In statistics, the Q-function is the tail probability of the normalized Gaussian distribution.[1][2] In other words, is the probability that a normalized Gaussian random variable will obtain a value larger than . Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.[3]
Definition and related functions
Formally, the Q-function is defined as
Thus, where is the cumulative distribution function of the normal Gaussian distribution.
The Q-function can be expressed in terms of the error function as
Bounds
The Q-function cannot be written using elementary functions. However, the bounds
become increasingly tight for large x, and are often useful.
Using the substitution and defining , the upper bound is derived as follows: