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 (Gaussian) 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. Formally, the Q-function is defined as Thus, where 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: The Q-function is not an elementary function. However, the Borjesson-Sundberg bounds, where is the density function of the standard normal distribution, 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 and the quotient rule, Solving for Q(x) provides the lower bound. The geometric mean of the upper and lower bound gives a suitable approximation for : Tighter bounds and approximations of can also be obtained by optimizing the following expression For , the best upper bound is given by and with maximum absolute relative error of 0.