This is an old revision of this page, as edited by 165.12.252.111(talk) at 00:50, 6 February 2015(Listed formula is not in source given). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.
Revision as of 00:50, 6 February 2015 by 165.12.252.111(talk)(Listed formula is not in source given)
In statistics, the Q-function is the tail probability of the standard normal distribution.[1][2] 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.
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 and more useful form of the Q-function known as Craig's formula, after its discoverer, is expressed as:[4]
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.
Improved exponential bounds and a pure exponential approximation are [5]
A tight approximation for the whole range of arguments is given by Karagiannidis & Lioumpas (2007) [6][failed verification]
Inverse Q
The inverse Q-function can be trivially related to the inverse error function:
Values
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.