Q-function: Difference between revisions
m →Bounds |
m →Bounds |
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:<math> |
:<math> |
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\begin{align} |
\begin{align} |
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Q(x) |
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&=\int_x^\infty\varphi(u)\,du\\ |
&=\int_x^\infty\varphi(u)\,du\\ |
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&<\int_x^\infty\frac ux\varphi(u)\,du |
&<\int_x^\infty\frac ux\varphi(u)\,du |
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:<math> |
:<math> |
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\begin{align} |
\begin{align} |
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\Bigl(1+\frac1{x^2}\Bigr) |
\Bigl(1+\frac1{x^2}\Bigr)Q(x) |
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&=\int_x^\infty \Bigl(1+\frac1{x^2}\Bigr)\varphi(u)\,du\\ |
&=\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 |
&>\int_x^\infty \Bigl(1+\frac1{u^2}\Bigr)\varphi(u)\,du |
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</math> |
</math> |
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Solving for <math> |
Solving for <math>Q(x)</math> provides the lower bound. |
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== References == |
== References == |
Revision as of 08:09, 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 and defining , the upper bound is derived as follows:
Similarly, using and the quotient rule,
Solving for provides the lower bound.