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 finite.

$\left (\frac{x}{1+x^2} \right ) \phi(x) < Q(x) < \frac{\phi(x)}{x}, \qquad x>0,$
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.
$Q(x)\leq e^{-\frac{x^2}{2}}, \qquad x>0$
• 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 for the whole range of arguments is given by Karagiannidis & Lioumpas (2007) [6][not in citation given]
$Q(x)\approx\frac{\left( 1-e^{-1.4x}\right) e^{-\frac{x^{2}}{2}}}{1.135\sqrt{2\pi}x}, x >0$

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.