Statistics function
In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, formula_1 is the probability that a normal (Gaussian) random variable will obtain a value larger than formula_2 standard deviations. Equivalently, formula_1 is the probability that a standard normal random variable takes a value larger than formula_2.
If formula_5 is a Gaussian random variable with mean formula_6 and variance formula_7, then formula_8 is standard normal and
formula_9
where formula_10.
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.
Definition and basic properties.
Formally, the "Q"-function is defined as
formula_11
Thus,
formula_12
where formula_13 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
formula_14
An alternative form of the "Q"-function known as Craig's formula, after its discoverer, is expressed as:
formula_15
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: 
formula_16
formula_17
where formula_18 is the density function of the standard normal distribution, and the bounds become increasingly tight for large "x".
Using the substitution "v" ="u"2/2, the upper bound is derived as follows:
formula_19
Similarly, using formula_20 and the quotient rule,
formula_21
Solving for "Q"("x") provides the lower bound.
The geometric mean of the upper and lower bound gives a suitable approximation for formula_1:
formula_23
 formula_25
For formula_26, the best upper bound is given by formula_27 and formula_28 with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by formula_29 and formula_30 with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by formula_31 and formula_32 with maximum absolute relative error of 1.17%.
formula_33
formula_34
 formula_35
formula_37
In particular, they presented a systematic methodology to solve the numerical coefficients formula_38 that yield a minimax approximation or bound: formula_39, formula_40, or formula_41 for formula_42. With the example coefficients tabulated in the paper for formula_43, the relative and absolute approximation errors are less than formula_44 and formula_45, respectively. The coefficients formula_38 for many variations of the exponential approximations and bounds up to formula_47 have been released to open access as a comprehensive dataset.
 formula_51
 The absolute error between formula_52 and formula_53 over the range formula_54 is minimized by evaluating
 formula_55
 Using formula_56 and numerically integrating, they found the minimum error occurred when formula_57 which gave a good approximation for formula_58
 Substituting these values and using the relationship between formula_1 and formula_53 from above gives
 formula_61
 Alternative coefficients are also available for the above 'Karagiannidis–Lioumpas approximation' for tailoring accuracy for a specific application or transforming it into a tight bound.
 formula_64
 The fitting coefficients formula_65 can be optimized over any desired range of arguments in order to minimize the sum of square errors (formula_66, formula_67, formula_68 for formula_69) or minimize the maximum absolute error (formula_70, formula_71, formula_72 for formula_69). This approximation offers some benefits such as a good trade-off between accuracy and analytical tractability (for example, the extension to any arbitrary power of formula_1 is trivial and does not alter the algebraic form of the approximation).
Inverse "Q".
The inverse "Q"-function can be related to the inverse error functions:
formula_75
The function formula_76 finds application in digital communications. It is usually expressed in dB and generally called Q-factor:
formula_77
where "y" is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for quadrature phase-shift keying (QPSK) in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to "y". 
Values.
The "Q"-function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, MATLAB and Mathematica. Some values of the "Q"-function are given below for reference.
 ;templatestyles src="Col-begin/styles.css"/ ;
Generalization to high dimensions.
The "Q"-function can be generalized to higher dimensions:
formula_78
where formula_79 follows the multivariate normal distribution with covariance formula_80 and the threshold is of the form
formula_81 for some positive vector formula_82 and positive constant formula_83. As in the one dimensional case, there is no simple analytical formula for the "Q"-function. Nevertheless, the "Q"-function can be approximated arbitrarily well as formula_84 becomes larger and larger.