Q-function

A plot of the Q-function.

In statistics, the Q-function is the tail probability of the standard normal distribution $\phi(x)$.^[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:

$x=\frac{y - \mu}{\sigma}$

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.^[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.

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 $\Phi(x)$ 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^[2]

$\begin{align} Q(x) &=\frac{1}{2}\left( \frac{2}{\sqrt{\pi}} \int_{x/\sqrt{2}}^\infty \exp\left(-t^2\right) \, dt \right)\\ &= \frac{1}{2} - \frac{1}{2} \operatorname{erf} \left( \frac{x}{\sqrt{2}} \right) ~~\text{ -or-}\\ &= \frac{1}{2}\operatorname{erfc} \left(\frac{x}{\sqrt{2}} \right). \end{align}$

An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:^[4]

$Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} \right) d\theta.$

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 Q(x) for negative values. This form is advantageous in that the range of integration is fixed and finite.

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

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

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

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

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

Similarly, using $\phi'(u) = - u \phi(u)$ and the quotient rule,

$\left(1+\frac1{x^2}\right)Q(x) =\int_x^\infty \left(1+\frac1{x^2}\right)\phi(u)\,du >\int_x^\infty \left(1+\frac1{u^2}\right)\phi(u)\,du =-\biggl.\frac{\phi(u)}u\biggr|_x^\infty =\frac{\phi(x)}x.$

Solving for Q(x) provides the lower bound.

• The Chernoff bound of the Q-function is

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

• Improved exponential bounds and a pure exponential approximation are ^[5]

$Q(x)\leq \tfrac{1}{4}e^{-x^2}+\tfrac{1}{4}e^{-\frac{x^2}{2}} \leq \tfrac{1}{2}e^{-\frac{x^2}{2}}, \qquad x>0$

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

• A tight approximation of $Q(x)$ for $x \in [0,\infty)$ is given by Karagiannidis & Lioumpas (2007)^{[6] Fixed} who showed for the appropriate choice of parameters $\{A, B\}$ that

$f(x; A, B) = \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x} \approx \operatorname{erfc} \left(x\right).$

The absolute error between $f(x; A, B)$ and $\operatorname{erfc}(x)$ over the range $[0, R]$ is minimized by evaluating

$\{A, B\} = \underset{\{A,B\}}{arg\ min} \frac{1}{R} \int_0^R | f(x; A, B) - \operatorname{erfc}(x) |dx.$

Using $R = 20$ and numerically integrating, they found the minimum error occurred when $\{A, B\} = \{1.98, 1.135\},$ which gave a good approximation for $\forall x \ge 0.$

Substituting these values and using the relationship between $Q(x)$ and $\operatorname{erfc}(x)$ from above gives

$Q(x)\approx\frac{\left( 1-e^{-1.4x}\right) e^{-\frac{x^{2}}{2}}}{1.135\sqrt{2\pi}x}, x \ge 0.$

Inverse Q

The inverse Q-function can be trivially related to the inverse error function:

$Q^{-1}(x) = \sqrt{2}\ \mathrm{erf}^{-1}(1-2x)$

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 1/2.0000 Q(0.1) 0.460172163 1/2.1731 Q(0.2) 0.420740291 1/2.3768 Q(0.3) 0.382088578 1/2.6172 Q(0.4) 0.344578258 1/2.9021 Q(0.5) 0.308537539 1/3.2411 Q(0.6) 0.274253118 1/3.6463 Q(0.7) 0.241963652 1/4.1329 Q(0.8) 0.211855399 1/4.7202 Q(0.9) 0.184060125 1/5.4330

Q(1.0) 0.158655254 1/6.3030 Q(1.1) 0.135666061 1/7.3710 Q(1.2) 0.115069670 1/8.6904 Q(1.3) 0.096800485 1/10.3305 Q(1.4) 0.080756659 1/12.3829 Q(1.5) 0.066807201 1/14.9684 Q(1.6) 0.054799292 1/18.2484 Q(1.7) 0.044565463 1/22.4389 Q(1.8) 0.035930319 1/27.8316 Q(1.9) 0.028716560 1/34.8231

Q(2.0) 0.022750132 1/43.9558 Q(2.1) 0.017864421 1/55.9772 Q(2.2) 0.013903448 1/71.9246 Q(2.3) 0.010724110 1/93.2478 Q(2.4) 0.008197536 1/121.9879 Q(2.5) 0.006209665 1/161.0393 Q(2.6) 0.004661188 1/214.5376 Q(2.7) 0.003466974 1/288.4360 Q(2.8) 0.002555130 1/391.3695 Q(2.9) 0.001865813 1/535.9593

Q(3.0) 0.001349898 1/740.7967 Q(3.1) 0.000967603 1/1033.4815 Q(3.2) 0.000687138 1/1455.3119 Q(3.3) 0.000483424 1/2068.5769 Q(3.4) 0.000336929 1/2967.9820 Q(3.5) 0.000232629 1/4298.6887 Q(3.6) 0.000159109 1/6285.0158 Q(3.7) 0.000107800 1/9276.4608 Q(3.8) 0.000072348 1/13822.0738 Q(3.9) 0.000048096 1/20791.6011 Q(4.0) 0.000031671 1/31574.3855

Generalization to high dimensions

The Q-function can be generalized to higher dimensions:^[7]

$Q(\mathbf{x})= \mathbb{P}(\mathbf{X}\geq \mathbf{x}),$

where $\mathbf{X}\sim \mathcal{N}(\mathbf{0},\, \Sigma)$ follows the multivariate normal distribution with covariance $\Sigma$ and the threshold is of the form $\mathbf{x}=\gamma\Sigma\mathbf{l}^*$ for some positive vector $\mathbf{l}^*>\mathbf{0}$ and positive constant $\gamma>0$. 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 $\gamma$ becomes larger and larger.^[8]

References

1. The Q-function, from cnx.org
2. Basic properties of the Q-function
3. Normal Distribution Function - from Wolfram MathWorld
4. John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellaions, Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575.
5. Chiani, M., Dardari, D., Simon, M.K. (2003). New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels. IEEE Transactions on Wireless Communications, 4(2), 840–845, doi=10.1109/TWC.2003.814350.
6. Karagiannidis, G. K., & Lioumpas, A. S. (2007). An improved approximation for the Gaussian Q-function. Communications Letters, IEEE, 11(8), 644-646.
7. Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal Res. Nat. Bur. Standards Sect. B 66: 93–96.
8. Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society: Series B (Statistical Methodology). doi:10.1111/rssb.12162.