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Q-function

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</style><div role="note" class="hatnote navigation-not-searchable">For the phase-space function representing a quantum state, see <a href="Husimi_Q_representation" title="Husimi Q representation">Husimi Q representation</a>.</div>

In <a href="Statistics" title="Statistics">statistics</a>, the Q-function is the <a href="Cumulative_distribution_function#Complementary_cumulative_distribution_function_(tail_distribution)" title="Cumulative distribution function">tail distribution function</a> of the <a href="Normal_distribution#Standard_normal_distribution" title="Normal distribution">standard normal distribution</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-jo-2">[2]</a> In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

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<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

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</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"> is the probability that a normal (Gaussian) <a href="Random_variable" title="Random variable">random variable</a> will obtain a value larger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> standard deviations. Equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

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<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

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</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"> is the probability that a standard normal random variable takes a value larger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is a Gaussian random variable with mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> and variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>

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</math><img src="./53a5c55e536acf250c1d3e0f754be5692b843ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\frac {Y-\mu }{\sigma }}}">

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<annotation encoding="application/x-tex">{\displaystyle X={\frac {Y-\mu }{\sigma }}}</annotation>

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</math><img src="./f30d73af6f8ef290506eb5fe95fd309a058d3c4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.93ex; height:5.343ex;" alt="{\displaystyle X={\frac {Y-\mu }{\sigma }}}" loading="lazy"> is <a href="Normal_distribution#Standard_normal_distribution" title="Normal distribution">standard normal</a> and

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<annotation encoding="application/x-tex">{\displaystyle P(Y>y)=P(X>x)=Q(x)}</annotation>

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</math><img src="./ffef0a19966873b2bcebf3e5a144256dfd0f36f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.719ex; height:2.843ex;" alt="{\displaystyle P(Y>y)=P(X>x)=Q(x)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {y-\mu }{\sigma }}}">

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<annotation encoding="application/x-tex">{\displaystyle x={\frac {y-\mu }{\sigma }}}</annotation>

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</math><img src="./00a390dea67bc4a681cb7f7b446b0fdb3dd44da7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.662ex; height:5.176ex;" alt="{\displaystyle x={\frac {y-\mu }{\sigma }}}" loading="lazy">.

Other definitions of the Q-function, all of which are simple transformations of the normal <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>, are also used occasionally.<a href="#cite_note-3">[3]</a>

Because of its relation to the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of the normal distribution, the Q-function can also be expressed in terms of the <a href="Error_function" title="Error function">error function</a>, which is an important function in applied mathematics and physics.

<div class="mw-heading mw-heading2"><h2 id="Definition_and_basic_properties">Definition and basic properties</h2></div>

Formally, the Q-function is defined as

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<annotation encoding="application/x-tex">{\displaystyle Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }\exp \left(-{\frac {u^{2}}{2}}\right)\,du.}</annotation>

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</math><img src="./ca22deb7d0539ca8e113f4beb6be8acb8dd673bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.037ex; height:6.676ex;" alt="{\displaystyle Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }\exp \left(-{\frac {u^{2}}{2}}\right)\,du.}" loading="lazy"></dd></dl>

Thus,

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<annotation encoding="application/x-tex">{\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,}</annotation>

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</math><img src="./cb960b7bc440681f7617f5571bb54c1437a51ed8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.429ex; height:2.843ex;" alt="{\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation>

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</math><img src="./79e4f01c93494fbb5dcd75761f4468121b00b294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="{\displaystyle \Phi (x)}" loading="lazy"> is the <a href="Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">cumulative distribution function of the standard normal Gaussian distribution</a>.

The Q-function can be expressed in terms of the <a href="Error_function" title="Error function">error function</a>, or the complementary error function, as<a href="#cite_note-jo-2">[2]</a>

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./319e67050923498582788fe262857654262eba2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:36.817ex; height:19.509ex;" alt="{\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}}}" loading="lazy"></dd></dl>

An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle Q(x)={\frac {1}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{2\sin ^{2}\theta }}\right)d\theta .}</annotation>

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</math><img src="./14be158f63c514eaa9657aa01cf556fae88bdd15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.823ex; height:6.676ex;" alt="{\displaystyle Q(x)={\frac {1}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{2\sin ^{2}\theta }}\right)d\theta .}" loading="lazy"></dd></dl>

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)<a href="#cite_note-5">[5]</a> for the Q-function of the sum of two non-negative variables, as follows:

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<annotation encoding="application/x-tex">{\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.}</annotation>

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</math><img src="./62c1d041885c56af665ed6909a01388509b02bd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.825ex; height:6.676ex;" alt="{\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.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Bounds_and_approximations">Bounds and approximations</h2></div>

<ul><li>The Q-function is not an <a href="Elementary_function" title="Elementary function">elementary function</a>. However, it can be upper and lower bounded as,<a href="#cite_note-Gordon-6">[6]</a><a href="#cite_note-Borjesson-7">[7]</a></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {x}{1+x^{2}}}\right)\phi (x)<Q(x)<{\frac {\phi (x)}{x}},\qquad x>0,}</annotation>

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</math><img src="./1020ae098d55f15b0a8ba5fbe95ce994e0e65fd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.007ex; height:6.343ex;" alt="{\displaystyle \left({\frac {x}{1+x^{2}}}\right)\phi (x)<Q(x)<{\frac {\phi (x)}{x}},\qquad x>0,}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)}</annotation>

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</math><img src="./546b660b2f3cfb5f34be7b3ed8371d54f5c74227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.524ex; height:2.843ex;" alt="{\displaystyle \phi (x)}" loading="lazy"> is the density function of the standard normal distribution, and the bounds become increasingly tight for large x.</dd></dl>

<dl><dd>Using the <a href="Integration_by_substitution" title="Integration by substitution">substitution</a> v =u2/2, the upper bound is derived as follows:</dd></dl>

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<annotation encoding="application/x-tex">{\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}}.}</annotation>

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</math><img src="./b73fa8e3328351d9d11cb1566f104d39846c6bb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:78.747ex; height:7.176ex;" alt="{\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}}.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Similarly, using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi '(u)=-u\phi (u)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi '(u)=-u\phi (u)}</annotation>

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<dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}.}">

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<annotation encoding="application/x-tex">{\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}}.}</annotation>

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</math><img src="./a44fa4cdc8b6084d47fdeec91f9ef53b3054e8c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:90.53ex; height:6.343ex;" alt="{\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}}.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Solving for Q(x) provides the lower bound.</dd></dl>

<dl><dd>The <a href="Geometric_mean" title="Geometric mean">geometric mean</a> of the upper and lower bound gives a suitable approximation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

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<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

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</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy">:</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle Q(x)\approx {\frac {\phi (x)}{\sqrt {1+x^{2}}}},\qquad x\geq 0.}</annotation>

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</math><img src="./019c6d534ad29d06165b86edd213f4c85752ba39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.539ex; height:7.009ex;" alt="{\displaystyle Q(x)\approx {\frac {\phi (x)}{\sqrt {1+x^{2}}}},\qquad x\geq 0.}" loading="lazy"></dd></dl></dd></dl>

<ul><li>Tighter bounds and approximations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

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<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

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</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"> can also be obtained by optimizing the following expression <a href="#cite_note-Borjesson-7">[7]</a></li></ul>

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<mfrac>

<mrow>

</mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {Q}}(x)={\frac {\phi (x)}{(1-a)x+a{\sqrt {x^{2}+b}}}}.}</annotation>

</semantics>

</math><img src="./d861dc5748980872bfb88320575fc14aea65e929.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.546ex; height:7.009ex;" alt="{\displaystyle {\tilde {Q}}(x)={\frac {\phi (x)}{(1-a)x+a{\sqrt {x^{2}+b}}}}.}" loading="lazy"></dd></dl></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation>

</semantics>

</math><img src="./a2608e2b392b079f5b763f27bf52883dbee3b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}" loading="lazy">, the best upper bound is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0.344}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=0.344}</annotation>

</semantics>

</math><img src="./305702cc1883b96630acb57e77fae0e47be667b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.625ex; height:2.176ex;" alt="{\displaystyle a=0.344}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=5.334}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=5.334}</annotation>

</semantics>

</math><img src="./2e6d4637614211ac9996ed86aed844314af6f910.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.393ex; height:2.176ex;" alt="{\displaystyle b=5.334}" loading="lazy"> with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0.339}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=0.339}</annotation>

</semantics>

</math><img src="./6fd7ec89d5697a8aec5d478fe764506276471d8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.625ex; height:2.176ex;" alt="{\displaystyle a=0.339}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=5.510}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=5.510}</annotation>

</semantics>

</math><img src="./69e0f4f1c8ae151637f31115d0b23359d8dcb6e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.393ex; height:2.176ex;" alt="{\displaystyle b=5.510}" loading="lazy"> with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1/\pi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=1/\pi }</annotation>

</semantics>

</math><img src="./8a81dc20d84f22133ebb7d363989e4e20369f063.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.985ex; height:2.843ex;" alt="{\displaystyle a=1/\pi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=2\pi }</annotation>

</semantics>

</math><img src="./e9303b5903b529cbbb06ae3ec420a4380e9925a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.59ex; height:2.176ex;" alt="{\displaystyle b=2\pi }" loading="lazy"> with maximum absolute relative error of 1.17%.</dd></dl>

<ul><li>The <a href="Chernoff_bound" title="Chernoff bound">Chernoff bound</a> of the Q-function is</li></ul>

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)\leq e^{-{\frac {x^{2}}{2}}},\qquad x>0}</annotation>

</semantics>

</math><img src="./b06cb873609c1ededd788d151cf83cb4e6dcc781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.34ex; height:4.509ex;" alt="{\displaystyle Q(x)\leq e^{-{\frac {x^{2}}{2}}},\qquad x>0}" loading="lazy"></dd></dl></dd></dl>

<ul><li>Improved exponential bounds and a pure exponential approximation are <a href="#cite_note-8">[8]</a></li></ul>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}</annotation>

</semantics>

</math><img src="./a6f3c4c05f7dd8bdbc7a69acd06b9d9d04c5aff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:44.614ex; height:4.843ex;" alt="{\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}" loading="lazy"></dd></dl></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)\approx {\frac {1}{12}}e^{-{\frac {x^{2}}{2}}}+{\frac {1}{4}}e^{-{\frac {2}{3}}x^{2}},\qquad x>0}</annotation>

</semantics>

</math><img src="./d572e6a10c35330ea00a7f6e59ad268429d16473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.209ex; height:5.509ex;" alt="{\displaystyle Q(x)\approx {\frac {1}{12}}e^{-{\frac {x^{2}}{2}}}+{\frac {1}{4}}e^{-{\frac {2}{3}}x^{2}},\qquad x>0}" loading="lazy"></dd></dl></dd></dl>

<ul><li>The above were generalized by Tanash & Riihonen (2020),<a href="#cite_note-9">[9]</a> who showed that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

</semantics>

</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"> can be accurately approximated or bounded by</li></ul>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {Q}}(x)=\sum _{n=1}^{N}a_{n}e^{-b_{n}x^{2}}.}</annotation>

</semantics>

</math><img src="./33b5695852ac76ebc6687d5d73e30d15bab16474.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.949ex; height:7.343ex;" alt="{\displaystyle {\tilde {Q}}(x)=\sum _{n=1}^{N}a_{n}e^{-b_{n}x^{2}}.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>In particular, they presented a systematic methodology to solve the numerical coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}}</annotation>

</semantics>

</math><img src="./a928be24d5559d0011262a33d6bf743396f7210f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.151ex; height:3.176ex;" alt="{\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}}" loading="lazy"> that yield a <a href="Minimax_approximation_algorithm" title="Minimax approximation algorithm">minimax</a> approximation or bound: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)\approx {\tilde {Q}}(x)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)\approx {\tilde {Q}}(x)}</annotation>

</semantics>

</math><img src="./a13679c099b89a7e9bff5a84e440133ddadb7029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.053ex; height:3.176ex;" alt="{\displaystyle Q(x)\approx {\tilde {Q}}(x)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)\leq {\tilde {Q}}(x)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)\leq {\tilde {Q}}(x)}</annotation>

</semantics>

</math><img src="./18cbf6fa6db25e34917abc34ad19a46337912eef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.053ex; height:3.176ex;" alt="{\displaystyle Q(x)\leq {\tilde {Q}}(x)}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)\geq {\tilde {Q}}(x)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)\geq {\tilde {Q}}(x)}</annotation>

</semantics>

</math><img src="./8cda849a2a5267eff4ad1965ceda306eb89e919c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.053ex; height:3.176ex;" alt="{\displaystyle Q(x)\geq {\tilde {Q}}(x)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation>

</semantics>

</math><img src="./a2608e2b392b079f5b763f27bf52883dbee3b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}" loading="lazy">. With the example coefficients tabulated in the paper for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=20}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N=20}</annotation>

</semantics>

</math><img src="./2a5c78ec997d223a010921f350195268f242c37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=20}" loading="lazy">, the relative and absolute approximation errors are less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2.831\cdot 10^{-6}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2.831\cdot 10^{-6}}</annotation>

</semantics>

</math><img src="./cc9de8624aac0df8a127eb0aeddc9d52e4ab1db6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.633ex; height:2.676ex;" alt="{\displaystyle 2.831\cdot 10^{-6}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.416\cdot 10^{-6}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1.416\cdot 10^{-6}}</annotation>

</semantics>

</math><img src="./b51f2ae1215677756447b65b08e882d9303c4a50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.633ex; height:2.676ex;" alt="{\displaystyle 1.416\cdot 10^{-6}}" loading="lazy">, respectively. The coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}}</annotation>

</semantics>

</math><img src="./a928be24d5559d0011262a33d6bf743396f7210f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.151ex; height:3.176ex;" alt="{\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}}" loading="lazy"> for many variations of the exponential approximations and bounds up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=25}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N=25}</annotation>

</semantics>

</math><img src="./b29dd09628a7e36df91dd19dc25db4908428e85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=25}" loading="lazy"> have been released to open access as a comprehensive dataset.<a href="#cite_note-10">[10]</a></dd></dl>

<ul><li>Another approximation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

</semantics>

</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in [0,\infty )}</annotation>

</semantics>

</math><img src="./7fdcab24f8202e4a37d346b76fd6a80edcddc03f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.242ex; height:2.843ex;" alt="{\displaystyle x\in [0,\infty )}" loading="lazy"> is given by <a href="George_Karagiannidis" title="George Karagiannidis">Karagiannidis</a> & Lioumpas (2007)<a href="#cite_note-11">[11]</a> who showed for the appropriate choice of parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A,B\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{A,B\}}</annotation>

</semantics>

</math><img src="./f71a70f6b5ef10db8ada87f9ea61a03b369918de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.866ex; height:2.843ex;" alt="{\displaystyle \{A,B\}}" loading="lazy"> that</li></ul>

<mfrac>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x;A,B)={\frac {\left(1-e^{-Ax}\right)e^{-x^{2}}}{B{\sqrt {\pi }}x}}\approx \operatorname {erfc} \left(x\right).}</annotation>

</semantics>

</math><img src="./17dd5d537fe6740d90fd1fce96a5508c1cfd6762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.153ex; height:7.509ex;" alt="{\displaystyle f(x;A,B)={\frac {\left(1-e^{-Ax}\right)e^{-x^{2}}}{B{\sqrt {\pi }}x}}\approx \operatorname {erfc} \left(x\right).}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>The absolute error between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x;A,B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x;A,B)}</annotation>

</semantics>

</math><img src="./f69329fd7e41758069c371b67a18fa00de228497.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.993ex; height:2.843ex;" alt="{\displaystyle f(x;A,B)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfc} (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x)}</annotation>

</semantics>

</math><img src="./0ce4532c4ac939dbe7ef6d2f1fee55f0978847cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.827ex; height:2.843ex;" alt="{\displaystyle \operatorname {erfc} (x)}" loading="lazy"> over the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,R]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,R]}</annotation>

</semantics>

</math><img src="./3ea1a7293db142452ce72c16d8f7c427fccf434c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.254ex; height:2.843ex;" alt="{\displaystyle [0,R]}" loading="lazy"> is minimized by evaluating</dd></dl>

<mrow>

</mrow>

<mrow>

</mrow>

</munder>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{A,B\}={\underset {\{A,B\}}{\arg \min }}{\frac {1}{R}}\int _{0}^{R}|f(x;A,B)-\operatorname {erfc} (x)|dx.}</annotation>

</semantics>

</math><img src="./68b6219b49e037faf4c93085c1cf8c8dc04f0bf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.928ex; height:6.843ex;" alt="{\displaystyle \{A,B\}={\underset {\{A,B\}}{\arg \min }}{\frac {1}{R}}\int _{0}^{R}|f(x;A,B)-\operatorname {erfc} (x)|dx.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=20}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=20}</annotation>

</semantics>

</math><img src="./0416cd43ec8842129e0190bf545b32256aad2220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.187ex; height:2.176ex;" alt="{\displaystyle R=20}" loading="lazy"> and numerically integrating, they found the minimum error occurred when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A,B\}=\{1.98,1.135\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{A,B\}=\{1.98,1.135\},}</annotation>

</semantics>

</math><img src="./c6cea6a52668d607fb5b56a25c0cbbf77e775718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.401ex; height:2.843ex;" alt="{\displaystyle \{A,B\}=\{1.98,1.135\},}" loading="lazy"> which gave a good approximation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\geq 0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall x\geq 0.}</annotation>

</semantics>

</math><img src="./12c51a75badf13201b757cb4b343a63927d746a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.53ex; height:2.343ex;" alt="{\displaystyle \forall x\geq 0.}" loading="lazy"></dd></dl>

<dl><dd>Substituting these values and using the relationship between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

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</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfc} (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x)}</annotation>

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</math><img src="./0ce4532c4ac939dbe7ef6d2f1fee55f0978847cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.827ex; height:2.843ex;" alt="{\displaystyle \operatorname {erfc} (x)}" loading="lazy"> from above gives</dd></dl>

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<annotation encoding="application/x-tex">{\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.}</annotation>

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</math><img src="./31c2aa0829a760080539f26f3dd6f29c7f24c001.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.315ex; height:10.009ex;" alt="{\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.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>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 href="#cite_note-12">[12]</a></dd></dl>

<ul><li>A tighter and more tractable approximation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

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<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

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</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"> for positive arguments <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [0,\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle x\in [0,\infty )}</annotation>

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</math><img src="./7fdcab24f8202e4a37d346b76fd6a80edcddc03f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.242ex; height:2.843ex;" alt="{\displaystyle x\in [0,\infty )}" loading="lazy"> is given by López-Benítez & Casadevall (2011)<a href="#cite_note-13">[13]</a> based on a second-order exponential function:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle Q(x)\approx e^{-ax^{2}-bx-c},\qquad x\geq 0.}</annotation>

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</math><img src="./b5642ef2b58eb2deb0b579f669d1c1daa74987a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.143ex; height:3.509ex;" alt="{\displaystyle Q(x)\approx e^{-ax^{2}-bx-c},\qquad x\geq 0.}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>The fitting coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b,c)}">

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<annotation encoding="application/x-tex">{\displaystyle (a,b,c)}</annotation>

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</math><img src="./ae973a762a92b9cd3eafe7f283890ccfa9b887e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.111ex; height:2.843ex;" alt="{\displaystyle (a,b,c)}" loading="lazy"> can be optimized over any desired range of arguments in order to minimize the sum of square errors (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0.3842}">

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<annotation encoding="application/x-tex">{\displaystyle a=0.3842}</annotation>

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</math><img src="./ec7e36ad336ea6b1fcd4a3abdd53daf2241db35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.787ex; height:2.176ex;" alt="{\displaystyle a=0.3842}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0.7640}">

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<annotation encoding="application/x-tex">{\displaystyle b=0.7640}</annotation>

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</math><img src="./eb05fb479fd21f7f373869e3b2923649141ae2a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.555ex; height:2.176ex;" alt="{\displaystyle b=0.7640}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=0.6964}">

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<annotation encoding="application/x-tex">{\displaystyle c=0.6964}</annotation>

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</math><img src="./620b0d34d4e148a44b51d037a298f0d2f3a6081e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.564ex; height:2.176ex;" alt="{\displaystyle c=0.6964}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [0,20]}">

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</math><img src="./385e4b36c43d3baf83dd06f9430503dc5dbaed18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.985ex; height:2.843ex;" alt="{\displaystyle x\in [0,20]}" loading="lazy">) or minimize the maximum absolute error (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0.4920}">

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<annotation encoding="application/x-tex">{\displaystyle a=0.4920}</annotation>

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</math><img src="./443ac49f5966e3308e861f2431bf613370b161db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.787ex; height:2.176ex;" alt="{\displaystyle a=0.4920}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=0.2887}">

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<annotation encoding="application/x-tex">{\displaystyle b=0.2887}</annotation>

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</math><img src="./b4f419ac0045714ed8dbca254c225ef37c06e337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.555ex; height:2.176ex;" alt="{\displaystyle b=0.2887}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=1.1893}">

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<annotation encoding="application/x-tex">{\displaystyle c=1.1893}</annotation>

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</math><img src="./7ce94251fc4eb7ec4c6c223da863dbea15aea3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.564ex; height:2.176ex;" alt="{\displaystyle c=1.1893}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [0,20]}">

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<annotation encoding="application/x-tex">{\displaystyle x\in [0,20]}</annotation>

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</math><img src="./385e4b36c43d3baf83dd06f9430503dc5dbaed18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.985ex; height:2.843ex;" alt="{\displaystyle x\in [0,20]}" loading="lazy">). 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)}">

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<annotation encoding="application/x-tex">{\displaystyle Q(x)}</annotation>

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</math><img src="./bde0d39b408d6f794d58388248774df9f4d96528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle Q(x)}" loading="lazy"> is trivial and does not alter the algebraic form of the approximation).</dd></dl>

<ul><li>A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [0,\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle x\in [0,\infty )}</annotation>

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</math><img src="./7fdcab24f8202e4a37d346b76fd6a80edcddc03f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.242ex; height:2.843ex;" alt="{\displaystyle x\in [0,\infty )}" loading="lazy"> was introduced by Abreu (2012)<a href="#cite_note-14">[14]</a> based on a simple algebraic expression with only two exponential terms:</li></ul>

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<annotation encoding="application/x-tex">{\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,}</annotation>

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</math><img src="./74d2d8b67d3f4cff5e634d51288efff5acddae68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.778ex; height:6.343ex;" alt="{\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,}" loading="lazy"></dd></dl></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle Q(x)\leq {\frac {1}{50}}e^{-x^{2}}+{\frac {1}{2(x+1)}}e^{-x^{2}/2},\qquad x\geq 0.}</annotation>

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</math><img src="./a50af8838c6c916fb74cee75a3a167777e25d303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.51ex; height:6.009ex;" alt="{\displaystyle Q(x)\leq {\frac {1}{50}}e^{-x^{2}}+{\frac {1}{2(x+1)}}e^{-x^{2}/2},\qquad x\geq 0.}" loading="lazy"></dd></dl></dd></dl>

These bounds are derived from a unified form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}}}</annotation>

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</math><img src="./1b35c171fd63326e13759d600729b55bee856097.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.712ex; height:6.676ex;" alt="{\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}}}" loading="lazy">, where the parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are chosen to satisfy specific conditions ensuring the lower (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\mathrm {L} }=12}">

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<annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {L} }=12}</annotation>

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</math><img src="./b5b9b11c3b77cc735e1345cbe20622f7b54c9a5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.913ex; height:2.509ex;" alt="{\displaystyle a_{\mathrm {L} }=12}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\mathrm {L} }={\sqrt {2\pi }}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{\mathrm {L} }={\sqrt {2\pi }}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {U} }=50}</annotation>

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</math><img src="./40d96ee67b77ee38e288e21dfff4af8d72dc0e85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.118ex; height:2.509ex;" alt="{\displaystyle a_{\mathrm {U} }=50}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\mathrm {U} }=2}">

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<annotation encoding="application/x-tex">{\displaystyle b_{\mathrm {U} }=2}</annotation>

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</math><img src="./2ce32971e532fef460efbfc0211550faf5623280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.723ex; height:2.509ex;" alt="{\displaystyle b_{\mathrm {U} }=2}" loading="lazy">) 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{n}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle Q^{n}(x)}</annotation>

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</math><img src="./21d10ac7dc0a3efa8751bfa01d592977ec7c7cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.196ex; height:2.843ex;" alt="{\displaystyle Q^{n}(x)}" loading="lazy"> for positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> using the binomial theorem, maintaining their simplicity and effectiveness.

<div class="mw-heading mw-heading2"><h2 id="Inverse_Q">Inverse Q</h2></div>

The inverse Q-function can be related to the <a href="Error_function#Inverse_functions" title="Error function">inverse error functions</a>:

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<annotation encoding="application/x-tex">{\displaystyle Q^{-1}(y)={\sqrt {2}}\ \mathrm {erf} ^{-1}(1-2y)={\sqrt {2}}\ \mathrm {erfc} ^{-1}(2y)}</annotation>

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</math><img src="./57012fb099b73ef4ca4e090ab7f2d0105aaa0ac7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.311ex; height:3.176ex;" alt="{\displaystyle Q^{-1}(y)={\sqrt {2}}\ \mathrm {erf} ^{-1}(1-2y)={\sqrt {2}}\ \mathrm {erfc} ^{-1}(2y)}" loading="lazy"></dd></dl>

The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{-1}(y)}">

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<annotation encoding="application/x-tex">{\displaystyle Q^{-1}(y)}</annotation>

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</math><img src="./21ca38469756972a7583798c73d53dc4f270bab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.136ex; height:3.176ex;" alt="{\displaystyle Q^{-1}(y)}" loading="lazy"> finds application in digital communications. It is usually expressed in <a href="Decibel#Field_quantities_and_root-power_quantities" title="Decibel">dB</a> and generally called Q-factor:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Q{\text{-}}factor} =20\log _{10}\!\left(Q^{-1}(y)\right)\!~\mathrm {dB} }</annotation>

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</math><img src="./1711c995998c014b3328d562c2c2b427e4c921e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.064ex; height:3.343ex;" alt="{\displaystyle \mathrm {Q{\text{-}}factor} =20\log _{10}\!\left(Q^{-1}(y)\right)\!~\mathrm {dB} }" loading="lazy"></dd></dl>

where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for <a href="Quadrature_phase-shift_keying" class="mw-redirect" title="Quadrature phase-shift keying">quadrature phase-shift keying</a> (QPSK) in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the <a href="Signal-to-noise_ratio#Decibels" title="Signal-to-noise ratio">signal to noise ratio</a> that yields a bit error rate equal to y.

<div class="mw-heading mw-heading2"><h2 id="Values">Values</h2></div>

The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as <a href="R_(programming_language)" title="R (programming language)">R</a> and those available in <a href="Python_(programming_language)" title="Python (programming language)">Python</a>, <a href="MATLAB" title="MATLAB">MATLAB</a> and <a href="Wolfram_Mathematica" class="mw-redirect" title="Wolfram Mathematica">Mathematica</a>. Some values of the Q-function are given below for reference.

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<th scope="row">Q(0.0)

</th>

<td>1/2.0000

</td></tr>

<tr>

<th scope="row">Q(0.1)

</th>

<td>1/2.1731

</td></tr>

<tr>

<th scope="row">Q(0.2)

</th>

<td>1/2.3768

</td></tr>

<tr>

<th scope="row">Q(0.3)

</th>

<td>1/2.6172

</td></tr>

<tr>

<th scope="row">Q(0.4)

</th>

<td>1/2.9021

</td></tr>

<tr>

<th scope="row">Q(0.5)

</th>

<td>1/3.2411

</td></tr>

<tr>

<th scope="row">Q(0.6)

</th>

<td>1/3.6463

</td></tr>

<tr>

<th scope="row">Q(0.7)

</th>

<td>1/4.1329

</td></tr>

<tr>

<th scope="row">Q(0.8)

</th>

<td>1/4.7202

</td></tr>

<tr>

<th scope="row">Q(0.9)

</th>

<td>1/5.4330

</td></tr></tbody></table>

</td>

<th scope="row">Q(1.0)

</th>

<td>1/6.3030

</td></tr>

<tr>

<th scope="row">Q(1.1)

</th>

<td>1/7.3710

</td></tr>

<tr>

<th scope="row">Q(1.2)

</th>

<td>1/8.6904

</td></tr>

<tr>

<th scope="row">Q(1.3)

</th>

<td>1/10.3305

</td></tr>

<tr>

<th scope="row">Q(1.4)

</th>

<td>1/12.3829

</td></tr>

<tr>

<th scope="row">Q(1.5)

</th>

<td>1/14.9684

</td></tr>

<tr>

<th scope="row">Q(1.6)

</th>

<td>1/18.2484

</td></tr>

<tr>

<th scope="row">Q(1.7)

</th>

<td>1/22.4389

</td></tr>

<tr>

<th scope="row">Q(1.8)

</th>

<td>1/27.8316

</td></tr>

<tr>

<th scope="row">Q(1.9)

</th>

<td>1/34.8231

</td></tr></tbody></table>

</td>

<th scope="row">Q(2.0)

</th>

<td>1/43.9558

</td></tr>

<tr>

<th scope="row">Q(2.1)

</th>

<td>1/55.9772

</td></tr>

<tr>

<th scope="row">Q(2.2)

</th>

<td>1/71.9246

</td></tr>

<tr>

<th scope="row">Q(2.3)

</th>

<td>1/93.2478

</td></tr>

<tr>

<th scope="row">Q(2.4)

</th>

<td>1/121.9879

</td></tr>

<tr>

<th scope="row">Q(2.5)

</th>

<td>1/161.0393

</td></tr>

<tr>

<th scope="row">Q(2.6)

</th>

<td>1/214.5376

</td></tr>

<tr>

<th scope="row">Q(2.7)

</th>

<td>1/288.4360

</td></tr>

<tr>

<th scope="row">Q(2.8)

</th>

<td>1/391.3695

</td></tr>

<tr>

<th scope="row">Q(2.9)

</th>

<td>1/535.9593

</td></tr></tbody></table>

</td>

<th scope="row">Q(3.0)

</th>

<td>1/740.7967

</td></tr>

<tr>

<th scope="row">Q(3.1)

</th>

<td>1/1033.4815

</td></tr>

<tr>

<th scope="row">Q(3.2)

</th>

<td>1/1455.3119

</td></tr>

<tr>

<th scope="row">Q(3.3)

</th>

<td>1/2068.5769

</td></tr>

<tr>

<th scope="row">Q(3.4)

</th>

<td>1/2967.9820

</td></tr>

<tr>

<th scope="row">Q(3.5)

</th>

<td>1/4298.6887

</td></tr>

<tr>

<th scope="row">Q(3.6)

</th>

<td>1/6285.0158

</td></tr>

<tr>

<th scope="row">Q(3.7)

</th>

<td>1/9276.4608

</td></tr>

<tr>

<th scope="row">Q(3.8)

</th>

<td>1/13822.0738

</td></tr>

<tr>

<th scope="row">Q(3.9)

</th>

<td>1/20791.6011

</td></tr>

<tr>

<th scope="row">Q(4.0)

</th>

<td>1/31574.3855

</td></tr></tbody></table>

</td></tr></tbody></table></div>

<div class="mw-heading mw-heading2"><h2 id="Generalization_to_high_dimensions">Generalization to high dimensions</h2></div>

The Q-function can be generalized to higher dimensions:<a href="#cite_note-15">[15]</a>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(\mathbf {x} )=\mathbb {P} (\mathbf {X} \geq \mathbf {x} ),}</annotation>

</semantics>

</math><img src="./bccfffb1e42af4439047db37b0aff59210ae11ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.562ex; height:2.843ex;" alt="{\displaystyle Q(\mathbf {x} )=\mathbb {P} (\mathbf {X} \geq \mathbf {x} ),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} \sim {\mathcal {N}}(\mathbf {0} ,\,\Sigma )}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} \sim {\mathcal {N}}(\mathbf {0} ,\,\Sigma )}</annotation>

</semantics>

</math><img src="./38c6e03747b04bda37f2475b68e099cdf445c000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.638ex; height:3.009ex;" alt="{\displaystyle \mathbf {X} \sim {\mathcal {N}}(\mathbf {0} ,\,\Sigma )}" loading="lazy"> follows the multivariate normal distribution with covariance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> and the threshold is of the form

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\gamma \Sigma \mathbf {l} ^{*}}</annotation>

</semantics>

</math><img src="./68c1ee118a9936cf521b02f428a9f149d91294a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.246ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =\gamma \Sigma \mathbf {l} ^{*}}" loading="lazy"> for some positive vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} ^{*}>\mathbf {0} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} ^{*}>\mathbf {0} }</annotation>

</semantics>

</math><img src="./5a95d8f65ac494b13f43ca6db1bc8c52da1a56fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.231ex; height:2.343ex;" alt="{\displaystyle \mathbf {l} ^{*}>\mathbf {0} }" loading="lazy"> and positive constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma >0}</annotation>

</semantics>

</math><img src="./775a6435e4270cddf2cd7dcb486c20f7f4bb8cee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma >0}" loading="lazy">. As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, the Q-function can be <a rel="nofollow" class="external text" href="http://www.mathworks.com/matlabcentral/fileexchange/53796">approximated arbitrarily well</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> becomes larger and larger.<a href="#cite_note-16">[16]</a><a href="#cite_note-bmc17-17">[17]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-bmc17-17"><a href="#cite_ref-bmc17_17-0">^</a> <cite id="CITEREFBotevMackinlayChen2017" class="citation book cs1">Botev, 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. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FWSC.2017.8247926">10.1109/WSC.2017.8247926</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-5386-3428-8</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4626481">4626481</a>.</cite>

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