Q-function

A plot of the Q-function.

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

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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.

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

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

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 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:

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

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 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

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

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

• A tight approximation of $Q(x)$ for $x\in [0,\infty )$ is given by Karagiannidis & Lioumpas (2007) ^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 $\forall x\geq 0.$

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

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

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