Q-function

In statistics, the Q-function is defined as 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.

Formally, the Q-function is defined as

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

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

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