In statistics, the Q-function is the tail probability of the standard normal distribution.[1][2] In other words, is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations above the mean. Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.[3]
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.
The Q-function can be expressed in terms of the error function, or the complementary error function, as[2]
An alternative form of the Q-function that is more useful is expressed as:[4]
This expression is valid only for positive values of , but it can be used in conjunction with to obtain for the negative values. This form is advantageous in that the range of integration is finite.
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.