Q-function

In statistics, the Q-function is the tail probability of the normalized Gaussian distribution. In other words, $Q(x)$ is the probability that a normalized Gaussian random variable will obtain a value larger than $x$. Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.

Definition and related functions

Formally, the Q-function is defined as

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

Thus, $Q(x)=1-\Phi (x),$ 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 as

$Q(x)={\frac {1}{2}}-{\frac {1}{2}}\operatorname {erf} {\Bigl (}{\frac {x}{\sqrt {2}}}{\Bigr )}.$

Bounds

The Q-function cannot be written using elementary functions. However, the bounds

${\frac {x}{1+x^{2}}}\cdot {\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}<Q(x)<{\frac {1}{x}}\cdot {\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2},\qquad x>0,$

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

Using the substitution $v=u^{2}/2$ and defining $\varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}$, the upper bound is derived as follows:

${\begin{aligned}1-\Phi (x)&=\int _{x}^{\infty }\varphi (u)\,du\\&<\int _{x}^{\infty }{\frac {u}{x}}\varphi (u)\,du=\int _{x^{2}/2}^{\infty }{\frac {e^{-v}}{x{\sqrt {2\pi }}}}\,dv=-{\biggl .}{\frac {e^{-v}}{x{\sqrt {2\pi }}}}{\biggr |}_{x^{2}/2}^{\infty }={\frac {\varphi (x)}{x}}.\end{aligned}}$

Similarly, using $\scriptstyle \varphi '(u)\,{=}\,-u\,\varphi (u)$ and the quotient rule,

${\begin{aligned}{\Bigl (}1+{\frac {1}{x^{2}}}{\Bigr )}(1-\Phi (x))&=\int _{x}^{\infty }{\Bigl (}1+{\frac {1}{x^{2}}}{\Bigr )}\varphi (u)\,du\\&>\int _{x}^{\infty }{\Bigl (}1+{\frac {1}{u^{2}}}{\Bigr )}\varphi (u)\,du=-{\biggl .}{\frac {\varphi (u)}{u}}{\biggr |}_{x}^{\infty }={\frac {\varphi (x)}{x}}.\end{aligned}}$

Solving for $\scriptstyle 1\,{-}\,\Phi (x)\,$ provides the lower bound.