# Q-function

In statistics, the Q-function is the tail distribution function 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. Equivalently, $Q(x)$ is the probability that a standard normal random variable takes a value larger than $x$.
If $Y$ is a Gaussian random variable with mean $\mu$ and variance $\sigma ^{2}$, then $X={\frac {Y-\mu }{\sigma }}$ is standard normal and
    $P(Y>y)=P(X>x)=Q(x)$
where $x={\frac {y-\mu }{\sigma }}$.
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 $\Phi (x)$ is the cumulative distribution function of the standard normal Gaussian distribution.
The Q-function can be expressed in terms of the error function, or the complementary error function, as
    ${\begin{aligned}Q(x)&={\frac {1}{2}}\left({\frac {2}{\sqrt {\pi }}}\int _{x/{\sqrt {2}}}^{\infty }\exp \left(-t^{2}\right)\,dt\right)\\&={\frac {1}{2}}-{\frac {1}{2}}\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)~~{\text{ -or-}}\\&={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right).\end{aligned}}$
An alternative form of the Q-function known as Craig's formula, after its discoverer, 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 Q(x) for negative values. This form is advantageous in that the range of integration is fixed and finite.
Craig's formula was later extended by Behnad (2020) for the Q-function of the sum of two non-negative variables, as follows:
    the Q-function plotted in the complex plane$Q(x+y)={\frac {1}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{2\sin ^{2}\theta }}-{\frac {y^{2}}{2\cos ^{2}\theta }}\right)d\theta ,\quad x,y\geqslant 0.$

## Bounds and approximations

* The Q-function is not an elementary function. However, it can be upper and lower bounded as,
       $\left({\frac {x}{1+x^{2}}}\right)\phi (x)<Q(x)<{\frac {\phi (x)}{x}},\qquad x>0,$
    where $\phi (x)$ is the density function of the standard normal distribution, and the bounds become increasingly tight for large x.
    Using the substitution v =u$^2$/2, the upper bound is derived as follows:
       $Q(x)=\int _{x}^{\infty }\phi (u)\,du<\int _{x}^{\infty }{\frac {u}{x}}\phi (u)\,du=\int _{\frac {x^{2}}{2}}^{\infty }{\frac {e^{-v}}{x{\sqrt {2\pi }}}}\,dv=-{\biggl .}{\frac {e^{-v}}{x{\sqrt {2\pi }}}}{\biggr |}_{\frac {x^{2}}{2}}^{\infty }={\frac {\phi (x)}{x}}.$
    Similarly, using $\phi '(u)=-u\phi (u)$ and the quotient rule,
       $\left(1+{\frac {1}{x^{2}}}\right)Q(x)=\int _{x}^{\infty }\left(1+{\frac {1}{x^{2}}}\right)\phi (u)\,du>\int _{x}^{\infty }\left(1+{\frac {1}{u^{2}}}\right)\phi (u)\,du=-{\biggl .}{\frac {\phi (u)}{u}}{\biggr |}_{x}^{\infty }={\frac {\phi (x)}{x}}.$
    Solving for Q(x) provides the lower bound.
    The geometric mean of the upper and lower bound gives a suitable approximation for $Q(x)$:
       $Q(x)\approx {\frac {\phi (x)}{\sqrt {1+x^{2}}}},\qquad x\geq 0.$
* Tighter bounds and approximations of $Q(x)$ can also be obtained by optimizing the following expression
       ${\tilde {Q}}(x)={\frac {\phi (x)}{(1-a)x+a{\sqrt {x^{2}+b}}}}.$
    For $x\geq 0$, the best upper bound is given by $a=0.344$ and $b=5.334$ with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by $a=0.339$ and $b=5.510$ with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by $a=1/\pi$ and $b=2\pi$ with maximum absolute relative error of 1.17%.
* 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
       $Q(x)\leq {\tfrac {1}{4}}e^{-x^{2}}+{\tfrac {1}{4}}e^{-{\frac {x^{2}}{2}}}\leq {\tfrac {1}{2}}e^{-{\frac {x^{2}}{2}}},\qquad x>0$
       $Q(x)\approx {\frac {1}{12}}e^{-{\frac {x^{2}}{2}}}+{\frac {1}{4}}e^{-{\frac {2}{3}}x^{2}},\qquad x>0$
* The above were generalized by Tanash & Riihonen (2020), who showed that $Q(x)$ can be accurately approximated or bounded by
       ${\tilde {Q}}(x)=\sum _{n=1}^{N}a_{n}e^{-b_{n}x^{2}}.$
    In particular, they presented a systematic methodology to solve the numerical coefficients $\{(a_{n},b_{n})\}_{n=1}^{N}$ that yield a minimax approximation or bound: $Q(x)\approx {\tilde {Q}}(x)$, $Q(x)\leq {\tilde {Q}}(x)$, or $Q(x)\geq {\tilde {Q}}(x)$ for $x\geq 0$. With the example coefficients tabulated in the paper for $N=20$, the relative and absolute approximation errors are less than $2.831\cdot 10^{-6}$ and $1.416\cdot 10^{-6}$, respectively. The coefficients $\{(a_{n},b_{n})\}_{n=1}^{N}$ for many variations of the exponential approximations and bounds up to $N=25$ have been released to open access as a comprehensive dataset.
* Another approximation of $Q(x)$ for $x\in [0,\infty )$ is given by Karagiannidis & Lioumpas (2007) 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 $\forall x\geq 0.$
    Substituting these values and using the relationship between $Q(x)$ and $\operatorname {erfc} (x)$ from above gives
       $Q(x)\approx {\frac {\left(1-e^{\frac {-1.98x}{\sqrt {2}}}\right)e^{-{\frac {x^{2}}{2}}}}{1.135{\sqrt {2\pi }}x}},x\geq 0.$
    Alternative coefficients are also available for the above 'Karagiannidis–Lioumpas approximation' for tailoring accuracy for a specific application or transforming it into a tight bound.
* A tighter and more tractable approximation of $Q(x)$ for positive arguments $x\in [0,\infty )$ is given by López-Benítez & Casadevall (2011) based on a second-order exponential function:
       $Q(x)\approx e^{-ax^{2}-bx-c},\qquad x\geq 0.$
    The fitting coefficients $(a,b,c)$ can be optimized over any desired range of arguments in order to minimize the sum of square errors ($a=0.3842$, $b=0.7640$, $c=0.6964$ for $x\in [0,20]$) or minimize the maximum absolute error ($a=0.4920$, $b=0.2887$, $c=1.1893$ for $x\in [0,20]$). This approximation offers some benefits such as a good trade-off between accuracy and analytical tractability (for example, the extension to any arbitrary power of $Q(x)$ is trivial and does not alter the algebraic form of the approximation).
* A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments $x\in [0,\infty )$ was introduced by Abreu (2012) based on a simple algebraic expression with only two exponential terms:
       $Q(x)\geq {\frac {1}{12}}e^{-x^{2}}+{\frac {1}{{\sqrt {2\pi }}(x+1)}}e^{-x^{2}/2},\qquad x\geq 0,$
       $Q(x)\leq {\frac {1}{50}}e^{-x^{2}}+{\frac {1}{2(x+1)}}e^{-x^{2}/2},\qquad x\geq 0.$
These bounds are derived from a unified form $Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}}$, where the parameters $a$ and $b$ are chosen to satisfy specific conditions ensuring the lower ($a_{\mathrm {L} }=12$, $b_{\mathrm {L} }={\sqrt {2\pi }}$) and upper ($a_{\mathrm {U} }=50$, $b_{\mathrm {U} }=2$) bounding properties. The resulting expressions are notable for their simplicity and tightness, offering a favorable trade-off between accuracy and mathematical tractability. These bounds are particularly useful in theoretical analysis, such as in communication theory over fading channels. Additionally, they can be extended to bound $Q^{n}(x)$ for positive integers $n$ using the binomial theorem, maintaining their simplicity and effectiveness.

## Inverse Q

The inverse Q-function can be related to the inverse error functions:
    $Q^{-1}(y)={\sqrt {2}}\ \mathrm {erf} ^{-1}(1-2y)={\sqrt {2}}\ \mathrm {erfc} ^{-1}(2y)$
The function $Q^{-1}(y)$ finds application in digital communications. It is usually expressed in dB and generally called Q-factor:
    $\mathrm {Q{\text{-}}factor} =20\log _{10}\!\left(Q^{-1}(y)\right)\!~\mathrm {dB}$
where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for quadrature phase-shift keying (QPSK) in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to y.

## Values

The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, 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:
    $Q(\mathbf {x} )=\mathbb {P} (\mathbf {X} \geq \mathbf {x} ),$
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.