Q-function

A plot of the Q-function.

In statistics, the Q-function is the tail probability of the standard normal distribution. 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:

${\frac {y-\eta }{\sigma }}$

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.

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

$Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }\exp \left(-{\frac {u^{2}}{2}}\right)\,du.$

Thus,

$Q(x)=1-Q(-x)=1-\Phi (x)\,\!,$

where Φ(x) is the cumulative distribution function of the normal Gaussian distribution.

The Q-function can be expressed in terms of the error function, or the complementary error function, as

$Q(x)={\frac {1}{2}}\left({\frac {2}{\sqrt {\pi }}}\int _{x/{\sqrt {2}}}^{\infty }\exp \left(-t^{2}\right)\,dt\right)={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right)={\frac {1}{2}}-{\frac {1}{2}}\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right).$

An alternative form of the Q-function, also known as Craig's formula after its discoverer, that is more useful is expressed as:

$Q(x)={\frac {1}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{2\sin ^{2}\theta }}\right)d\theta .$

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 for the negative values. This form is advantageous in that the range of integration is finite.

• The Q-function is not an elementary function. However, the bounds

$\left({\frac {x}{1+x^{2}}}\right)\varphi (x)<Q(x)<{\frac {\varphi (x)}{x}},\qquad x>0,$

become increasingly tight for large x, and are often useful.

Using the substitution v =u²/2 and defining the upper bound is derived as follows:

$Q(x)=\int _{x}^{\infty }\varphi (u)\,du<\int _{x}^{\infty }{\frac {u}{x}}\varphi (u)\,du=\int _{\frac {x^{2}}{2}}^{\infty }{\frac {e^{-v}}{x{\sqrt {2\pi }}}}\,dv=-{\biggl .}{\frac {e^{-v}}{x{\sqrt {2\pi }}}}{\biggr |}_{\frac {x^{2}}{2}}^{\infty }={\frac {\varphi (x)}{x}}.$

Similarly, using φ′(u) = −uφ(u) and the quotient rule,

$\left(1+{\frac {1}{x^{2}}}\right)Q(x)=\int _{x}^{\infty }\left(1+{\frac {1}{x^{2}}}\right)\varphi (u)\,du>\int _{x}^{\infty }\left(1+{\frac {1}{u^{2}}}\right)\varphi (u)\,du=-{\biggl .}{\frac {\varphi (u)}{u}}{\biggr |}_{x}^{\infty }={\frac {\varphi (x)}{x}}.$

Solving for Q(x) provides the lower bound.

• Chernoff bound of Q-function is

$Q(x)\leq {\tfrac {1}{2}}e^{-{\frac {x^{2}}{2}}},\qquad x>0$

• A pure exponential approximation is given by Chiani, Dardari & Simon (2012)

$Q(x)\approx {\frac {1}{12}}e^{-{\frac {x^{2}}{2}}}+{\frac {1}{4}}e^{-{\frac {2}{3}}x^{2}}\qquad x>0$

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.

Q(0.0) = 0.500000000
Q(0.1) = 0.460172163
Q(0.2) = 0.420740291
Q(0.3) = 0.382088578
Q(0.4) = 0.344578258
Q(0.5) = 0.308537539
Q(0.6) = 0.274253118
Q(0.7) = 0.241963652
Q(0.8) = 0.211855399
Q(0.9) = 0.184060125

Q(1.0) = 0.158655254
Q(1.1) = 0.135666061
Q(1.2) = 0.115069670
Q(1.3) = 0.096800485
Q(1.4) = 0.080756659
Q(1.5) = 0.066807201
Q(1.6) = 0.054799292
Q(1.7) = 0.044565463
Q(1.8) = 0.035930319
Q(1.9) = 0.028716560

Q(2.0) = 0.022750132
Q(2.1) = 0.017864421
Q(2.2) = 0.013903448
Q(2.3) = 0.010724110
Q(2.4) = 0.008197536
Q(2.5) = 0.006209665
Q(2.7) = 0.003466974
Q(2.8) = 0.002555170
Q(2.9) = 0.001865813

Q(3.0) = 0.001349898
Q(3.1) = 0.000967603
Q(3.2) = 0.000687138
Q(3.3) = 0.000483424
Q(3.4) = 0.000336929
Q(3.5) = 0.000232629
Q(3.6) = 0.000159109
Q(3.7) = 0.000107800
Q(3.8) = 0.000072348
Q(3.9) = 0.000048096
Q(4.0) = 0.000031671

References

• Chiani, M.; Dardari, D.; Simon, M. K. (2003), "New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels", IEEE Trans. on Wireless Communications, 4 (2): 840–845, doi:10.1109/TWC.2003.814350.