Q-function: Difference between revisions
A distribution is a probability measure, specifically a function from a sigma-algebra of events to [0,1] which gives the probability of each event. The density of the standard normal is NOT A DISTRIBUTION. It determines a distribution, but is not one. |
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[[Image:Q-function.png|thumb|right|400px|A plot of the Q-function.]]
In [[statistics]], the '''Q-function''' is the [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|tail distribution function]] of the [[Normal distribution#Standard normal distribution|standard normal distribution]]
If <math>Y</math> is a Gaussian random variable with mean <math>\mu</math> and variance <math>\sigma^2</math>, then <math>X = \frac{Y-\mu}{\sigma}</math> is [[Normal distribution#Standard normal distribution|standard normal]] and
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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.
*The ''Q''-function is not an [[elementary function]]. However, the bounds, where <math>\phi(x)</math> is the density function of the standard normal distribution,
::<math>\left (\frac{x}{1+x^2} \right ) \phi(x) < Q(x) < \frac{\phi(x)}{x}, \qquad x>0,</math>
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Revision as of 19:51, 4 February 2018
In statistics, the Q-function is the tail distribution function of the standard normal distribution.[1][2] In other words, is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations above the mean. Equivalently, is the probability that a standard normal random variable takes a value larger than .
If is a Gaussian random variable with mean and variance , then is standard normal and
where .
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 complementary error function, as[2]
An alternative form of the Q-function known as Craig's formula, after its discoverer, 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 Q(x) for negative values. This form is advantageous in that the range of integration is fixed and finite.
- The Q-function is not an elementary function. However, the bounds, where is the density function of the standard normal distribution,
- become increasingly tight for large x, and are often useful.
- Using the substitution v =u2/2, the upper bound is derived as follows:
- Similarly, using and the quotient rule,
- Solving for Q(x) provides the lower bound.
- The Chernoff bound of the Q-function is
- Improved exponential bounds and a pure exponential approximation are [5]
- A tight approximation of for is given by Karagiannidis & Lioumpas (2007)[6] who showed for the appropriate choice of parameters that
- The absolute error between and over the range is minimized by evaluating
- Using and numerically integrating, they found the minimum error occurred when which gave a good approximation for
- Substituting these values and using the relationship between and from above gives
Inverse Q
The inverse Q-function can be related to the inverse error functions:
The function finds application in digital communications. It is usually expressed in dB and generally called Q-factor:
where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for QPSK in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to y.
Values
The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, MATLAB and Mathematica. Some values of the Q-function are given below for reference.
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Generalization to high dimensions
The Q-function can be generalized to higher dimensions:[7]
where follows the multivariate normal distribution with covariance and the threshold is of the form for some positive vector and positive constant . As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, the Q-function can be approximated arbitrarily well as becomes larger and larger.[8]
References
- ^ The Q-function, from cnx.org
- ^ a b Basic properties of the Q-function Archived March 25, 2009, at the Wayback Machine
- ^ 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 constellations, Proceedings of the 1991 IEEE Military Communication Conference, vol. 2, pp. 571–575.
- ^ Chiani, M., Dardari, D., Simon, M.K. New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels (2003). IEEE Transactions on Wireless Communications, 4(2), pp. 840–845. doi=10.1109/TWC.2003.814350.
- ^ Karagiannidis, G. K., & Lioumpas, A. S. An improved approximation for the Gaussian Q-function. 2007. Communications Letters, IEEE, 11(8), pp. 644-646.
- ^ Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal Res. Nat. Bur. Standards Sect. B. 66: 93–96.
- ^ Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. doi:10.1111/rssb.12162.