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In statistics, the Q-function is the tail distribution function of the standard normal distribution.cite-ref-1[1]cite-ref-jo-2-0[2] In other words, Q ( x ) {\displaystyle Q(x)} is the probability that a normal (Gaussian) random variable will obtain a value larger than x {\displaystyle x} standard deviations. Equivalently, Q ( x ) {\displaystyle Q(x)} is the probability that a standard normal random variable takes a value larger than x {\displaystyle x} .

If Y {\displaystyle Y} is a Gaussian random variable with mean μ μ {\displaystyle \mu } and variance σ σ 2 {\displaystyle \sigma ^{2}} , then X = Y − − μ μ σ σ {\displaystyle X={\frac {Y-\mu }{\sigma }}} is standard normal and

P ( Y > y ) = P ( X > x ) = Q ( x ) {\displaystyle P(Y>y)=P(X>x)=Q(x)}

where x = y − − μ μ σ σ {\displaystyle 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.cite-ref-3[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

• Bounds and approximations

• Inverse Q

• Values

• Generalization to high dimensions

• References

Formally, the Q-function is defined as

Q ( x ) = 1 2 π π ∫ ∫ x ∞ ∞ exp ⁡ ⁡ ( − − u 2 2 ) d u . {\displaystyle 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 − − Φ Φ ( x ) , {\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,}

where Φ Φ ( x ) {\displaystyle \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, ascite-ref-jo-2-1[2]

Q ( x ) = 1 2 ( 2 π π ∫ ∫ x / 2 ∞ ∞ exp ⁡ ⁡ ( − − t 2 ) d t ) = 1 2 − − 1 2 erf ⁡ ⁡ ( x 2 ) -or- = 1 2 erfc ⁡ ⁡ ( x 2 ) . {\displaystyle {\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:cite-ref-4[4]

Q ( x ) = 1 π π ∫ ∫ 0 π π 2 exp ⁡ ⁡ ( − − x 2 2 sin 2 ⁡ ⁡ θ θ ) d θ θ . {\displaystyle 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)cite-ref-5[5] for the Q-function of the sum of two non-negative variables, as follows:

Q ( x + y ) = 1 π π ∫ ∫ 0 π π 2 exp ⁡ ⁡ ( − − x 2 2 sin 2 ⁡ ⁡ θ θ − − y 2 2 cos 2 ⁡ ⁡ θ θ ) d θ θ , x , y ⩾ ⩾ 0. {\displaystyle 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.}

• The Q-function is not an elementary function. However, it can be upper and lower bounded as,cite-ref-gordon-6-0[6]cite-ref-borjesson-7-0[7]

( x 1 + x 2 ) ϕ ϕ ( x ) < Q ( x ) < ϕ ϕ ( x ) x , x > 0 , {\displaystyle \left({\frac {x}{1+x^{2}}}\right)\phi (x)<Q(x)<{\frac {\phi (x)}{x}},\qquad x>0,}

where ϕ ϕ ( x ) {\displaystyle \phi (x)} is the density function of the standard normal distribution, and the bounds become increasingly tight for large x.

Using the substitution v =u2/2, the upper bound is derived as follows:

Q ( x ) = ∫ ∫ x ∞ ∞ ϕ ϕ ( u ) d u < ∫ ∫ x ∞ ∞ u x ϕ ϕ ( u ) d u = ∫ ∫ x 2 2 ∞ ∞ e − − v x 2 π π d v = − − e − − v x 2 π π | x 2 2 ∞ ∞ = ϕ ϕ ( x ) x . {\displaystyle 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 ϕ ϕ ′ ( u ) = − − u ϕ ϕ ( u ) {\displaystyle \phi '(u)=-u\phi (u)} and the quotient rule,

( 1 + 1 x 2 ) Q ( x ) = ∫ ∫ x ∞ ∞ ( 1 + 1 x 2 ) ϕ ϕ ( u ) d u > ∫ ∫ x ∞ ∞ ( 1 + 1 u 2 ) ϕ ϕ ( u ) d u = − − ϕ ϕ ( u ) u | x ∞ ∞ = ϕ ϕ ( x ) x . {\displaystyle \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 ) {\displaystyle Q(x)} :

Q ( x ) ≈ ≈ ϕ ϕ ( x ) 1 + x 2 , x ≥ ≥ 0. {\displaystyle Q(x)\approx {\frac {\phi (x)}{\sqrt {1+x^{2}}}},\qquad x\geq 0.}

• Tighter bounds and approximations of Q ( x ) {\displaystyle Q(x)} can also be obtained by optimizing the following expression cite-ref-borjesson-7-1[7]

Q ~ ~ ( x ) = ϕ ϕ ( x ) ( 1 − − a ) x + a x 2 + b . {\displaystyle {\tilde {Q}}(x)={\frac {\phi (x)}{(1-a)x+a{\sqrt {x^{2}+b}}}}.}

For x ≥ ≥ 0 {\displaystyle x\geq 0} , the best upper bound is given by a = 0.344 {\displaystyle a=0.344} and b = 5.334 {\displaystyle b=5.334} with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by a = 0.339 {\displaystyle a=0.339} and b = 5.510 {\displaystyle b=5.510} with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by a = 1 / π π {\displaystyle a=1/\pi } and b = 2 π π {\displaystyle b=2\pi } with maximum absolute relative error of 1.17%.

• The Chernoff bound of the Q-function is

Q ( x ) ≤ ≤ e − − x 2 2 , x > 0 {\displaystyle Q(x)\leq e^{-{\frac {x^{2}}{2}}},\qquad x>0}

• Improved exponential bounds and a pure exponential approximation are cite-ref-8[8]

Q ( x ) ≤ ≤ 1 4 e − − x 2 + 1 4 e − − x 2 2 ≤ ≤ 1 2 e − − x 2 2 , x > 0 {\displaystyle 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 ) ≈ ≈ 1 12 e − − x 2 2 + 1 4 e − − 2 3 x 2 , x > 0 {\displaystyle 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),cite-ref-9[9] who showed that Q ( x ) {\displaystyle Q(x)} can be accurately approximated or bounded by

Q ~ ~ ( x ) = ∑ ∑ n = 1 N a n e − − b n x 2 . {\displaystyle {\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 {\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}} that yield a minimax approximation or bound: Q ( x ) ≈ ≈ Q ~ ~ ( x ) {\displaystyle Q(x)\approx {\tilde {Q}}(x)} , Q ( x ) ≤ ≤ Q ~ ~ ( x ) {\displaystyle Q(x)\leq {\tilde {Q}}(x)} , or Q ( x ) ≥ ≥ Q ~ ~ ( x ) {\displaystyle Q(x)\geq {\tilde {Q}}(x)} for x ≥ ≥ 0 {\displaystyle x\geq 0} . With the example coefficients tabulated in the paper for N = 20 {\displaystyle N=20} , the relative and absolute approximation errors are less than 2.831 ⋅ ⋅ 10 − − 6 {\displaystyle 2.831\cdot 10^{-6}} and 1.416 ⋅ ⋅ 10 − − 6 {\displaystyle 1.416\cdot 10^{-6}} , respectively. The coefficients { ( a n , b n ) } n = 1 N {\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}} for many variations of the exponential approximations and bounds up to N = 25 {\displaystyle N=25} have been released to open access as a comprehensive dataset.cite-ref-10[10]

• Another approximation of Q ( x ) {\displaystyle Q(x)} for x ∈ ∈ [ 0 , ∞ ∞ ) {\displaystyle x\in [0,\infty )} is given by Karagiannidis & Lioumpas (2007)cite-ref-11[11] who showed for the appropriate choice of parameters { A , B } {\displaystyle \{A,B\}} that

f ( x ; A , B ) = ( 1 − − e − − A x ) e − − x 2 B π π x ≈ ≈ erfc ⁡ ⁡ ( x ) . {\displaystyle 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 ) {\displaystyle f(x;A,B)} and erfc ⁡ ⁡ ( x ) {\displaystyle \operatorname {erfc} (x)} over the range [ 0 , R ] {\displaystyle [0,R]} is minimized by evaluating

{ A , B } = arg ⁡ ⁡ min { A , B } 1 R ∫ ∫ 0 R | f ( x ; A , B ) − − erfc ⁡ ⁡ ( x ) | d x . {\displaystyle \{A,B\}={\underset {\{A,B\}}{\arg \min }}{\frac {1}{R}}\int _{0}^{R}|f(x;A,B)-\operatorname {erfc} (x)|dx.}

Using R = 20 {\displaystyle R=20} and numerically integrating, they found the minimum error occurred when { A , B } = { 1.98 , 1.135 } , {\displaystyle \{A,B\}=\{1.98,1.135\},} which gave a good approximation for ∀ ∀ x ≥ ≥ 0. {\displaystyle \forall x\geq 0.}

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

Q ( x ) ≈ ≈ ( 1 − − e − − 1.98 x 2 ) e − − x 2 2 1.135 2 π π x , x ≥ ≥ 0. {\displaystyle 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.cite-ref-12[12]

• A tighter and more tractable approximation of Q ( x ) {\displaystyle Q(x)} for positive arguments x ∈ ∈ [ 0 , ∞ ∞ ) {\displaystyle x\in [0,\infty )} is given by López-Benítez & Casadevall (2011)cite-ref-13[13] based on a second-order exponential function:

Q ( x ) ≈ ≈ e − − a x 2 − − b x − − c , x ≥ ≥ 0. {\displaystyle Q(x)\approx e^{-ax^{2}-bx-c},\qquad x\geq 0.}

The fitting coefficients ( a , b , c ) {\displaystyle (a,b,c)} can be optimized over any desired range of arguments in order to minimize the sum of square errors ( a = 0.3842 {\displaystyle a=0.3842} , b = 0.7640 {\displaystyle b=0.7640} , c = 0.6964 {\displaystyle c=0.6964} for x ∈ ∈ [ 0 , 20 ] {\displaystyle x\in [0,20]} ) or minimize the maximum absolute error ( a = 0.4920 {\displaystyle a=0.4920} , b = 0.2887 {\displaystyle b=0.2887} , c = 1.1893 {\displaystyle c=1.1893} for x ∈ ∈ [ 0 , 20 ] {\displaystyle 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 ) {\displaystyle 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 ∈ ∈ [ 0 , ∞ ∞ ) {\displaystyle x\in [0,\infty )} was introduced by Abreu (2012)cite-ref-14[14] based on a simple algebraic expression with only two exponential terms:

Q ( x ) ≥ ≥ 1 12 e − − x 2 + 1 2 π π ( x + 1 ) e − − x 2 / 2 , x ≥ ≥ 0 , {\displaystyle 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 ) ≤ ≤ 1 50 e − − x 2 + 1 2 ( x + 1 ) e − − x 2 / 2 , x ≥ ≥ 0. {\displaystyle 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 B ( x ; a , b ) = exp ⁡ ⁡ ( − − x 2 ) a + exp ⁡ ⁡ ( − − x 2 / 2 ) b ( x + 1 ) {\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}}} , where the parameters a {\displaystyle a} and b {\displaystyle b} are chosen to satisfy specific conditions ensuring the lower ( a L = 12 {\displaystyle a_{\mathrm {L} }=12} , b L = 2 π π {\displaystyle b_{\mathrm {L} }={\sqrt {2\pi }}} ) and upper ( a U = 50 {\displaystyle a_{\mathrm {U} }=50} , b U = 2 {\displaystyle 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 ) {\displaystyle Q^{n}(x)} for positive integers n {\displaystyle n} using the binomial theorem, maintaining their simplicity and effectiveness.

The inverse Q-function can be related to the inverse error functions:

Q − − 1 ( y ) = 2 e r f − − 1 ( 1 − − 2 y ) = 2 e r f c − − 1 ( 2 y ) {\displaystyle Q^{-1}(y)={\sqrt {2}}\ \mathrm {erf} ^{-1}(1-2y)={\sqrt {2}}\ \mathrm {erfc} ^{-1}(2y)}

The function Q − − 1 ( y ) {\displaystyle Q^{-1}(y)} finds application in digital communications. It is usually expressed in dB and generally called Q-factor:

Q - f a c t o r = 20 log 10 ( Q − − 1 ( y ) ) d B {\displaystyle \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.

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.

The Q-function can be generalized to higher dimensions:cite-ref-15[15]

Q ( x ) = P ( X ≥ ≥ x ) , {\displaystyle Q(\mathbf {x} )=\mathbb {P} (\mathbf {X} \geq \mathbf {x} ),}

where X ∼ ∼ N ( 0 , Σ Σ ) {\displaystyle \mathbf {X} \sim {\mathcal {N}}(\mathbf {0} ,\,\Sigma )} follows the multivariate normal distribution with covariance Σ Σ {\displaystyle \Sigma } and the threshold is of the form x = γ γ Σ Σ l ∗ ∗ {\displaystyle \mathbf {x} =\gamma \Sigma \mathbf {l} ^{*}} for some positive vector l ∗ ∗ > 0 {\displaystyle \mathbf {l} ^{*}>\mathbf {0} } and positive constant γ γ > 0 {\displaystyle \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 γ γ {\displaystyle \gamma } becomes larger and larger.cite-ref-16[16]cite-ref-bmc17-17-0[17]

cite-note-11. ↑ "The Q-function". cnx.org. Archived from the original on 2012-02-29.

cite-note-jo-22. ↑ "Basic properties of the Q-function" (PDF). 2009-03-05. Archived from the original (PDF) on 2009-03-25.

cite-note-33. ↑ Normal Distribution Function – from Wolfram MathWorld

cite-note-44. ↑ citerefcraig1991Craig, J.W. (1991). "A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations" (PDF). MILCOM 91 - Conference record. pp. 571–575. doi:10.1109/MILCOM.1991.258319. ISBN 0-87942-691-8. S2CID 16034807. Archived from the original (PDF) on 2012-04-03. Retrieved 2011-11-16.

cite-note-55. ↑ citerefbehnad2020Behnad, Aydin (2020). "A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis". IEEE Transactions on Communications. 68 (7): 4117–4125. doi:10.1109/TCOMM.2020.2986209. S2CID 216500014.

cite-note-gordon-66. ↑ citerefgordon1941Gordon, R.D. (1941). "Values of Mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the argument". Ann. Math. Stat. 12 (3): 364–366. doi:10.1214/aoms/1177731721.

cite-note-borjesson-77. ↑ citerefborjessonsundberg1979Borjesson, P.; Sundberg, C.-E. (1979). "Simple Approximations of the Error Function Q(x) for Communications Applications". IEEE Transactions on Communications. 27 (3): 639–643. doi:10.1109/TCOM.1979.1094433.

cite-note-88. ↑ citerefchianidardarisimon2003Chiani, M.; Dardari, D.; Simon, M.K. (2003). "New exponential bounds and approximations for the computation of error probability in fading channels" (PDF). IEEE Transactions on Wireless Communications. 24 (5): 840–845. doi:10.1109/TWC.2003.814350. Archived from the original (PDF) on 2014-10-20. Retrieved 2014-10-20.

cite-note-99. ↑ citereftanashriihonen2020Tanash, I.M.; Riihonen, T. (2020). "Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials". IEEE Transactions on Communications. 68 (10): 6514–6524. arXiv:2007.06939. doi:10.1109/TCOMM.2020.3006902. S2CID 220514754.

cite-note-1010. ↑ citereftanashriihonen2020Tanash, I.M.; Riihonen, T. (2020). "Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]". Zenodo. doi:10.5281/zenodo.4112978.

cite-note-1111. ↑ citerefkaragiannidislioumpas2007Karagiannidis, George; Lioumpas, Athanasios (2007). "An Improved Approximation for the Gaussian Q-Function" (PDF). IEEE Communications Letters. 11 (8): 644–646. doi:10.1109/LCOMM.2007.070470. S2CID 4043576.

cite-note-1212. ↑ citereftanashriihonen2021Tanash, I.M.; Riihonen, T. (2021). "Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function". IEEE Communications Letters. 25 (5): 1468–1471. arXiv:2101.07631. doi:10.1109/LCOMM.2021.3052257. S2CID 231639206.

cite-note-1313. ↑ citereflopez-benitezcasadevall2011Lopez-Benitez, Miguel; Casadevall, Fernando (2011). "Versatile, Accurate, and Analytically Tractable Approximation for the Gaussian Q-Function" (PDF). IEEE Transactions on Communications. 59 (4): 917–922. doi:10.1109/TCOMM.2011.012711.100105. S2CID 1145101.

cite-note-1414. ↑ citerefabreu2012Abreu, Giuseppe (2012). "Very Simple Tight Bounds on the Q-Function". IEEE Transactions on Communications. 60 (9): 2415–2420. doi:10.1109/TCOMM.2012.080612.110075.

cite-note-1515. ↑ citerefsavage1962Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal of Research of the National Bureau of Standards Section B. 66 (3): 93–96. doi:10.6028/jres.066B.011. Zbl 0105.12601.

cite-note-1616. ↑ citerefbotev2016Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. 79: 125–148. arXiv:1603.04166. Bibcode:2016arXiv160304166B. doi:10.1111/rssb.12162. S2CID 88515228.

cite-note-bmc17-1717. ↑ citerefbotevmackinlaychen2017Botev, Z. I.; Mackinlay, D.; Chen, Y.-L. (2017). "Logarithmically efficient estimation of the tail of the multivariate normal distribution". 2017 Winter Simulation Conference (WSC). IEEE. pp. 1903–191. doi:10.1109/WSC.2017.8247926. ISBN 978-1-5386-3428-8. S2CID 4626481.