Q-function: Difference between revisions
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:<math>Q(x) =\tfrac{1}{2} - \tfrac{1}{2} \operatorname{erf} \left( \frac{x}{\sqrt{2}} \right)=\tfrac{1}{2}\operatorname{erfc} \left(\frac{x}{\sqrt{2}} \right).</math> |
:<math>Q(x) =\tfrac{1}{2} - \tfrac{1}{2} \operatorname{erf} \left( \frac{x}{\sqrt{2}} \right)=\tfrac{1}{2}\operatorname{erfc} \left(\frac{x}{\sqrt{2}} \right).</math> |
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An alternative form of the ''Q''-function that is more useful is expressed as:<ref>[http://wsl.stanford.edu/~ee359/craig.pdf |
An alternative form of the ''Q''-function that is more useful is expressed as:<ref>[http://wsl.stanford.edu/~ee359/craig.pdf John W. Craig, ''A new, simple and exact result for calculating the probability of error for two-dimensional signal constellaions'', Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575.]</ref> |
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:<math>Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} \right) d\theta.</math> |
:<math>Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} \right) d\theta.</math> |
Revision as of 06:40, 19 December 2012
In statistics, the Q-function is the tail probability of the standard normal distribution.[1][2] In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations above the mean. 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 Φ(x) is the cumulative distribution function of the normal Gaussian distribution.
The Q-function can be expressed in terms of the error function, or the complementary error function, as[2]
An alternative form of the Q-function that is more useful is expressed as:[4]
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 for the negative values. This form is advantageous in that the range of integration is finite.
Bounds
Before introducing bounds we first define the following function:
- The Q-function is not an elementary function. 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 φ′(u) = −uφ(u) and the quotient rule,
- Solving for Q(x) 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 and 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) |
References
- ^ The Q-function, from cnx.org
- ^ a b Basic properties of the Q-function
- ^ Normal Distribution Function - from Wolfram MathWorld
- ^ John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellaions, Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575.