翻訳と辞書 Words near each other ・ Q-Collection ・ Q-Connector ・ Q-construction ・ Q-D-Š ・ Q-dance ・ Q-derivative ・ Q-difference polynomial ・ Q-expansion principle ・ Q-exponential ・ Q-exponential distribution ・ Q-Factor (LGBT) ・ Q-Feel ・ Q-FISH ・ Q-Flex ・ Q-Free ・ Q-function ・ Q-Games ・ Q-gamma function ・ Q-Gaussian distribution ・ Q-Genz ・ Q-go ・ Q-guidance ・ Q-Hahn polynomials ・ Q-in-Law ・ Q-Jacobi polynomials ・ Q-Konhauser polynomials ・ Q-Krawtchouk polynomials ・ Q-Laguerre polynomials ・ Q-LAN ・ Q-learning Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Q-function ： ウィキペディア英語版 Q-function In statistics, the Q-function is the tail probability of the standard normal distribution $\backslash phi(x)$.〔(The Q-function ), from cnx.org〕〔(Basic properties of the Q-function )〕 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. If the underlying random variable is y, then the proper argument to the tail probability is derived as: :$x=\backslash frac$ which expresses the number of standard deviations away from the mean. Other definitions of the ''Q''-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.〔(Normal Distribution Function - from Wolfram MathWorld )〕 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 :$Q(x)\; =\; \backslash frac\backslash right)\; \backslash ,\; du.$ Thus, :$Q(x)\; =\; 1\; -\; Q(-x)\; =\; 1\; -\; \backslash Phi(x)\backslash ,\backslash !,$ where $\backslash Phi(x)$ 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〔 :$$ \begin Q(x) &=\frac\left( \frac}^\infty \exp\left(-t^2\right) \, dt \right)\\ &= \frac - \frac \operatorname \left( \frac\\ &= \frac\operatorname \left(\frac An alternative form of the ''Q''-function known as Craig's formula, after its discoverer, is expressed as:〔(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. )〕 :$Q(x)\; =\; \backslash frac\; \backslash int\_0^\}\; \backslash exp\; \backslash left(\; -\; \backslash frac\; \backslash right)\; d\backslash theta.$ 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 finite. *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'' =''u''2/2, the upper bound is derived as follows: :: :Similarly, using $\backslash phi\text{'}(u)\; =\; -\; u\; \backslash phi(u)$ and the quotient rule, ::$\backslash left(1+\backslash frac1\backslash right)Q(x)\; =\backslash int\_x^\backslash infty\; \backslash left(1+\backslash frac1\backslash right)\backslash phi(u)\backslash ,du\; >\backslash int\_x^\backslash infty\; \backslash left(1+\backslash frac1\backslash right)\backslash phi(u)\backslash ,du\; =-\backslash biggl.\backslash fracu\backslash biggr|\_x^\backslash infty$ =\fracx. :Solving for ''Q''(''x'') provides the lower bound. *The Chernoff bound of the ''Q''-function is ::$Q(x)\backslash leq\; e^\},\; \backslash qquad\; x>0$ *Improved exponential bounds and a pure exponential approximation are 〔(Chiani, M., Dardari, D., Simon, M.K. (2003). ''New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels''. IEEE Transactions on Wireless Communications, 4(2), 840–845, doi=10.1109/TWC.2003.814350. )〕 ::$Q(x)\backslash leq\; \backslash tfrace^+\backslash tfrace^\}\; \backslash leq\; \backslash tfrace^\},\; \backslash qquad\; x>0$ :: $Q(x)\backslash approx\; \backslash frace^\}+\backslash frace^\; x^2\},\; \backslash qquad\; x>0$ *A tight approximation of $Q(x)$ for $x\; \backslash in$ is minimized by evaluating : $\backslash \; =\; \backslash underset\; \backslash frac\; \backslash int\_0^R\; |\; f(x;\; A,\; B)\; -\; \backslash operatorname(x)\; |dx.$ : Using $R\; =\; 20$ and numerically integrating, they found the minimum error occurred when $\backslash \; =\; \backslash ,$ which gave a good approximation for $\backslash forall\; x\; \backslash ge\; 0.$ : Substituting these values and using the relationship between $Q(x)$ and $\backslash operatorname(x)$ from above gives : $Q(x)\backslash approx\backslash frac\}\}\},\; x\; \backslash ge\; 0.$ Inverse ''Q'' The inverse ''Q''-function can be trivially related to the inverse error function: :$Q^(x)\; =\; \backslash sqrt\backslash \; \backslash mathrm^(1-2x)$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Q-function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.