Q-function: Difference between revisions
{{cn}} for last edit, remove see also for error function and place this a second time in the text |
m Dated {{Refimprove}}{{Citation needed}}. (Build p613) |
||
Line 8: | Line 8: | ||
:<math> |
:<math> |
||
Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty \exp\Bigl(-\frac{u^2}{2}\Bigr) \, du.</math> |
Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty \exp\Bigl(-\frac{u^2}{2}\Bigr) \, du.</math> |
||
Thus, |
Thus, |
||
:<math>Q(x) = 1 - Q(-x) = 1 - \Phi(x)\,\!,</math> |
:<math>Q(x) = 1 - Q(-x) = 1 - \Phi(x)\,\!,</math> |
||
where <math>\Phi(x)</math> is the [[Standard_normal_distribution#Cumulative_distribution_function|cumulative distribution function of the normal Gaussian distribution]]. |
where <math>\Phi(x)</math> is the [[Standard_normal_distribution#Cumulative_distribution_function|cumulative distribution function of the normal Gaussian distribution]]. |
||
Line 17: | Line 17: | ||
</math> |
</math> |
||
An alternative form of the Q-function that is more useful is expressed as:{{ |
An alternative form of the Q-function that is more useful is expressed as:{{Citation needed|date=September 2011}} |
||
:<math> |
:<math> |
||
Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} \right) d\theta. |
Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} \right) d\theta. |
||
Line 68: | Line 68: | ||
<!-- This table was calculated in Matlab as follows: |
<!-- This table was calculated in Matlab as follows: |
||
x=0:0.1:4; |
x=0:0.1:4; |
||
for i=1:length(x), |
for i=1:length(x), |
||
fprintf('Q(%.1f) = %.9f<br/>\n',x(i),qfunc(x(i))); |
fprintf('Q(%.1f) = %.9f<br/>\n',x(i),qfunc(x(i))); |
||
end; |
end; |
||
--> |
--> |
||
Line 120: | Line 120: | ||
{{col-end}} |
{{col-end}} |
||
{{Refimprove|date=September 2011}} |
|||
{{refimprove}} |
|||
== References == |
== References == |
||
<references /> |
<references /> |
Revision as of 09:12, 5 September 2011
In statistics, the Q-function is the tail probability of the standard normal distribution.[1][2] In other words, is the probability that a standard normal 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]
Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics.
Definition and basic properties
Formally, the Q-function is defined as
Thus,
where is the cumulative distribution function of the normal Gaussian distribution.
The Q-function can be expressed in terms of the error function, or the compelementary error function, as[2]
An alternative form of the Q-function that is more useful is expressed as:[citation needed]
Bounds
- The Q-function is not an elementary function. 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.
- Chernoff bound of Q-function is
Values
The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as Matlab, Mathematica. Some values of the Q-function are given below for reference.
Q(0.0) = 0.500000000 |
Q(1.0) = 0.158655254 |
Q(2.0) = 0.022750132 |
Q(3.0) = 0.001349898 |
This article needs additional citations for verification. (September 2011) |