties of the Q-function
- Normal Distribution Function - from Wolfram MathWorld
- John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellaions, Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575.
- Karagiannidis, G. K., & Lioumpas, A. S. (2007). An improved approximation for the Gaussian Q-function. Communications Letters, IEEE, 11(8), 644-646.