Q-function: Difference between revisions
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more stuff adapted from normal distribution, i think it's more appropriate here. |
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In statistics, the '''Q-function''' is the [[tail probability]] of the normalized [[Gaussian distribution]].<ref>[http://cnx.org/content/m11537/latest/ The Q-function<!-- Bot generated title -->]</ref><ref>http://www.eng.tau.ac.il/~jo/academic/Q.pdf</ref> In other words, <math>Q(x)</math> is the probability that a normalized Gaussian random variable will obtain a value larger than <math>x</math>. Other definitions of the Q-function, all of which are simple transformations of the normal [[cumulative distribution function]], are also used occasionally.<ref>[http://mathworld.wolfram.com/NormalDistributionFunction.html Normal Distribution Function - from Wolfram MathWorld<!-- Bot generated title -->]</ref> |
In statistics, the '''Q-function''' is the [[tail probability]] of the normalized [[Gaussian distribution]].<ref>[http://cnx.org/content/m11537/latest/ The Q-function<!-- Bot generated title -->]</ref><ref>http://www.eng.tau.ac.il/~jo/academic/Q.pdf</ref> In other words, <math>Q(x)</math> is the probability that a normalized Gaussian random variable will obtain a value larger than <math>x</math>. Other definitions of the Q-function, all of which are simple transformations of the normal [[cumulative distribution function]], are also used occasionally.<ref>[http://mathworld.wolfram.com/NormalDistributionFunction.html Normal Distribution Function - from Wolfram MathWorld<!-- Bot generated title -->]</ref> |
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== Definition and related functions == |
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Formally, the Q-function is defined as |
Formally, the Q-function is defined as |
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:<math> |
:<math> |
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Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty \exp\Bigl(-\frac{u^2}{2}\Bigr) \, du. |
Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty \exp\Bigl(-\frac{u^2}{2}\Bigr) \, du. |
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</math> |
</math> |
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Thus, <math>Q(x) = 1 - \Phi(x),</math> where <math>\Phi(x)</math> is the cumulative distribution function of the normal Gaussian distribution. |
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The Q-function can be expressed in terms of the [[error function]] as |
The Q-function can be expressed in terms of the [[error function]] as |
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=\frac{1}{2} - \frac{1}{2} \operatorname{erf} \Bigl( \frac{x}{\sqrt{2}} \Bigr). |
=\frac{1}{2} - \frac{1}{2} \operatorname{erf} \Bigl( \frac{x}{\sqrt{2}} \Bigr). |
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</math> |
</math> |
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== Bounds == |
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The Q-function cannot be written using [[elementary function]]s. However, the bounds |
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:<math> |
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\frac{x}{1+x^2} \cdot \frac{1}{\sqrt{2\pi}} e^{-x^2/2} < Q(x) < \frac{1}{x} \cdot \frac{1}{\sqrt{2 \pi}}e^{-x^2/2}, \qquad x>0, |
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</math> |
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become increasingly tight for large ''x'', and are often useful. |
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Using the [[integration by substitution|substitution]] ''v'' = ''u''²/2 and defining <math>\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}</math>, the upper bound is derived as follows: |
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:<math> |
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\begin{align} |
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1-\Phi(x) |
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&=\int_x^\infty\varphi(u)\,du\\ |
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&<\int_x^\infty\frac ux\varphi(u)\,du |
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=\int_{x^2/2}^\infty\frac{e^{-v}}{x\sqrt{2\pi}}\,dv |
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=-\biggl.\frac{e^{-v}}{x\sqrt{2\pi}}\biggr|_{x^2/2}^\infty |
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=\frac{\varphi(x)}{x}. |
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\end{align} |
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</math> |
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Similarly, using <math>\scriptstyle\varphi'(u)\,{=}\,-u\,\varphi(u)</math> and the [[quotient rule]], |
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:<math> |
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\begin{align} |
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\Bigl(1+\frac1{x^2}\Bigr)(1-\Phi(x)) |
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&=\int_x^\infty \Bigl(1+\frac1{x^2}\Bigr)\varphi(u)\,du\\ |
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&>\int_x^\infty \Bigl(1+\frac1{u^2}\Bigr)\varphi(u)\,du |
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=-\biggl.\frac{\varphi(u)}u\biggr|_x^\infty |
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=\frac{\varphi(x)}x. |
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\end{align} |
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</math> |
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Solving for <math>\scriptstyle 1\,{-}\,\Phi(x)\,</math> provides the lower bound. |
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== References == |
== References == |
Revision as of 08:07, 7 April 2009
In statistics, the Q-function is the tail probability of the normalized Gaussian distribution.[1][2] In other words, is the probability that a normalized Gaussian random variable will obtain a value larger than . Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.[3]
Definition and related functions
Formally, the Q-function is defined as
Thus, where is the cumulative distribution function of the normal Gaussian distribution.
The Q-function can be expressed in terms of the error function as
Bounds
The Q-function cannot be written using elementary functions. However, the bounds
become increasingly tight for large x, and are often useful.
Using the substitution v = u²/2 and defining , the upper bound is derived as follows:
Similarly, using and the quotient rule,
Solving for provides the lower bound.